{"id":33,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/2-ifthen-and-it-is-not-the-case-that\/"},"modified":"2025-10-13T17:08:13","modified_gmt":"2025-10-13T17:08:13","slug":"2-ifthen-and-it-is-not-the-case-that","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/2-ifthen-and-it-is-not-the-case-that\/","title":{"raw":"\u201cIf\u2026then\u2026.\u201d and \u201cIt is not the case that\u2026.\u201d","rendered":"\u201cIf\u2026then\u2026.\u201d and \u201cIt is not the case that\u2026.\u201d"},"content":{"raw":"<h2>2.1 \u00a0The Conditional<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What are parentheses used for in logical statements?<\/li>\r\n \t<li>How are conditional statements written in propositional logic?<\/li>\r\n \t<li>Can symbols be used to represent \"If...Then...\" statements?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><strong>Conditional Sentence<\/strong> - a sentence containing an antecedent (if) and a consequent (then)<\/li>\r\n \t<li><strong>Antecedent<\/strong> - the first sentence in a conditional statement; it is the statement before the \u2192 symbol<\/li>\r\n \t<li><strong>Consequent<\/strong> - the second sentence in a conditional statement; it is the statement after the \u2192 symbol<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nAs we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. \u00a0Our examples included\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Obama wins the election, then Obama will be President.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If it rains tomorrow, then the picnic will be canceled.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The Earth is not the center of the universe.<\/p>\r\nWe could treat these like atomic sentences, but then we would lose a great deal of important information. \u00a0For example, the first sentence tells us something about the relationship between the atomic sentences \u201cObama wins the election\u201d and \u201cObama will be President\u201d. \u00a0And the third sentence above will, one supposes, have an interesting relationship to the sentence, \u201cThe Earth is the center of the universe\u201d. \u00a0To make these relations explicit, we will have to understand what \u201cif\u2026then\u2026\u201d and \u00a0\u201cnot\u201d mean. \u00a0Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. \u00a0Let us start with the first one.\r\n\r\nThe sentence, \u201cIf Obama wins the election, then Obama will be President\u201d contains two atomic sentences, \u201cObama wins the election\u201d and \u201cObama will be President\u201d. \u00a0We could thus represent this sentence by letting\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Obama wins the election<\/p>\r\nbe represented in our logical language by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\r\nAnd by letting\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Obama will be president<\/p>\r\nbe represented by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nThen, the whole expression could be represented by writing\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>then <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nIt will be useful, however, to replace the English phrase \u201cif\u2026then...\u201d by a single symbol in our language. \u00a0The most commonly used such symbol is \u201c\u2192\u201d. \u00a0Thus, we would write\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>\u2192<strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nOne last thing needs to be observed, however. \u00a0We might want to combine this complex sentence with other sentences. \u00a0In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. \u00a0There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. \u00a0Thus, we will write our expression\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\r\nThis kind of sentence is called a \u201cconditional\u201d. \u00a0It is also sometimes called a \u201cmaterial conditional\u201d. \u00a0The first constituent sentence (the one before the arrow, which in this example is \u201c<strong><span class=\"strong\">P<\/span><\/strong>\u201d) is called the \u201cantecedent\u201d. Antecedent means 'the condition'; think of it like 'if I study (condition), I pass the test (result).'\r\nThe second sentence (the one after the arrow, which in this example is \u201c<strong><span class=\"strong\">Q<\/span><\/strong>\u201d) is called the \u201cconsequent\u201d.\r\n\r\nWe know how to write the conditional, but what does it mean? \u00a0As before, we will take the meaning to be given by the truth conditions\u2014that is, a description of when the sentence is either true or false. \u00a0We do this with a truth table. \u00a0But now, our sentence has two parts that are atomic sentences, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0Note that either atomic sentence could be true or false. \u00a0That means, we have to consider four possible kinds of situations. \u00a0We must consider when <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and when it is false, but then we need to consider those two kinds of situations twice: \u00a0once for when <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true and once for when <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is false. \u00a0Thus, the left hand side of our truth table will look like this:\r\n<table class=\"grid\" style=\"height: 75px;\" width=\"125\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\r\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong>\u00a0Q\r\n<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>T\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>F\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>T\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>F\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are four kinds of ways the world could be that we must consider.\r\n\r\nNote that, since there are two possible truth values (true and false), whenever we consider another atomic sentence, there are twice as many ways the world could be that we should consider. \u00a0Thus, for n atomic sentences, our truth table must have 2<sup><span class=\"super\">n<\/span><\/sup>\u00a0rows. \u00a0In the case of a conditional formed out of two atomic sentences, like our example of <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, our truth table will have 2<sup><span class=\"super\">2 <\/span><\/sup>rows, which is 4 rows. \u00a0We see this is the case above.\r\n\r\nNow, we must decide upon what the conditional means. \u00a0To some degree this is up to us. \u00a0What matters is that once we define the semantics of the conditional, we stick to our definition. \u00a0But we want to capture as much of the meaning of the English \u201cif\u2026then\u2026\u201d as we can, while remaining absolutely precise in our language.\r\n\r\nLet us consider each kind of way the world could be. \u00a0For the first row of the truth table, we have that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true. \u00a0Suppose the world is such that Obama wins the election, and also Obama will be President. \u00a0Then, would I have spoken truly if I said, \u201cIf Obama wins the election, then Obama will be President\u201d? \u00a0Most people agree that I would have. \u00a0Similarly, suppose that Obama wins the election, but Obama will not be President. \u00a0Would the sentence \u201cIf Obama wins the election, then Obama will be President\u201d still be true? \u00a0Most agree that it would be false now. \u00a0So the first rows of our truth table are uncontroversial.\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\r\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><strong>\u00a0Q\r\n<\/strong><\/td>\r\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">(<strong>P<\/strong><strong><span class=\"strong\">\u2192<\/span><\/strong><strong>Q<\/strong>)<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>T\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>F\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>T\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>F\r\n<\/strong><\/em><\/td>\r\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSome students, however, find it hard to determine what truth values should go in the next two rows. \u00a0Note now that our principle of bivalence requires us to fill in these rows. \u00a0We cannot leave them blank. \u00a0If we did, we would be saying that sometimes a conditional can have no truth value; that is, we would be saying that sometimes, some sentences have no truth value. \u00a0But our principle of bivalence requires that\u2014in all kinds of situations\u2014every sentence is either true or false, never both, never neither. \u00a0So, if we are going to respect the principle of bivalence, then we have to put either <strong><span class=\"em strong\">T<\/span>\u00a0<\/strong>or <strong><span class=\"em strong\">F<\/span>\u00a0<\/strong>in for each of the last two rows.\r\n\r\nIt is helpful at this point to change our example. \u00a0Let us consider two different examples to illustrate how best to fill out the remainder of the truth table for the conditional.\r\n\r\nFirst, suppose I say the following to you: \u00a0\u201cIf you give me $50, then I will buy you a ticket to the concert tonight.\u201d \u00a0Let\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">You give me $50<\/p>\r\nbe represented in our logic by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">R<\/span><\/strong><\/p>\r\nand let\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">I will buy you a ticket to the concert tonight.<\/p>\r\nbe represented by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong><\/p>\r\nOur sentence then is\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\r\nAnd its truth table\u2014as far as we understand right now\u2014is:\r\n<div class=\"keep\">\r\n<table class=\"grid\" style=\"width: 175px; height: 75px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<th class=\"border\" style=\"text-align: center; width: 208.517px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">R\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0<\/span><\/th>\r\n<th class=\"border-right\" style=\"text-align: center; width: 199.5px; height: 15px;\"><span class=\"strong\">S<\/span><\/th>\r\n<th class=\"border\" style=\"text-align: center; width: 238.583px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(R\u2192S)<\/span><\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\">T<\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\nThat is, if you give me the money and I buy you the ticket, my claim that \u201cIf you give me $50, then I will buy you a ticket to the concert tonight\u201d is true. \u00a0And, if you give me the money and I don\u2019t buy you the ticket, I lied, and my claim is false. \u00a0But now, suppose you do not give me $50, but I buy you a ticket for the concert as a gift. \u00a0Was my claim false? \u00a0No. \u00a0I simply bought you the ticket as a gift, but, presumably would have bought it if you gave me the money, also. \u00a0Similarly, if you don\u2019t give me money, and I do not buy you a ticket, that seems perfectly consistent with my claim.\r\n\r\nSo, the best way to fill out the truth table is as follows.\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">R \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">S<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(R\u2192S)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSecond, consider another sentence, which has the advantage that it is very clear with respect to these last two rows. \u00a0Assume that <span class=\"em strong\">a<\/span>\u00a0is a particular natural number, only you and I don\u2019t know what number it is (the natural numbers are the whole positive numbers: \u00a01, 2, 3, 4\u2026). \u00a0Consider now the following sentence.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 4, then <span class=\"em strong\">a<\/span>\u00a0is evenly divisible by 2.<\/p>\r\n(By \u201cevenly divisible,\u201d I mean divisible without remainder.) \u00a0The first thing to ask yourself is: is this sentence true? \u00a0I hope we can all agree that it is\u2014even though we do not know what <strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is. \u00a0Let\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 4<\/p>\r\nbe represented in our logic by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">U<\/span><\/strong><\/p>\r\nand let\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2<\/p>\r\nbe represented by\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">V<\/span><\/strong><\/p>\r\nOur sentence then is\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>U<\/strong>\u2192<strong>V<\/strong>)<\/span><\/p>\r\nAnd its truth table\u2014as far as we understand right now\u2014is:\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">U \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">V<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(U\u2192V)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow consider a case in which <span class=\"em strong\">a<\/span>\u00a0is 6. \u00a0This is like the third row of the truth table. \u00a0It is not the case that 6 is evenly divisible by 4, but it is the case that 6 is evenly divisible by 2. \u00a0And consider the case in which <span class=\"em strong\">a<\/span>\u00a0is 7. \u00a0This is like the fourth row of the truth table; 7 would be evenly divisible by neither 4 nor 2. \u00a0But we agreed that the conditional is true\u2014regardless of the value of <span class=\"em strong\">a<\/span>! \u00a0So, the truth table must be:[footnote]One thing is a little funny about this second example with unknown number a. We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. First, this oddity arises because of mathematical facts, not facts of our propositional logic\u2014that is, we need to know what \u201cdivisible\u201d means, what \u201c4\u201d and \u201c2\u201d mean, and so on, in order to understand the sentence. So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. Second, some conditionals can be false. In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.[\/footnote]\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">U \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">V<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(U\u2192V)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nFollowing this pattern, we should also fill out our table about the election with:\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P\r\n<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">Q<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong em\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nIf you are dissatisfied by this, it might be helpful to think of these last two rows as vacuous cases. \u00a0A conditional tells us about what happens if the antecedent is true. \u00a0But when the antecedent is false, we simply default to true.\r\n\r\nWe are now ready to offer, in a more formal way, the syntax and semantics for the conditional.\r\n\r\nThe syntax of the conditional is that, if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>\u03a6<\/strong>\u2192<strong>\u03a8<\/strong>)<\/span><\/p>\r\nis a sentence.\r\n\r\nThe semantics of the conditional are given by a truth table. \u00a0For any sentences <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <span class=\"strong\"><strong>\u03a8<\/strong>:<\/span>\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6\u2192\u03a8)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nRemember that this truth table is now a definition. \u00a0It defines the meaning of \u201c<span class=\"strong\">\u2192<\/span>\u201d. \u00a0We are agreeing to use the symbol \u201c<span class=\"strong\">\u2192<\/span>\u201d to mean this from here on out.\r\n\r\nThe elements of the propositional logic, like \u201c<span class=\"strong\">\u2192<\/span>\u201d, that we add to our language in order to form more complex sentences, are called \u201ctruth functional connectives\u201d. \u00a0I hope it is clear why: \u00a0the meaning of this symbol is given in a truth function. \u00a0(If you are unfamiliar or uncertain about the idea of a function, think of a function as like a machine that takes in one or more inputs, and always then gives exactly one output. \u00a0For the conditional, the inputs are two truth values; and the output is one truth value. \u00a0For example, put <span class=\"em strong\"><strong>T<\/strong> <strong>F<\/strong><\/span><strong>\u00a0<\/strong>into the truth function called \u201c<span class=\"strong\">\u2192<\/span>\u201d, and you get out <strong><span class=\"em strong\">F<\/span><\/strong>.)\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What defines a conditional?<\/li>\r\n \t<li>How is the Principle of Bivalence applied in truth tables for conditional statements?<\/li>\r\n \t<li>What are formal ways to represent the syntax and semantics of sentence?<\/li>\r\n \t<li>Describe a decision where an 'if\u2026then\u2026' statement influenced your choice. Was the condition met?<\/li>\r\n \t<li><strong>Critical Thinking Task:\u00a0<\/strong>Contemplate some specific cases with false antecedents and true consequents.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>2.2 \u00a0Alternative phrasings\u00a0in English for the conditional. \u00a0Only if.<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>How can the word \"only\" change the logical meaning of a sentence?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\nEnglish includes many alternative phrasings that appear to be equivalent to the conditional. \u00a0Furthermore, in English and other natural languages, the order of the conditional will sometimes be reversed. \u00a0We can capture the general sense of these cases by recognizing that each of the following phrasings would be translated as <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>. \u00a0 (In these examples, we mix English and our propositional logic, in order to illustrate the variations succinctly.)\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><span class=\"strong\">P<\/span><\/strong>, then <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, if <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">On the condition that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, on the condition that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Given that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, given that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Provided that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, provided that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">When <strong><span class=\"strong\">P<\/span><\/strong>, then <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, when <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>implies <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is implied by <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is sufficient for <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is necessary for <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\nAn oddity of English is that the word \u201conly\u201d changes the meaning of \u201cif\u201d. \u00a0You can see this if you consider the following two sentences.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Fifi is a cat, if Fifi is a mammal.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Fifi is a cat only if Fifi is a mammal.<\/p>\r\nSuppose we know Fifi is an organism, but, we don\u2019t know what kind of organism Fifi is. \u00a0Fifi could be a dog, a cat, a gray whale, a ladybug, a sponge. \u00a0It seems clear that the first sentence is not necessarily true. \u00a0If Fifi is a gray whale, for example, then it is true that Fifi is a mammal, but false that Fifi is a cat; and so, the first sentence would be false. \u00a0But the second sentence looks like it must be true (given what you and I know about cats and mammals).\r\n\r\nWe should thus be careful to recognize that \u201conly if\u201d does not mean the same thing as \u201cif\u201d. \u00a0(If it did, these two sentences would have the same truth value in all situations.) \u00a0In fact, it seems that \u201conly if\u201d can best be expressed by a conditional where the \u201conly if\u201d appears before the consequent (remember, the consequent is the second part of the conditional\u2014the part that the arrows points at). \u00a0Thus, sentences of this form:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>only if <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Only if <strong><span class=\"strong\">Q<\/span><\/strong>, <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\r\nare best expressed by the formula\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What is the difference between \"if\" and \"only if\" in logical expressions?<\/li>\r\n \t<li><strong>Critical Thinking Task:\u00a0<\/strong>Design a hypothetical scenario when the phrases \"if\" and \"only if\" have different meanings.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>2.3 \u00a0Test your understanding of the conditional<\/h2>\r\nPeople sometimes find conditionals confusing. \u00a0In part, this seems to be because some people confuse them with another kind of truth-functional connective, which we will learn about later, called the \u201cbiconditional\u201d<span class=\"em\">.<\/span>\u00a0 Also, sometimes \u201cif\u2026then\u2026\u201d is used in English in a different way (see section 17.7 if you are curious about alternative possible meanings). \u00a0But from now on, we will understand the conditional as described above. \u00a0To test whether you have properly grasped the conditional, consider the following puzzle.[footnote]See Wason (1966).[\/footnote]\r\n\r\nWe have a set of four cards in Figure 2.1. \u00a0Each card has the following property: \u00a0it has a shape on one side, and a letter on the other side. \u00a0We shuffle and mix the cards, flipping some over while we shuffle. \u00a0Then, we lay out the four cards:\r\n\r\n[caption id=\"attachment_32\" align=\"aligncenter\" width=\"719\"]<img class=\"wp-image-31 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514.png\" alt=\"cards r square q star\" width=\"719\" height=\"329\" \/> Figure 2.1[\/caption]\r\n\r\nGiven our constraint that each card has a letter on one side and a shape on the other, we know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side; and so on.\r\n\r\nConsider now the following claim:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card.<\/p>\r\nHere is our puzzle: \u00a0what is the minimum number of cards that we must turn over to test whether this claim is true of all four cards; and which cards are they that we must turn over? \u00a0Of course we could turn them all over, but the puzzle asks you to identify all and only the cards that will test the claim.\r\n\r\nStop reading now, and see if you can decide on the answer. \u00a0Be warned, people generally perform poorly on this puzzle. \u00a0Think about it for a while. \u00a0The answer is given below in problem 1.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li><strong>Self Reflection: <\/strong>How did you do on the conditional test that uses the four cards? What do you think needs to happen to deepen your understanding of conditionals?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>2.4 \u00a0Alternative symbolizations for the conditional<\/h2>\r\nSome logic books, and some logicians, use alternative symbolizations for the various truth-functional connectives. \u00a0The meanings (that is, the truth tables) are always the same, but the symbol used may be different. \u00a0For this reason, we will take the time in this text to briefly recognize alternative symbolizations.\r\n\r\nThe conditional is sometimes represented with the following symbol: \u00a0\u201c<span class=\"strong\">\u2283<\/span>\u201d. \u00a0Thus, in such a case, <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0would be written\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2283<strong>Q<\/strong>)<\/span><\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li>Explain the meaning of the \"\u2283\" symbol in conditional statements.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<h2>2.5 \u00a0Negation<\/h2>\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>What purpose does the word \"not\" serve in logic?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Key Terms<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ul>\r\n \t<li><strong>Negation<\/strong> - opposing the truth value of a statement<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nIn chapter 1, we considered as an example the sentence,\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The Earth is not the center of the universe.<\/p>\r\nAt first glance, such a sentence might appear to be fundamentally unlike a conditional. \u00a0It does not contain two sentences, but only one. \u00a0There is a \u201cnot\u201d in the sentence, but it is not connecting two sentences. \u00a0However, we can still think of this sentence as being constructed with a truth functional connective, if we are willing to accept that this sentence is equivalent to the following sentence.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that the Earth is the center of the universe.<\/p>\r\nIf this sentence is equivalent to the one above, then we can treat \u201cIt is not the case\u201d as a truth functional connective. \u00a0It is traditional to replace this cumbersome English phrase with a single symbol, \u201c<span class=\"strong\">\u00ac<\/span>\u201d. \u00a0 Then, mixing our propositional logic with English, we would have\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<\/span>The Earth is the center of the universe.<\/p>\r\nAnd if we let <strong><span class=\"strong\">W<\/span>\u00a0<\/strong>be a sentence in our language that has the meaning <em>The Earth is the center of the universe<\/em>, we would write\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>W<\/strong><\/span><\/p>\r\nThis connective is called \u201cnegation\u201d. \u00a0Its syntax is: \u00a0if <strong>\u03a6<span class=\"strong\">\u00a0<\/span><\/strong>is a sentence, then\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<\/span><strong>\u03a6<\/strong><\/p>\r\nis a sentence. \u00a0We call such a sentence a \u201cnegation sentence\u201d.\r\n\r\nThe semantics of a negation sentence is also obvious, and is given by the following truth table.\r\n<table class=\"grid\" style=\"height: 82px;\" width=\"150\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<th class=\"border-right\" style=\"height: 15px; width: 202.683px;\" colspan=\"1\" rowspan=\"1\">\u03a6<\/th>\r\n<th class=\"border\" style=\"height: 15px; width: 266.983px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u00ac<\/span>\u03a6<\/th>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 203.083px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 267.383px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 203.083px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 267.383px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nTo deny a true sentence is to speak a falsehood. \u00a0To deny a false sentence is to say something true.\r\n\r\nOur syntax always is recursive<span class=\"em\">.<\/span>\u00a0 This means that syntactic rules can be applied repeatedly, to the product of the rule. \u00a0In other words, our syntax tells us that if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is a sentence, then <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence. \u00a0But now note that the same rule applies again: \u00a0if <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence, then <span class=\"strong\">\u00ac\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence. \u00a0And so on. \u00a0Similarly, if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are sentences, the syntax for the conditional tells us that <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0is a sentence. \u00a0But then so is <span class=\"strong\">\u00ac(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, and so is <span class=\"strong\">(\u00ac(<strong>P<\/strong>\u2192<strong>Q<\/strong>) \u2192 (<strong>P<\/strong>\u2192<strong>Q<\/strong>))<\/span>. \u00a0And so on. \u00a0If we have just a single atomic sentence, our recursive syntax will allow us to form infinitely many different sentences with negation and the conditional.\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Check for Understanding<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n<ol>\r\n \t<li>Explain the meaning of the \u201c\u00ac\u201d symbol in conditional statements.<\/li>\r\n \t<li>What does it mean if the syntax of a statement is \"recursive\"?<\/li>\r\n \t<li>Explain the meaning of the \u201c~\u201d symbol in conditional statements.<\/li>\r\n \t<li>Create three real-life examples where negation plays a key role (e.g., 'It is not the case that\u2026').<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>2.6 \u00a0Alternative symbolizations for negation<\/h2>\r\nSome texts may use \u201c<span class=\"strong\">~<\/span>\u201d for negation. \u00a0Thus, <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>would be expressed with\r\n<p style=\"padding-left: 120px;\"><span class=\"strong\">~<strong>P<\/strong><\/span><\/p>\r\n\r\n<h2>2.7 \u00a0Problems<\/h2>\r\n<ol class=\"lst-kix_list_24-0 start\" start=\"1\">\r\n \t<li>The answer to our card game was: you need only turn over cards 3 and 4. \u00a0This might seem confusing to many people at first. \u00a0But remember the meaning of the conditional: \u00a0it can only be false if the first part is true and the second part is false. \u00a0The sentence we want to test is \u201cFor each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card\u201d. \u00a0Let <span class=\"strong\">Q<\/span>\u00a0stand for \u201cthe card has a Q on the letter side of the card.\u201d \u00a0Let <span class=\"strong\">S<\/span>\u00a0stand for \u201cthe card has a square on the shape side of the card.\u201d \u00a0Then we could make a truth table to express the meaning of the claim being tested:<\/li>\r\n<\/ol>\r\n<table class=\"grid\" style=\"width: 175px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">Q \u00a0 \u00a0 \u00a0\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"strong\">S<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192S)<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nLook back at the cards. The first card has an R on the letter side. \u00a0So, sentence <span class=\"strong\">Q<\/span>\u00a0is false. \u00a0But then we are in a situation like the last two rows of the truth table, and the conditional cannot be false. \u00a0We do not need to check that card. \u00a0The second card has a square on it. \u00a0That means <span class=\"strong\">S<\/span>\u00a0is true for that card. \u00a0But then we are in a situation represented by either the first or third row of the truth table. \u00a0Again, the claim that <span class=\"strong\">(Q\u2192S)<\/span>\u00a0cannot be false in either case with respect to that card, so there is no point in checking that card. \u00a0The third card shows a Q. \u00a0It corresponds to a situation that is like either the first or second row of the truth table. \u00a0We cannot tell then whether <span class=\"strong\">(Q\u2192S) <\/span>is true or false of that card, without turning the card over. \u00a0Similarly, the last card shows a situation where <span class=\"strong\">S<\/span>\u00a0is false, so we are in a kind of situation represented by either the second or last row of the truth table. \u00a0We must turn the card over to determine if <span class=\"strong\">(Q\u2192S) <\/span>is true or false of that card.\r\n\r\nTry this puzzle again. \u00a0Consider the following claim about those same four cards: \u00a0If there is a star on the shape side of the card, then there is an R on the letter side of the card. \u00a0What is the minimum number of cards that you must turn over to check this claim? \u00a0What cards are they?\r\n<ol class=\"lst-kix_list_24-0\" start=\"2\">\r\n \t<li>Consider the following four cards in figure 2.2. \u00a0Each card has a letter on one side, and a shape on the other side.<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_32\" align=\"aligncenter\" width=\"719\"]<img class=\"wp-image-32 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766.png\" alt=\"cards p triangle hex diamond\" width=\"719\" height=\"338\" \/> Figure 2.2[\/caption]\r\n\r\nFor each of the following claims, in order to determine if the claim is true of all four cards, describe (1) The minimum number of cards you must turn over to check the claim, and (2) what those cards are.\r\n<ol class=\"lower-alpha\">\r\n \t<li>There is not a Q on the letter side of the card.<\/li>\r\n \t<li>There is not an octagon on the shape side of the card.<\/li>\r\n \t<li>If there is a triangle on the shape side of the card, then there is a P on the letter side of the card.<\/li>\r\n \t<li>There is an R on the letter side of the card only if there is a diamond on the shape side of the card.<\/li>\r\n \t<li>There is a hexagon on the shape side of the card, on the condition that there is a P on the letter side of the card.<\/li>\r\n \t<li>There is a diamond on the shape side of the card only if there is a P on the letter side of the card.<\/li>\r\n<\/ol>\r\n3. Which of the following have correct syntax? \u00a0Which have incorrect syntax?\r\n<ol class=\"lower-alpha\">\r\n \t<li><span class=\"strong\">P<\/span>\u2192<span class=\"strong\">Q<\/span><\/li>\r\n \t<li>\u00ac<span class=\"strong\">(P<\/span>\u2192<span class=\"strong\">Q)<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">Q)<\/span><\/li>\r\n \t<li><span class=\"strong\">(P<\/span>\u00ac\u2192<span class=\"strong\">Q)<\/span><\/li>\r\n \t<li><span class=\"strong\">(P<\/span>\u2192<span class=\"strong\">\u00acQ)<\/span><\/li>\r\n \t<li><span class=\"strong\">\u00ac\u00acP<\/span><\/li>\r\n \t<li><span class=\"strong\">\u00acP\u00ac<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acP\u00acQ)<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">\u00acQ)<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">\u00acQ)\u00ac<\/span><\/li>\r\n<\/ol>\r\n4. Use the following translation key to translate the following sentences into a propositional logic.\r\n<table class=\"grid\" style=\"width: 250px;\">\r\n<thead>\r\n<tr>\r\n<th style=\"text-align: center;\" colspan=\"2\">Translation Key<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td style=\"text-align: left;\">Logic<\/td>\r\n<td style=\"text-align: left;\">English<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: left;\"><strong>P<\/strong><\/td>\r\n<td style=\"text-align: left;\">Abe is\u00a0able.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"text-align: left;\"><strong>Q<\/strong><\/td>\r\n<td style=\"text-align: left;\">Abe is honest.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<ol class=\"lower-alpha\">\r\n \t<li>If Abe is honest, Abe is able.<\/li>\r\n \t<li>Abe is honest only if Abe is able.<\/li>\r\n \t<li>Abe is able, if Abe is honest.<\/li>\r\n \t<li>Only if Able is able, is Abe honest.<\/li>\r\n \t<li>Abe is not able.<\/li>\r\n \t<li>It's not the case that Abe isn't able.<\/li>\r\n \t<li>Abe is not able only if Abe is not honest.<\/li>\r\n \t<li>Abe is able, provided that Abe is not honest.<\/li>\r\n \t<li>If Abe is not able then Abe is not honest.<\/li>\r\n \t<li>It is not the case that, if Abe is able, then Abe is honest.<\/li>\r\n<\/ol>\r\n5.\u00a0Make up your own translation key to translate the following sentences into a propositional logic. Then, use your key to translate the sentences into the propositional logic. Your translation key should contain only atomic sentences. \u00a0 These should be all and only the atomic sentences needed to translate the following sentences of English. \u00a0Don\u2019t let it bother you that some of the sentences must be false.\r\n<ol class=\"lower-alpha\">\r\n \t<li>Josie is a cat.<\/li>\r\n \t<li>Josie is a mammal.<\/li>\r\n \t<li>Josie is not a mammal.<\/li>\r\n \t<li>If Josie is not a cat, then Josie is not a mammal.<\/li>\r\n \t<li>Josie is a fish.<\/li>\r\n \t<li>Provided that Josie is a mammal, Josie is not a fish.<\/li>\r\n \t<li>Josie is a cat only if Josie is a mammal.<\/li>\r\n \t<li>Josie is a fish only if Josie is not a mammal.<\/li>\r\n \t<li>It\u2019s not the case that Josie is not a mammal.<\/li>\r\n \t<li>Josie is not a cat, if Josie is a fish.<\/li>\r\n<\/ol>\r\n6. This problem will make use of the principle that our syntax is recursive. \u00a0Translating these sentences is more challenging. \u00a0Make up your own translation key to translate the following sentences into a propositional logic. \u00a0Your translation key should contain only atomic sentences; these should be all and only the atomic sentences needed to translate the following sentences of English.\r\n<ol class=\"lower-alpha\">\r\n \t<li>It is not the case that Luis won\u2019t pass the exam.<\/li>\r\n \t<li>If Luis studies, Luis will pass the exam.<\/li>\r\n \t<li>It is not the case that if Luis studies, then Luis will pass the exam.<\/li>\r\n \t<li>If Luis does not study, then Luis will not pass the exam.<\/li>\r\n \t<li>If Luis studies, Luis will pass the exam\u2014provided that he wakes in time.<\/li>\r\n \t<li>If Luis passes the exam, then if James studies, James will pass the exam.<\/li>\r\n \t<li>It is not the case that if Luis passes the exam, then if James studies, James will pass the exam.<\/li>\r\n \t<li>If Luis does not pass the exam, then if James studies, James will pass the exam.<\/li>\r\n \t<li>If Luis does not pass the exam, then it is not the case that if James studies, James will pass the exam.<\/li>\r\n \t<li>If Luis does not pass the exam, then if James does not study, James won\u2019t pass the exam.<\/li>\r\n<\/ol>\r\n7. Make up your own translation key in order to translate the following sentences into English. \u00a0Write out the English equivalents in English sentences that seem (as much as is possible) natural.\r\n<ol class=\"lower-alpha\">\r\n \t<li><span class=\"strong\">(R<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\r\n \t<li>\u00ac\u00ac<span class=\"strong\">R<\/span><\/li>\r\n \t<li><span class=\"strong\">(S<\/span>\u2192<span class=\"strong\">R)<\/span><\/li>\r\n \t<li><span class=\"strong\">\u00ac(S<\/span>\u2192<span class=\"strong\">R)<\/span><\/li>\r\n \t<li><span class=\"strong\">(<\/span>\u00ac<span class=\"strong\">S<\/span>\u2192\u00ac<span class=\"strong\">\u00acR)<\/span><\/li>\r\n \t<li>\u00ac<span class=\"strong\">\u00ac(R<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acR<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\r\n \t<li><span class=\"strong\">(R<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\r\n \t<li><span class=\"strong\">(\u00acR<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\r\n \t<li><span class=\"strong\">\u00ac(\u00acR<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\r\n<\/ol>\r\n<div>\r\n\r\n<hr \/>\r\n\r\n<a id=\"ftnt3\" href=\"#ftnt_ref3\">[3]<\/a>\u00a0One thing is a little funny about this second example with unknown number <span class=\"em strong\">a<\/span>. \u00a0We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. \u00a0First, this oddity arises because of mathematical facts, not facts of our propositional logic\u2014that is, we need to know what \u201cdivisible\u201d means, what \u201c4\u201d and \u201c2\u201d mean, and so on, in order to understand the sentence. \u00a0So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. \u00a0Second, some conditionals can be false. \u00a0In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.\r\n\r\n<a id=\"ftnt4\" href=\"#ftnt_ref4\">[4]<\/a> See Wason (1966).\r\n\r\n<\/div>","rendered":"<h2>2.1 \u00a0The Conditional<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What are parentheses used for in logical statements?<\/li>\n<li>How are conditional statements written in propositional logic?<\/li>\n<li>Can symbols be used to represent &#8220;If&#8230;Then&#8230;&#8221; statements?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><strong>Conditional Sentence<\/strong> &#8211; a sentence containing an antecedent (if) and a consequent (then)<\/li>\n<li><strong>Antecedent<\/strong> &#8211; the first sentence in a conditional statement; it is the statement before the \u2192 symbol<\/li>\n<li><strong>Consequent<\/strong> &#8211; the second sentence in a conditional statement; it is the statement after the \u2192 symbol<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>As we noted in chapter 1, there are sentences of a natural language, like English, that are not atomic sentences. \u00a0Our examples included<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If Obama wins the election, then Obama will be President.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If it rains tomorrow, then the picnic will be canceled.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The Earth is not the center of the universe.<\/p>\n<p>We could treat these like atomic sentences, but then we would lose a great deal of important information. \u00a0For example, the first sentence tells us something about the relationship between the atomic sentences \u201cObama wins the election\u201d and \u201cObama will be President\u201d. \u00a0And the third sentence above will, one supposes, have an interesting relationship to the sentence, \u201cThe Earth is the center of the universe\u201d. \u00a0To make these relations explicit, we will have to understand what \u201cif\u2026then\u2026\u201d and \u00a0\u201cnot\u201d mean. \u00a0Thus, it would be useful if our logical language was able to express these kinds of sentences in a way that made these elements explicit. \u00a0Let us start with the first one.<\/p>\n<p>The sentence, \u201cIf Obama wins the election, then Obama will be President\u201d contains two atomic sentences, \u201cObama wins the election\u201d and \u201cObama will be President\u201d. \u00a0We could thus represent this sentence by letting<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Obama wins the election<\/p>\n<p>be represented in our logical language by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\n<p>And by letting<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Obama will be president<\/p>\n<p>be represented by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>Then, the whole expression could be represented by writing<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>then <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>It will be useful, however, to replace the English phrase \u201cif\u2026then&#8230;\u201d by a single symbol in our language. \u00a0The most commonly used such symbol is \u201c\u2192\u201d. \u00a0Thus, we would write<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>\u2192<strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>One last thing needs to be observed, however. \u00a0We might want to combine this complex sentence with other sentences. \u00a0In that case, we need a way to identify that this is a single sentence when it is combined with other sentences. \u00a0There are several ways to do this, but the most familiar (although not the most elegant) is to use parentheses. \u00a0Thus, we will write our expression<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\n<p>This kind of sentence is called a \u201cconditional\u201d. \u00a0It is also sometimes called a \u201cmaterial conditional\u201d. \u00a0The first constituent sentence (the one before the arrow, which in this example is \u201c<strong><span class=\"strong\">P<\/span><\/strong>\u201d) is called the \u201cantecedent\u201d. Antecedent means &#8216;the condition&#8217;; think of it like &#8216;if I study (condition), I pass the test (result).&#8217;<br \/>\nThe second sentence (the one after the arrow, which in this example is \u201c<strong><span class=\"strong\">Q<\/span><\/strong>\u201d) is called the \u201cconsequent\u201d.<\/p>\n<p>We know how to write the conditional, but what does it mean? \u00a0As before, we will take the meaning to be given by the truth conditions\u2014that is, a description of when the sentence is either true or false. \u00a0We do this with a truth table. \u00a0But now, our sentence has two parts that are atomic sentences, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0Note that either atomic sentence could be true or false. \u00a0That means, we have to consider four possible kinds of situations. \u00a0We must consider when <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and when it is false, but then we need to consider those two kinds of situations twice: \u00a0once for when <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true and once for when <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is false. \u00a0Thus, the left hand side of our truth table will look like this:<\/p>\n<table class=\"grid\" style=\"height: 75px; width: 125px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong>\u00a0Q<br \/>\n<\/strong><\/td>\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>T<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>F<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>T<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 151.45px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\" style=\"width: 192.067px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><em><strong>F<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 112.383px; height: 15px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are four kinds of ways the world could be that we must consider.<\/p>\n<p>Note that, since there are two possible truth values (true and false), whenever we consider another atomic sentence, there are twice as many ways the world could be that we should consider. \u00a0Thus, for n atomic sentences, our truth table must have 2<sup><span class=\"super\">n<\/span><\/sup>\u00a0rows. \u00a0In the case of a conditional formed out of two atomic sentences, like our example of <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, our truth table will have 2<sup><span class=\"super\">2 <\/span><\/sup>rows, which is 4 rows. \u00a0We see this is the case above.<\/p>\n<p>Now, we must decide upon what the conditional means. \u00a0To some degree this is up to us. \u00a0What matters is that once we define the semantics of the conditional, we stick to our definition. \u00a0But we want to capture as much of the meaning of the English \u201cif\u2026then\u2026\u201d as we can, while remaining absolutely precise in our language.<\/p>\n<p>Let us consider each kind of way the world could be. \u00a0For the first row of the truth table, we have that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true. \u00a0Suppose the world is such that Obama wins the election, and also Obama will be President. \u00a0Then, would I have spoken truly if I said, \u201cIf Obama wins the election, then Obama will be President\u201d? \u00a0Most people agree that I would have. \u00a0Similarly, suppose that Obama wins the election, but Obama will not be President. \u00a0Would the sentence \u201cIf Obama wins the election, then Obama will be President\u201d still be true? \u00a0Most agree that it would be false now. \u00a0So the first rows of our truth table are uncontroversial.<\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><strong>\u00a0Q<br \/>\n<\/strong><\/td>\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">(<strong>P<\/strong><strong><span class=\"strong\">\u2192<\/span><\/strong><strong>Q<\/strong>)<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>T<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>F<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>T<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" style=\"width: 146px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\" style=\"width: 251px;\" colspan=\"1\" rowspan=\"1\"><em><strong>F<br \/>\n<\/strong><\/em><\/td>\n<td class=\"border\" style=\"width: 291px;\"><span class=\"space\">\u00a0<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Some students, however, find it hard to determine what truth values should go in the next two rows. \u00a0Note now that our principle of bivalence requires us to fill in these rows. \u00a0We cannot leave them blank. \u00a0If we did, we would be saying that sometimes a conditional can have no truth value; that is, we would be saying that sometimes, some sentences have no truth value. \u00a0But our principle of bivalence requires that\u2014in all kinds of situations\u2014every sentence is either true or false, never both, never neither. \u00a0So, if we are going to respect the principle of bivalence, then we have to put either <strong><span class=\"em strong\">T<\/span>\u00a0<\/strong>or <strong><span class=\"em strong\">F<\/span>\u00a0<\/strong>in for each of the last two rows.<\/p>\n<p>It is helpful at this point to change our example. \u00a0Let us consider two different examples to illustrate how best to fill out the remainder of the truth table for the conditional.<\/p>\n<p>First, suppose I say the following to you: \u00a0\u201cIf you give me $50, then I will buy you a ticket to the concert tonight.\u201d \u00a0Let<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">You give me $50<\/p>\n<p>be represented in our logic by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">R<\/span><\/strong><\/p>\n<p>and let<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">I will buy you a ticket to the concert tonight.<\/p>\n<p>be represented by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong><\/p>\n<p>Our sentence then is<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\n<p>And its truth table\u2014as far as we understand right now\u2014is:<\/p>\n<div class=\"keep\">\n<table class=\"grid\" style=\"width: 175px; height: 75px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<th class=\"border\" style=\"text-align: center; width: 208.517px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">R\u00a0\u00a0\u00a0\u00a0\u00a0 \u00a0\u00a0<\/span><\/th>\n<th class=\"border-right\" style=\"text-align: center; width: 199.5px; height: 15px;\"><span class=\"strong\">S<\/span><\/th>\n<th class=\"border\" style=\"text-align: center; width: 238.583px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(R\u2192S)<\/span><\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\">T<\/td>\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border\" style=\"text-align: center; width: 209.017px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\" style=\"text-align: center; width: 200.5px; height: 15px;\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" style=\"text-align: center; width: 239.083px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>That is, if you give me the money and I buy you the ticket, my claim that \u201cIf you give me $50, then I will buy you a ticket to the concert tonight\u201d is true. \u00a0And, if you give me the money and I don\u2019t buy you the ticket, I lied, and my claim is false. \u00a0But now, suppose you do not give me $50, but I buy you a ticket for the concert as a gift. \u00a0Was my claim false? \u00a0No. \u00a0I simply bought you the ticket as a gift, but, presumably would have bought it if you gave me the money, also. \u00a0Similarly, if you don\u2019t give me money, and I do not buy you a ticket, that seems perfectly consistent with my claim.<\/p>\n<p>So, the best way to fill out the truth table is as follows.<\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">R \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">S<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(R\u2192S)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Second, consider another sentence, which has the advantage that it is very clear with respect to these last two rows. \u00a0Assume that <span class=\"em strong\">a<\/span>\u00a0is a particular natural number, only you and I don\u2019t know what number it is (the natural numbers are the whole positive numbers: \u00a01, 2, 3, 4\u2026). \u00a0Consider now the following sentence.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 4, then <span class=\"em strong\">a<\/span>\u00a0is evenly divisible by 2.<\/p>\n<p>(By \u201cevenly divisible,\u201d I mean divisible without remainder.) \u00a0The first thing to ask yourself is: is this sentence true? \u00a0I hope we can all agree that it is\u2014even though we do not know what <strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is. \u00a0Let<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 4<\/p>\n<p>be represented in our logic by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">U<\/span><\/strong><\/p>\n<p>and let<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><em><span class=\"em strong\">a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2<\/p>\n<p>be represented by<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">V<\/span><\/strong><\/p>\n<p>Our sentence then is<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>U<\/strong>\u2192<strong>V<\/strong>)<\/span><\/p>\n<p>And its truth table\u2014as far as we understand right now\u2014is:<\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">U \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">V<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(U\u2192V)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Now consider a case in which <span class=\"em strong\">a<\/span>\u00a0is 6. \u00a0This is like the third row of the truth table. \u00a0It is not the case that 6 is evenly divisible by 4, but it is the case that 6 is evenly divisible by 2. \u00a0And consider the case in which <span class=\"em strong\">a<\/span>\u00a0is 7. \u00a0This is like the fourth row of the truth table; 7 would be evenly divisible by neither 4 nor 2. \u00a0But we agreed that the conditional is true\u2014regardless of the value of <span class=\"em strong\">a<\/span>! \u00a0So, the truth table must be:<a class=\"footnote\" title=\"One thing is a little funny about this second example with unknown number a. We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. First, this oddity arises because of mathematical facts, not facts of our propositional logic\u2014that is, we need to know what \u201cdivisible\u201d means, what \u201c4\u201d and \u201c2\u201d mean, and so on, in order to understand the sentence. So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. Second, some conditionals can be false. In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.\" id=\"return-footnote-33-1\" href=\"#footnote-33-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">U \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">V<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(U\u2192V)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><span class=\"strong\">\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Following this pattern, we should also fill out our table about the election with:<\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P<br \/>\n<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">Q<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(P\u2192Q)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong em\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>If you are dissatisfied by this, it might be helpful to think of these last two rows as vacuous cases. \u00a0A conditional tells us about what happens if the antecedent is true. \u00a0But when the antecedent is false, we simply default to true.<\/p>\n<p>We are now ready to offer, in a more formal way, the syntax and semantics for the conditional.<\/p>\n<p>The syntax of the conditional is that, if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>\u03a6<\/strong>\u2192<strong>\u03a8<\/strong>)<\/span><\/p>\n<p>is a sentence.<\/p>\n<p>The semantics of the conditional are given by a truth table. \u00a0For any sentences <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <span class=\"strong\"><strong>\u03a8<\/strong>:<\/span><\/p>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6\u2192\u03a8)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Remember that this truth table is now a definition. \u00a0It defines the meaning of \u201c<span class=\"strong\">\u2192<\/span>\u201d. \u00a0We are agreeing to use the symbol \u201c<span class=\"strong\">\u2192<\/span>\u201d to mean this from here on out.<\/p>\n<p>The elements of the propositional logic, like \u201c<span class=\"strong\">\u2192<\/span>\u201d, that we add to our language in order to form more complex sentences, are called \u201ctruth functional connectives\u201d. \u00a0I hope it is clear why: \u00a0the meaning of this symbol is given in a truth function. \u00a0(If you are unfamiliar or uncertain about the idea of a function, think of a function as like a machine that takes in one or more inputs, and always then gives exactly one output. \u00a0For the conditional, the inputs are two truth values; and the output is one truth value. \u00a0For example, put <span class=\"em strong\"><strong>T<\/strong> <strong>F<\/strong><\/span><strong>\u00a0<\/strong>into the truth function called \u201c<span class=\"strong\">\u2192<\/span>\u201d, and you get out <strong><span class=\"em strong\">F<\/span><\/strong>.)<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What defines a conditional?<\/li>\n<li>How is the Principle of Bivalence applied in truth tables for conditional statements?<\/li>\n<li>What are formal ways to represent the syntax and semantics of sentence?<\/li>\n<li>Describe a decision where an &#8216;if\u2026then\u2026&#8217; statement influenced your choice. Was the condition met?<\/li>\n<li><strong>Critical Thinking Task:\u00a0<\/strong>Contemplate some specific cases with false antecedents and true consequents.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>2.2 \u00a0Alternative phrasings\u00a0in English for the conditional. \u00a0Only if.<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>How can the word &#8220;only&#8221; change the logical meaning of a sentence?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<p>English includes many alternative phrasings that appear to be equivalent to the conditional. \u00a0Furthermore, in English and other natural languages, the order of the conditional will sometimes be reversed. \u00a0We can capture the general sense of these cases by recognizing that each of the following phrasings would be translated as <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>. \u00a0 (In these examples, we mix English and our propositional logic, in order to illustrate the variations succinctly.)<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If <strong><span class=\"strong\">P<\/span><\/strong>, then <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, if <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">On the condition that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, on the condition that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Given that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, given that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Provided that <strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, provided that <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">When <strong><span class=\"strong\">P<\/span><\/strong>, then <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>, when <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>implies <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is implied by <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is sufficient for <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is necessary for <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p>An oddity of English is that the word \u201conly\u201d changes the meaning of \u201cif\u201d. \u00a0You can see this if you consider the following two sentences.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Fifi is a cat, if Fifi is a mammal.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Fifi is a cat only if Fifi is a mammal.<\/p>\n<p>Suppose we know Fifi is an organism, but, we don\u2019t know what kind of organism Fifi is. \u00a0Fifi could be a dog, a cat, a gray whale, a ladybug, a sponge. \u00a0It seems clear that the first sentence is not necessarily true. \u00a0If Fifi is a gray whale, for example, then it is true that Fifi is a mammal, but false that Fifi is a cat; and so, the first sentence would be false. \u00a0But the second sentence looks like it must be true (given what you and I know about cats and mammals).<\/p>\n<p>We should thus be careful to recognize that \u201conly if\u201d does not mean the same thing as \u201cif\u201d. \u00a0(If it did, these two sentences would have the same truth value in all situations.) \u00a0In fact, it seems that \u201conly if\u201d can best be expressed by a conditional where the \u201conly if\u201d appears before the consequent (remember, the consequent is the second part of the conditional\u2014the part that the arrows points at). \u00a0Thus, sentences of this form:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>only if <strong><span class=\"strong\">Q<\/span><\/strong>.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Only if <strong><span class=\"strong\">Q<\/span><\/strong>, <strong><span class=\"strong\">P<\/span><\/strong>.<\/p>\n<p>are best expressed by the formula<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What is the difference between &#8220;if&#8221; and &#8220;only if&#8221; in logical expressions?<\/li>\n<li><strong>Critical Thinking Task:\u00a0<\/strong>Design a hypothetical scenario when the phrases &#8220;if&#8221; and &#8220;only if&#8221; have different meanings.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>2.3 \u00a0Test your understanding of the conditional<\/h2>\n<p>People sometimes find conditionals confusing. \u00a0In part, this seems to be because some people confuse them with another kind of truth-functional connective, which we will learn about later, called the \u201cbiconditional\u201d<span class=\"em\">.<\/span>\u00a0 Also, sometimes \u201cif\u2026then\u2026\u201d is used in English in a different way (see section 17.7 if you are curious about alternative possible meanings). \u00a0But from now on, we will understand the conditional as described above. \u00a0To test whether you have properly grasped the conditional, consider the following puzzle.<a class=\"footnote\" title=\"See Wason (1966).\" id=\"return-footnote-33-2\" href=\"#footnote-33-2\" aria-label=\"Footnote 2\"><sup class=\"footnote\">[2]<\/sup><\/a><\/p>\n<p>We have a set of four cards in Figure 2.1. \u00a0Each card has the following property: \u00a0it has a shape on one side, and a letter on the other side. \u00a0We shuffle and mix the cards, flipping some over while we shuffle. \u00a0Then, we lay out the four cards:<\/p>\n<figure id=\"attachment_32\" aria-describedby=\"caption-attachment-32\" style=\"width: 719px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-31 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514.png\" alt=\"cards r square q star\" width=\"719\" height=\"329\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514.png 719w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514-300x137.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514-65x30.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514-225x103.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-r-square-q-star-e1467752397514-350x160.png 350w\" sizes=\"auto, (max-width: 719px) 100vw, 719px\" \/><figcaption id=\"caption-attachment-32\" class=\"wp-caption-text\">Figure 2.1<\/figcaption><\/figure>\n<p>Given our constraint that each card has a letter on one side and a shape on the other, we know that card 1 has a shape on the unseen side; card 2 has a letter on the unseen side; and so on.<\/p>\n<p>Consider now the following claim:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">For each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card.<\/p>\n<p>Here is our puzzle: \u00a0what is the minimum number of cards that we must turn over to test whether this claim is true of all four cards; and which cards are they that we must turn over? \u00a0Of course we could turn them all over, but the puzzle asks you to identify all and only the cards that will test the claim.<\/p>\n<p>Stop reading now, and see if you can decide on the answer. \u00a0Be warned, people generally perform poorly on this puzzle. \u00a0Think about it for a while. \u00a0The answer is given below in problem 1.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li><strong>Self Reflection: <\/strong>How did you do on the conditional test that uses the four cards? What do you think needs to happen to deepen your understanding of conditionals?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>2.4 \u00a0Alternative symbolizations for the conditional<\/h2>\n<p>Some logic books, and some logicians, use alternative symbolizations for the various truth-functional connectives. \u00a0The meanings (that is, the truth tables) are always the same, but the symbol used may be different. \u00a0For this reason, we will take the time in this text to briefly recognize alternative symbolizations.<\/p>\n<p>The conditional is sometimes represented with the following symbol: \u00a0\u201c<span class=\"strong\">\u2283<\/span>\u201d. \u00a0Thus, in such a case, <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0would be written<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2283<strong>Q<\/strong>)<\/span><\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li>Explain the meaning of the &#8220;\u2283&#8221; symbol in conditional statements.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<h2>2.5 \u00a0Negation<\/h2>\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Pre-Reading Questions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>What purpose does the word &#8220;not&#8221; serve in logic?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Key Terms<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ul>\n<li><strong>Negation<\/strong> &#8211; opposing the truth value of a statement<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>In chapter 1, we considered as an example the sentence,<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The Earth is not the center of the universe.<\/p>\n<p>At first glance, such a sentence might appear to be fundamentally unlike a conditional. \u00a0It does not contain two sentences, but only one. \u00a0There is a \u201cnot\u201d in the sentence, but it is not connecting two sentences. \u00a0However, we can still think of this sentence as being constructed with a truth functional connective, if we are willing to accept that this sentence is equivalent to the following sentence.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that the Earth is the center of the universe.<\/p>\n<p>If this sentence is equivalent to the one above, then we can treat \u201cIt is not the case\u201d as a truth functional connective. \u00a0It is traditional to replace this cumbersome English phrase with a single symbol, \u201c<span class=\"strong\">\u00ac<\/span>\u201d. \u00a0 Then, mixing our propositional logic with English, we would have<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<\/span>The Earth is the center of the universe.<\/p>\n<p>And if we let <strong><span class=\"strong\">W<\/span>\u00a0<\/strong>be a sentence in our language that has the meaning <em>The Earth is the center of the universe<\/em>, we would write<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<strong>W<\/strong><\/span><\/p>\n<p>This connective is called \u201cnegation\u201d. \u00a0Its syntax is: \u00a0if <strong>\u03a6<span class=\"strong\">\u00a0<\/span><\/strong>is a sentence, then<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">\u00ac<\/span><strong>\u03a6<\/strong><\/p>\n<p>is a sentence. \u00a0We call such a sentence a \u201cnegation sentence\u201d.<\/p>\n<p>The semantics of a negation sentence is also obvious, and is given by the following truth table.<\/p>\n<table class=\"grid\" style=\"height: 82px; width: 150px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<th class=\"border-right\" style=\"height: 15px; width: 202.683px;\" colspan=\"1\" rowspan=\"1\">\u03a6<\/th>\n<th class=\"border\" style=\"height: 15px; width: 266.983px;\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u00ac<\/span>\u03a6<\/th>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 203.083px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" style=\"height: 15px; width: 267.383px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 203.083px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" style=\"height: 15px; width: 267.383px;\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>To deny a true sentence is to speak a falsehood. \u00a0To deny a false sentence is to say something true.<\/p>\n<p>Our syntax always is recursive<span class=\"em\">.<\/span>\u00a0 This means that syntactic rules can be applied repeatedly, to the product of the rule. \u00a0In other words, our syntax tells us that if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is a sentence, then <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence. \u00a0But now note that the same rule applies again: \u00a0if <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence, then <span class=\"strong\">\u00ac\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>is a sentence. \u00a0And so on. \u00a0Similarly, if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are sentences, the syntax for the conditional tells us that <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0is a sentence. \u00a0But then so is <span class=\"strong\">\u00ac(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, and so is <span class=\"strong\">(\u00ac(<strong>P<\/strong>\u2192<strong>Q<\/strong>) \u2192 (<strong>P<\/strong>\u2192<strong>Q<\/strong>))<\/span>. \u00a0And so on. \u00a0If we have just a single atomic sentence, our recursive syntax will allow us to form infinitely many different sentences with negation and the conditional.<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Check for Understanding<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<ol>\n<li>Explain the meaning of the \u201c\u00ac\u201d symbol in conditional statements.<\/li>\n<li>What does it mean if the syntax of a statement is &#8220;recursive&#8221;?<\/li>\n<li>Explain the meaning of the \u201c~\u201d symbol in conditional statements.<\/li>\n<li>Create three real-life examples where negation plays a key role (e.g., &#8216;It is not the case that\u2026&#8217;).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>2.6 \u00a0Alternative symbolizations for negation<\/h2>\n<p>Some texts may use \u201c<span class=\"strong\">~<\/span>\u201d for negation. \u00a0Thus, <span class=\"strong\">\u00ac<strong>P<\/strong><\/span><strong>\u00a0<\/strong>would be expressed with<\/p>\n<p style=\"padding-left: 120px;\"><span class=\"strong\">~<strong>P<\/strong><\/span><\/p>\n<h2>2.7 \u00a0Problems<\/h2>\n<ol class=\"lst-kix_list_24-0 start\" start=\"1\">\n<li>The answer to our card game was: you need only turn over cards 3 and 4. \u00a0This might seem confusing to many people at first. \u00a0But remember the meaning of the conditional: \u00a0it can only be false if the first part is true and the second part is false. \u00a0The sentence we want to test is \u201cFor each of these four cards, if the card has a Q on the letter side of the card, then it has a square on the shape side of the card\u201d. \u00a0Let <span class=\"strong\">Q<\/span>\u00a0stand for \u201cthe card has a Q on the letter side of the card.\u201d \u00a0Let <span class=\"strong\">S<\/span>\u00a0stand for \u201cthe card has a square on the shape side of the card.\u201d \u00a0Then we could make a truth table to express the meaning of the claim being tested:<\/li>\n<\/ol>\n<table class=\"grid\" style=\"width: 175px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">Q \u00a0 \u00a0 \u00a0\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"strong\">S<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(Q<\/span><span class=\"strong\">\u2192S)<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Look back at the cards. The first card has an R on the letter side. \u00a0So, sentence <span class=\"strong\">Q<\/span>\u00a0is false. \u00a0But then we are in a situation like the last two rows of the truth table, and the conditional cannot be false. \u00a0We do not need to check that card. \u00a0The second card has a square on it. \u00a0That means <span class=\"strong\">S<\/span>\u00a0is true for that card. \u00a0But then we are in a situation represented by either the first or third row of the truth table. \u00a0Again, the claim that <span class=\"strong\">(Q\u2192S)<\/span>\u00a0cannot be false in either case with respect to that card, so there is no point in checking that card. \u00a0The third card shows a Q. \u00a0It corresponds to a situation that is like either the first or second row of the truth table. \u00a0We cannot tell then whether <span class=\"strong\">(Q\u2192S) <\/span>is true or false of that card, without turning the card over. \u00a0Similarly, the last card shows a situation where <span class=\"strong\">S<\/span>\u00a0is false, so we are in a kind of situation represented by either the second or last row of the truth table. \u00a0We must turn the card over to determine if <span class=\"strong\">(Q\u2192S) <\/span>is true or false of that card.<\/p>\n<p>Try this puzzle again. \u00a0Consider the following claim about those same four cards: \u00a0If there is a star on the shape side of the card, then there is an R on the letter side of the card. \u00a0What is the minimum number of cards that you must turn over to check this claim? \u00a0What cards are they?<\/p>\n<ol class=\"lst-kix_list_24-0\" start=\"2\">\n<li>Consider the following four cards in figure 2.2. \u00a0Each card has a letter on one side, and a shape on the other side.<\/li>\n<\/ol>\n<figure id=\"attachment_32\" aria-describedby=\"caption-attachment-32\" style=\"width: 719px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-32 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766.png\" alt=\"cards p triangle hex diamond\" width=\"719\" height=\"338\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766.png 719w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766-300x141.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766-65x31.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766-225x106.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2025\/04\/cards-p-triangle-hex-diamond-e1467752421766-350x165.png 350w\" sizes=\"auto, (max-width: 719px) 100vw, 719px\" \/><figcaption id=\"caption-attachment-32\" class=\"wp-caption-text\">Figure 2.2<\/figcaption><\/figure>\n<p>For each of the following claims, in order to determine if the claim is true of all four cards, describe (1) The minimum number of cards you must turn over to check the claim, and (2) what those cards are.<\/p>\n<ol class=\"lower-alpha\">\n<li>There is not a Q on the letter side of the card.<\/li>\n<li>There is not an octagon on the shape side of the card.<\/li>\n<li>If there is a triangle on the shape side of the card, then there is a P on the letter side of the card.<\/li>\n<li>There is an R on the letter side of the card only if there is a diamond on the shape side of the card.<\/li>\n<li>There is a hexagon on the shape side of the card, on the condition that there is a P on the letter side of the card.<\/li>\n<li>There is a diamond on the shape side of the card only if there is a P on the letter side of the card.<\/li>\n<\/ol>\n<p>3. Which of the following have correct syntax? \u00a0Which have incorrect syntax?<\/p>\n<ol class=\"lower-alpha\">\n<li><span class=\"strong\">P<\/span>\u2192<span class=\"strong\">Q<\/span><\/li>\n<li>\u00ac<span class=\"strong\">(P<\/span>\u2192<span class=\"strong\">Q)<\/span><\/li>\n<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">Q)<\/span><\/li>\n<li><span class=\"strong\">(P<\/span>\u00ac\u2192<span class=\"strong\">Q)<\/span><\/li>\n<li><span class=\"strong\">(P<\/span>\u2192<span class=\"strong\">\u00acQ)<\/span><\/li>\n<li><span class=\"strong\">\u00ac\u00acP<\/span><\/li>\n<li><span class=\"strong\">\u00acP\u00ac<\/span><\/li>\n<li><span class=\"strong\">(\u00acP\u00acQ)<\/span><\/li>\n<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">\u00acQ)<\/span><\/li>\n<li><span class=\"strong\">(\u00acP<\/span>\u2192<span class=\"strong\">\u00acQ)\u00ac<\/span><\/li>\n<\/ol>\n<p>4. Use the following translation key to translate the following sentences into a propositional logic.<\/p>\n<table class=\"grid\" style=\"width: 250px;\">\n<thead>\n<tr>\n<th style=\"text-align: center;\" colspan=\"2\">Translation Key<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td style=\"text-align: left;\">Logic<\/td>\n<td style=\"text-align: left;\">English<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left;\"><strong>P<\/strong><\/td>\n<td style=\"text-align: left;\">Abe is\u00a0able.<\/td>\n<\/tr>\n<tr>\n<td style=\"text-align: left;\"><strong>Q<\/strong><\/td>\n<td style=\"text-align: left;\">Abe is honest.<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<ol class=\"lower-alpha\">\n<li>If Abe is honest, Abe is able.<\/li>\n<li>Abe is honest only if Abe is able.<\/li>\n<li>Abe is able, if Abe is honest.<\/li>\n<li>Only if Able is able, is Abe honest.<\/li>\n<li>Abe is not able.<\/li>\n<li>It&#8217;s not the case that Abe isn&#8217;t able.<\/li>\n<li>Abe is not able only if Abe is not honest.<\/li>\n<li>Abe is able, provided that Abe is not honest.<\/li>\n<li>If Abe is not able then Abe is not honest.<\/li>\n<li>It is not the case that, if Abe is able, then Abe is honest.<\/li>\n<\/ol>\n<p>5.\u00a0Make up your own translation key to translate the following sentences into a propositional logic. Then, use your key to translate the sentences into the propositional logic. Your translation key should contain only atomic sentences. \u00a0 These should be all and only the atomic sentences needed to translate the following sentences of English. \u00a0Don\u2019t let it bother you that some of the sentences must be false.<\/p>\n<ol class=\"lower-alpha\">\n<li>Josie is a cat.<\/li>\n<li>Josie is a mammal.<\/li>\n<li>Josie is not a mammal.<\/li>\n<li>If Josie is not a cat, then Josie is not a mammal.<\/li>\n<li>Josie is a fish.<\/li>\n<li>Provided that Josie is a mammal, Josie is not a fish.<\/li>\n<li>Josie is a cat only if Josie is a mammal.<\/li>\n<li>Josie is a fish only if Josie is not a mammal.<\/li>\n<li>It\u2019s not the case that Josie is not a mammal.<\/li>\n<li>Josie is not a cat, if Josie is a fish.<\/li>\n<\/ol>\n<p>6. This problem will make use of the principle that our syntax is recursive. \u00a0Translating these sentences is more challenging. \u00a0Make up your own translation key to translate the following sentences into a propositional logic. \u00a0Your translation key should contain only atomic sentences; these should be all and only the atomic sentences needed to translate the following sentences of English.<\/p>\n<ol class=\"lower-alpha\">\n<li>It is not the case that Luis won\u2019t pass the exam.<\/li>\n<li>If Luis studies, Luis will pass the exam.<\/li>\n<li>It is not the case that if Luis studies, then Luis will pass the exam.<\/li>\n<li>If Luis does not study, then Luis will not pass the exam.<\/li>\n<li>If Luis studies, Luis will pass the exam\u2014provided that he wakes in time.<\/li>\n<li>If Luis passes the exam, then if James studies, James will pass the exam.<\/li>\n<li>It is not the case that if Luis passes the exam, then if James studies, James will pass the exam.<\/li>\n<li>If Luis does not pass the exam, then if James studies, James will pass the exam.<\/li>\n<li>If Luis does not pass the exam, then it is not the case that if James studies, James will pass the exam.<\/li>\n<li>If Luis does not pass the exam, then if James does not study, James won\u2019t pass the exam.<\/li>\n<\/ol>\n<p>7. Make up your own translation key in order to translate the following sentences into English. \u00a0Write out the English equivalents in English sentences that seem (as much as is possible) natural.<\/p>\n<ol class=\"lower-alpha\">\n<li><span class=\"strong\">(R<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\n<li>\u00ac\u00ac<span class=\"strong\">R<\/span><\/li>\n<li><span class=\"strong\">(S<\/span>\u2192<span class=\"strong\">R)<\/span><\/li>\n<li><span class=\"strong\">\u00ac(S<\/span>\u2192<span class=\"strong\">R)<\/span><\/li>\n<li><span class=\"strong\">(<\/span>\u00ac<span class=\"strong\">S<\/span>\u2192\u00ac<span class=\"strong\">\u00acR)<\/span><\/li>\n<li>\u00ac<span class=\"strong\">\u00ac(R<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\n<li><span class=\"strong\">(\u00acR<\/span>\u2192<span class=\"strong\">S)<\/span><\/li>\n<li><span class=\"strong\">(R<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\n<li><span class=\"strong\">(\u00acR<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\n<li><span class=\"strong\">\u00ac(\u00acR<\/span>\u2192<span class=\"strong\">\u00acS)<\/span><\/li>\n<\/ol>\n<div>\n<hr \/>\n<p><a id=\"ftnt3\" href=\"#ftnt_ref3\">[3]<\/a>\u00a0One thing is a little funny about this second example with unknown number <span class=\"em strong\">a<\/span>. \u00a0We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. \u00a0First, this oddity arises because of mathematical facts, not facts of our propositional logic\u2014that is, we need to know what \u201cdivisible\u201d means, what \u201c4\u201d and \u201c2\u201d mean, and so on, in order to understand the sentence. \u00a0So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. \u00a0Second, some conditionals can be false. \u00a0In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example.<\/p>\n<p><a id=\"ftnt4\" href=\"#ftnt_ref4\">[4]<\/a> See Wason (1966).<\/p>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-33-1\">One thing is a little funny about this second example with unknown number a. We will not be able to find a number that is evenly divisible by 4 and not evenly divisible by 2, so the world will never be like the second row of this truth table describes. Two things need to be said about this. First, this oddity arises because of mathematical facts, not facts of our propositional logic\u2014that is, we need to know what \u201cdivisible\u201d means, what \u201c4\u201d and \u201c2\u201d mean, and so on, in order to understand the sentence. So, when we see that the second row is not possible, we are basing that on our knowledge of mathematics, not on our knowledge of propositional logic. Second, some conditionals can be false. In defining the conditional, we need to consider all possible conditionals; so, we must define the conditional for any case where the antecedent is true and the consequent is false, even if that cannot happen for this specific example. <a href=\"#return-footnote-33-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><li id=\"footnote-33-2\">See Wason (1966). <a href=\"#return-footnote-33-2\" class=\"return-footnote\" aria-label=\"Return to footnote 2\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"cc-by-nc-sa"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-33","chapter","type-chapter","status-publish","hentry","license-cc-by-nc-sa"],"part":27,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/33","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":36,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/33\/revisions"}],"predecessor-version":[{"id":335,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/33\/revisions\/335"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/parts\/27"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/33\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=33"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=33"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=33"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=33"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}