{"id":29,"date":"2016-12-16T18:14:04","date_gmt":"2016-12-16T18:14:04","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/1-developing-a-precise-language\/"},"modified":"2025-10-13T17:04:18","modified_gmt":"2025-10-13T17:04:18","slug":"1-developing-a-precise-language","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/1-developing-a-precise-language\/","title":{"raw":"Developing a Precise Language","rendered":"Developing a Precise Language"},"content":{"raw":"<h2>1.1 Starting with sentences<\/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 is your prior knowledge related to logic?<\/li>\r\n \t<li>How can a sentence be declared as \"whole\"?<\/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>Propositional Logic<\/strong> - the reasoning used to create whole sentences known as propositions.<\/li>\r\n \t<li><strong>Declarative Sentence<\/strong> - a precise set of language that can be either true or false.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nWe begin the study of logic by building a precise logical language. \u00a0This will allow us to do at least two things: \u00a0first, to say some things more precisely than we otherwise would be able to do; second, to study reasoning. \u00a0We will use a natural language\u2014English\u2014as our guide, but our logical language will be far simpler, far weaker, but more rigorous than English.\r\n\r\nWe must decide where to start. \u00a0We could pick just about any part of English to try to emulate: \u00a0names, adjectives, prepositions, general nouns, and so on. \u00a0But it is traditional, and as we will see, quite handy, to begin with whole sentences. \u00a0For this reason, the first language we will develop is called \u201cthe <strong>propositional logic<\/strong>\u201d.<span class=\"em\">\u00a0 <\/span>It is also sometimes called \u201cthe sentential logic\u201d or even \u201cthe sentential calculus\u201d. These all mean the same thing: \u00a0the logic of sentences. \u00a0In this <strong>propositional logic<\/strong>, the smallest independent parts of the language are sentences (throughout this book, I will assume that sentences and propositions are the same thing in our logic, and I will use the terms \u201csentence\u201d and \u201cproposition\u201d interchangeably).\r\n\r\nThere are of course many kinds of sentences. \u00a0To take examples from our natural language, these include:\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">What time is it?<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Open the window.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Damn you!<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">I promise to pay you back.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">It rained in City Park on June 26, 2015.<\/p>\r\nWe could multiply such examples. \u00a0Sentences in English can be used to ask questions, give commands, curse or insult, form contracts, and express emotions. \u00a0But, the last example above is of special interest because it aims to describe the world. \u00a0Such sentences, which are sometimes called \u201cdeclarative sentences\u201d, will be our model sentences for our logical language. \u00a0We know a <strong>declarative sentence<\/strong> when we encounter it because it can be either true or false.\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>Please explain the logic of whole sentences.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>1.2 Precision in sentences<\/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 makes a sentence clearly true or false?<\/li>\r\n \t<li>Is it possible for a declarative sentence to be both true and false?<\/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>Truth Value <\/strong>- a clear assessment of whether a statement is true or false in reality.<\/li>\r\n \t<li><strong>Principle of Bivalence\u00a0<\/strong>- each sentence of our language must be either true or false, not both, not neither.<\/li>\r\n \t<li><strong>Principle of Non-Contradiction\u00a0<\/strong>- a claim must be asserted or denied.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nWe want our logic of declarative sentences to be precise. \u00a0But what does this mean? \u00a0We can help clarify how we might pursue this by looking at sentences in a natural language that are perplexing, apparently because they are not precise. \u00a0Here are three.\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Chris is kind of tall.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">When Jasmine had a baby, her mother gave her a pen.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">This sentence is false.<\/p>\r\nWe have already observed that an important feature of our declarative sentences is that they can be true or false. \u00a0We call this the \u201ctruth value\u201d of the sentence. \u00a0These three sentences are perplexing because their truth values are unclear. \u00a0The first sentence is vague, it is not clear under what conditions it would be true, and under what conditions it would be false. \u00a0If Chris is six feet tall, is he kind of tall? \u00a0There is no clear answer. \u00a0The second sentence is ambiguous<span class=\"em\">.<\/span>\u00a0 If \u201cpen\u201d means writing implement, and Jasmine\u2019s mother bought a playpen for the baby, then the sentence is false. \u00a0But until we know what \u201cpen\u201d means in this sentence, we cannot tell if the sentence is true.\r\n\r\nThe third sentence is strange. \u00a0Many logicians have spent many years studying this sentence, which is traditionally called \u201cthe Liar\u201d. \u00a0It is related to an old paradox about a Cretan who said, \u201cAll Cretans are liars\u201d. \u00a0The strange thing about the Liar is that its truth value seems to explode. \u00a0If it is true, then it is false. \u00a0If it is false, then it is true. \u00a0Some philosophers think this sentence is, therefore, neither true nor false; some philosophers think it is both true and false. \u00a0In either case, it is confusing. \u00a0How could a sentence that looks like a declarative sentence have both or no truth value?\r\n\r\nSince ancient times, philosophers have believed that we will deceive ourselves, and come to believe untruths, if we do not accept a principle sometimes called \u201c<strong>the principle of <\/strong><strong>bivalence<\/strong>\u201d, or a related principle called \u201c<strong>the principle of non-contradiction<\/strong>\u201d. \u00a0Bivalence is the view that there are only two truth values (true and false) and that they exclude each other. \u00a0The principle of non-contradiction states that you have made a mistake if you both assert and deny a claim. \u00a0One or the other of these principles seems to be violated by the Liar.\r\n\r\nWe can take these observations for our guide: \u00a0we want our language to have no vagueness and no ambiguity. \u00a0In our propositional logic, this means we want it to be the case that each sentence is either true or false. \u00a0It will not be kind of true, or partially true, or true from one perspective and not true from another. \u00a0We also want to avoid things like the Liar. \u00a0We do not need to agree on whether the Liar is both true and false, or neither true nor false. \u00a0Either would be unfortunate. \u00a0So, we will specify that our sentences have neither vice.\r\n\r\nWe can formulate our own revised version of the principle of bivalence, which states that:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>Principle of Bivalence<\/strong>: \u00a0Each sentence of our language must be either true or false, not both, not neither.<\/p>\r\nThis requirement may sound trivial, but in fact it constrains what we do from now on in interesting and even surprising ways. \u00a0Even as we build more complex logical languages later, this principle will be fundamental.\r\n\r\nSome readers may be thinking: \u00a0what if I reject bivalence, or the <strong>principle of non-contradiction<\/strong>? \u00a0There is a long line of philosophers who would like to argue with you, and propose that either move would be a mistake, and perhaps even incoherent. \u00a0Set those arguments aside. \u00a0If you have doubts about bivalence, or the principle of non-contradiction, stick with logic. That is because we could develop a logic in which there were more than two truth values. \u00a0Logics have been created and studied in which we allow for three truth values, or continuous truth values, or stranger possibilities. \u00a0The issue for us is that we must start somewhere, and the principle of bivalence is an intuitive way and\u2014it would seem\u2014the simplest way to start with respect to truth values. \u00a0Learn basic logic first, and then you can explore these alternatives.\r\n\r\nThis points us to an important feature, and perhaps a mystery, of logic. \u00a0In part, what a logical language shows us is the consequences of our assumptions. \u00a0That might sound trivial, but, in fact, it is anything but. \u00a0From very simple assumptions, we will discover new, and ultimately shocking, facts. \u00a0So, if someone wants to study a logical language where we reject the principle of bivalence, they can do so. The difference between what they are doing, and what we will do in the following chapters, is that they will discover the consequences of rejecting the principle of bivalence, whereas we will discover the consequences of adhering to it. \u00a0In either case, it would be wise to learn traditional logic first, before attempting to study or develop an alternative logic.\r\n\r\nWe should note at this point that we are not going to try to explain what \u201ctrue\u201d and \u201cfalse\u201d mean, other than saying that \u201cfalse\u201d means <span class=\"em\">not true.<\/span>\u00a0 When we add something to our language without explaining its meaning, we call it a \u201cprimitive\u201d<span class=\"em\">.<\/span>\u00a0 Philosophers have done much to try to understand what truth is, but it remains quite difficult to define truth in any way that is not controversial. Fortunately, taking <span class=\"em\">true<\/span>\u00a0as a primitive will not get us into trouble, and it appears unlikely to make logic mysterious. \u00a0We all have some grasp of what \u201ctrue\u201d means, and this grasp will be sufficient for our development of the propositional logic.\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>Why is the statement \"This sentence is false\" a unique statement in logic?<\/li>\r\n \t<li>Does logic allow for more than two truth values? Why or why not?<\/li>\r\n \t<li>What is the idea behind a \"primitive\" statement?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>1.3 Atomic sentences<\/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 some examples of sentences that are clearly true or false?<\/li>\r\n \t<li>Why are some sentences more fundamental than others?<\/li>\r\n \t<li>Please explain the logic of the word \"atom.\"<\/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>Atomic Sentence\u00a0<\/strong>- a sentence that can have no parts that are sentences.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nOur language will be concerned with declarative sentences, sentences that are either true or false, never both, and never neither. \u00a0Here are some example sentences.\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">2+2=4.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little<strong>*<\/strong> is tall.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If Obama wins the election, then Obama will be President.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If it rains tomorrow, then the picnic will be canceled.<\/p>\r\nThese are all declarative sentences. \u00a0These all appear to satisfy our principle of bivalence. \u00a0But they differ in important ways. \u00a0The first two sentences do not have sentences as parts.\u00a0 For example, try to break up the first sentence. \u00a0\u201c2+2\u201d is a function. \u00a0\u201c4\u201d is a name. \u00a0\u201c=4\u201d is a meaningless fragment, as is \u201c2+\u201d. \u00a0Only the whole expression, \u201c2+2=4\u201d, is a sentence with a truth value. \u00a0The second sentence is similar in this regard. \u00a0\u201cMalcolm Little\u201d is a name. \u00a0\u201cis tall\u201d is an adjective phrase (we will discover later that logicians call this a \u201cpredicate\u201d). \u00a0\u201cMalcolm Little is\u201d or \u201cis tall\u201d are fragments, they have no truth value.[footnote]There is a complex issue here that we will discuss later. \u00a0But, in brief: \u00a0\u201cis\u201d is ambiguous; it has several meanings. \u00a0\u201cMalcolm Little is\u201d is a sentence if it is meant to assert the existence of Malcolm Little. \u00a0The \u201cis\u201d that appears in the sentence, \u201cMalcolm Little is tall\u201d, however, is what we call the \u201c\u2018is\u2019 of predication\u201d.\u00a0 In that sentence, \u201cis\u201d is used to assert that a property is had by Malcolm Little (the property of being tall); and here \u201cis tall\u201d is what we are calling a \u201cpredicate\u201d.\u00a0 So, the \u201cis\u201d of predication has no clear meaning when appearing without the rest of the predicate; it does not assert existence.[\/footnote]<sup class=\"super\">\u00a0<\/sup>Only \u201cMalcolm Little is tall\u201d is a complete sentence.\r\n\r\nThe first two example sentences above are of a kind we call \u201c<strong>atomic sentences<\/strong>\u201d. \u00a0The word \u201catom\u201d comes from the ancient Greek word \u201catomos\u201d, meaning <span class=\"em\">cannot be cut.<\/span>\u00a0 When the ancient Greeks reasoned about matter, for example, some of them believed that if you took some substance, say a rock, and cut it into pieces, then cut the pieces into pieces, and so on, eventually you would get to something that could not be cut. \u00a0This would be the smallest possible thing. \u00a0(The fact that we now talk of having \u201csplit the atom\u201d just goes to show that we changed the meaning of the word \u201catom\u201d. \u00a0We came to use it as a name for a particular kind of thing, which then turned out to have parts, such as electrons, protons, and neutrons.) In logic, the idea of an atomic sentence is of a sentence that can have no parts that are sentences. In other words, atomic sentences are like LEGO bricks\u2014they build bigger ideas but stand alone.\r\n\r\nIn reasoning about these atomic sentences, we could continue to use English. \u00a0But for reasons that become clear as we proceed, there are many advantages to coming up with our own way of writing our sentences. \u00a0It is traditional in logic to use upper case letters from <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>on (<strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>, <strong><span class=\"strong\">R<\/span><\/strong>, <strong><span class=\"strong\">S<\/span><\/strong>\u2026.) to stand for atomic sentences. \u00a0Thus, instead of writing\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little is tall.<\/p>\r\nWe could write\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\r\nIf we want to know how to translate <span class=\"strong\">P<\/span>\u00a0to English, we can provide a translation key. \u00a0Similarly, instead of writing\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little is a great orator.<\/p>\r\nWe could write\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nAnd so on. \u00a0Of course, written in this way, all we can see about such a sentence is that it is a sentence, and that perhaps <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are different sentences. \u00a0But for now, these will be sufficient.\r\n\r\nNote that not all sentences are atomic. \u00a0The third sentence in our four examples above contains parts that are sentences. \u00a0It contains the atomic sentence, \u201cObama wins the election\u201d and also the atomic sentence, \u201cObama will be President\u201d. \u00a0We could represent this whole sentence with a single letter. \u00a0That is, we could let\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If Obama wins the election, Obama will be president.<\/p>\r\nbe represented in our logical language by\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong><\/p>\r\nHowever, this would have the disadvantage that it would hide some of the sentences that are inside this sentence, and also it would hide their relationship. \u00a0Our language would tell us more if we could capture the relation between the parts of this sentence, instead of hiding them. \u00a0We will do this in chapter 2.\r\n\r\n<strong>*<\/strong>Malcolm Little, later known as Malcolm X, was a civil rights leader\u2014his story adds depth to discussions of truth and belief.\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>Why are some sentences atomic and some not?<\/li>\r\n \t<li>What do upper case letters (<strong>P<\/strong>, <strong>Q<\/strong>,\u00a0<strong>R<\/strong>, <strong>S<\/strong>, ...) represent in the language of logic?<\/li>\r\n \t<li>Is it logical to use upper case letters to represent whole sentences? Please explain.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>1.4 Syntax and semantics<\/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 do we examine the shape of a sentence?<\/li>\r\n \t<li>What makes a sentence clearly true or false?<\/li>\r\n \t<li>How do we communicate the veracity of a propositional statement?<\/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>Syntax<\/strong> - the \"shape\" of an expression in a language.<\/li>\r\n \t<li><strong>Semantics\u00a0<\/strong>- the meaning of an expression in a language.<\/li>\r\n \t<li><strong>Truth Table\u00a0<\/strong>- describes the conditions in which a sentence is true or false in a table format.<\/li>\r\n \t<li><strong>Metalangauge<\/strong> - literally the \"after language,\" but which we take to mean our language about our language.<\/li>\r\n \t<li><strong>Object Language\u00a0<\/strong>- the particular propositional logic that we create.<\/li>\r\n \t<li><strong>Contingent Sentence\u00a0<\/strong>- a sentence that can be either true or false.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><\/div>\r\nAn important and useful principle for understanding a language is the difference between syntax and semantics. \u00a0\u201c<strong>Syntax<\/strong>\u201d refers to the \u201cshape\u201d of an expression in our language. \u00a0It does not concern itself with what the elements of the language mean, but just specifies how they can be written out.\r\n\r\nWe can make a similar distinction (though not exactly the same) in a natural language. \u00a0This expression in English has an uncertain meaning, but it has the right \u201cshape\u201d to be a sentence:\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Colorless green ideas sleep furiously.<\/p>\r\nIn other words, in English, this sentence is syntactically correct, although it may express some kind of meaning error.\r\n\r\nAn expression made with the parts of our language must have correct syntax in order for it to be a sentence. \u00a0Sometimes, we also call an expression with the right syntactic form a \u201cwell-formed formula\u201d.\r\n\r\nWe contrast syntax with semantics. \u00a0\u201c<strong>Semantics<\/strong>\u201d refers to the meaning of an expression of our language. \u00a0Semantics depends upon the relation of that element of the language to something else. \u00a0For example, the truth value of the sentence, \u201cThe Earth has one moon\u201d depends not upon the English language, but upon something exterior to the language. \u00a0Since the self-standing elements of our propositional logic are sentences, and the most important property of these is their truth value, the only semantic feature of sentences that will concern us in our propositional logic is their truth value.\r\n\r\nWhenever we introduce a new element into the propositional logic, we will specify its syntax and its semantics. In the propositional logic, the syntax is generally trivial, but the <strong>semantics<\/strong> is less so.\u00a0 We have so far introduced atomic sentences. \u00a0The syntax for an atomic sentence is trivial. \u00a0If <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is an atomic sentence, then it is syntactically correct to write down\r\n<p class=\"marg-left indent\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\r\nBy saying that this is syntactically correct, we are not saying that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0Rather, we are saying that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is a sentence.\r\n\r\nIf semantics in the propositional logic concerns only truth value, then we know that there are only two possible semantic values for <strong><span class=\"strong\">P<\/span><\/strong>; it can be either true or false. \u00a0We have a way of writing this that will later prove helpful. \u00a0It is called a \u201c<strong>truth table<\/strong>\u201d. \u00a0For an atomic sentence, the truth table is trivial, but when we look at other kinds of sentences their truth tables will be more complex.\r\n\r\nThe idea of a truth table is to describe the conditions in which a sentence is true or false. \u00a0We do this by identifying all the atomic sentences that compose that sentence. \u00a0Then, on the left side, we stipulate all the possible truth values of these atomic sentences and write these out. \u00a0On the right side, we then identify under what conditions the sentence (that is composed of the other atomic sentences) is true or false.\r\n\r\nThe idea is that the sentence on the right is dependent on the sentence(s) on the left. \u00a0So the truth table is filled in like this:\r\n<table class=\"grid aligncenter\" style=\"height: 102px; width: 400px;\" width=\"563\">\r\n<tbody>\r\n<tr style=\"height: 31px;\">\r\n<td class=\"border\" style=\"height: 31px; width: 320.633px;\" colspan=\"1\" rowspan=\"1\">Atomic sentence(s) that compose the dependent sentence on the right<\/td>\r\n<td class=\"border\" style=\"height: 31px; width: 339.933px;\" colspan=\"1\" rowspan=\"1\">Dependent sentence composed of the atomic sentences on the left<\/td>\r\n<\/tr>\r\n<tr style=\"height: 46px;\">\r\n<td class=\"border\" style=\"height: 46px; width: 320.633px;\" colspan=\"1\" rowspan=\"1\">All possible combinations of truth values of the composing atomic sentences<\/td>\r\n<td class=\"border\" style=\"height: 46px; width: 339.933px;\" colspan=\"1\" rowspan=\"1\">Resulting truth values for each possible combination of truth values of the composing atomic sentences<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe stipulate all the possible truth values on the bottom left because the propositional logic alone will not determine whether an atomic sentence is true or false; thus, we will simply have to consider both possibilities. \u00a0Note that there are many ways that an atomic sentence can be true, and there are many ways that it can be false. \u00a0For example, the sentence, \u201cJeremiah is American\u201d might be true if Jeremiah was born in New York, in Texas, in Ohio, and so on. \u00a0The sentence might be false because Jeremiah was born to Italian parents in Italy, to French parents in France, and so on. \u00a0So, we group all these cases together into two kinds of cases.\r\n\r\nThese are two rows of the truth table for an atomic sentence. \u00a0Each row of the truth table represents a kind of way that the world could be. \u00a0So here is the left side of a truth table with only a single atomic sentence, <strong><span class=\"strong\">P<\/span>. \u00a0<\/strong>We will write \u201c<strong><span class=\"em strong\">T<\/span><\/strong>\u201d for <span class=\"em\">true<\/span>\u00a0and \u201c<strong><span class=\"em strong\">F<\/span><\/strong>\u201d for <span class=\"em\">false.<\/span>\r\n<table class=\"grid\" style=\"height: 45px; width: 365px;\" width=\"75\">\r\n<thead>\r\n<tr class=\"border\" style=\"height: 15px;\">\r\n<td class=\"border\" style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\r\n<td style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThere are only two relevant kinds of ways that the world can be, when we are considering the semantics of an atomic sentence. \u00a0The world can be one of the many conditions such that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, or it can be one of the many conditions such that\u00a0<strong>P<\/strong> is false.\r\n\r\nTo complete the truth table, we place the dependent sentence on the top right side, and describe its truth value in relation to the truth value of its parts. \u00a0We want to identify the semantics of <strong><span class=\"strong\">P<\/span><\/strong>, which has only one part, <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0The truth table thus has the final form:\r\n<table class=\"grid\" style=\"height: 63px;\" width=\"75\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><strong>\r\n<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis truth table tells us the meaning of <strong><span class=\"strong\">P<\/span><\/strong>, as far as our propositional logic can tell us about it. \u00a0Thus, it gives us the complete semantics for <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0(As we will see later, truth tables have three uses: \u00a0to provide the semantics for a kind of sentence; to determine under what conditions a complex sentence is true or false; and to determine if an argument is good. \u00a0Here we are describing only this first use.)\r\n\r\nIn this truth table, the first row combined together all the kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0In the second column we see that for all of these kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, unsurprisingly, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0The second row combines together all the kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0In those, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0As we noted above, in the case of an atomic sentence, the truth table is trivial. Nonetheless, the basic concept is very useful, as we will begin to see in the next chapter.\r\n\r\nOne last tool will be helpful to us. \u00a0Strictly speaking, what we have done above is give the syntax and semantics for a particular atomic sentence, <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0We need a way to make general claims about all the sentences of our language, and then give the syntax and semantics for any atomic sentences. \u00a0We do this using variables, and here we will use Greek letters for those variables, such as <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span><\/strong>. \u00a0Things said using these variables is called our \u201c<strong>metalanguage<\/strong>\u201d<span class=\"em\">,<\/span>\u00a0which means literally the <span class=\"em\">after language<\/span>, but which we take to mean, <span class=\"em\">our language about our language.<\/span>\u00a0 The particular propositional logic that we create is called our \u201c<strong>object<\/strong> <strong>language<\/strong>\u201d. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are sentences of our object language. \u00a0<strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are elements of our metalanguage. \u00a0To specify now the syntax of atomic sentences (that is, of all atomic sentences) we can say: \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is an atomic sentence, then\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\r\nis a sentence. \u00a0This tells us that simply writing <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>down (whatever atomic sentence it may be), as we have just done, is to write down something that is syntactically correct.\r\n\r\nTo specify now the semantics of atomic sentences (that is, of all atomic sentences) we can say: \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is an atomic sentence, then the semantics of <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is given by\r\n<table class=\"grid\" style=\"height: 58px;\" width=\"75\">\r\n<tbody>\r\n<tr class=\"border-bottom\" style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\r\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote an important and subtle point. \u00a0The atomic sentences of our propositional logic will be what we call \u201c<strong>contingent<\/strong>\u201d <strong>sentences.<\/strong> \u00a0A contingent sentence can be either true or false. \u00a0We will see later that some complex sentences of our propositional logic must be true, and some complex sentences of our propositional logic must be false. \u00a0But for the propositional logic, every atomic sentence is (as far as we can tell using the propositional logic alone) contingent. \u00a0This observation matters because it greatly helps to clarify where logic begins, and where the methods of another discipline ends. \u00a0For example, suppose we have an atomic sentence like:\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Force is equal to mass times acceleration.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Igneous rocks formed under pressure.<\/p>\r\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Germany inflated its currency in 1923 in order to reduce its reparations debt.<\/p>\r\nLogic cannot tell us whether these sentences are true or false. \u00a0We will turn to physicists, and use their methods, to evaluate the first claim. \u00a0We will turn to geologists, and use their methods, to evaluate the second claim. \u00a0We will turn to historians, and use their methods, to evaluate the third claim. \u00a0But the logician can tell the physicist, geologist, and historian what follows from their claims.\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>Why is correct syntax fundamental for whole sentences?<\/li>\r\n \t<li>What makes semantics different than syntax?<\/li>\r\n \t<li>What is the only semantical feature of sentences of importance in logic?<\/li>\r\n \t<li>How can a statement be eligible for a truth table?<\/li>\r\n \t<li>What makes the truth table \"trivial\" for an atomic sentence?<\/li>\r\n \t<li>Why is the issue of contingency important for every atomic sentence?<\/li>\r\n \t<li><strong>Critical Thinking Task: <\/strong>Use a book, song, or movie to symbolize your understanding of the difference between syntax and semantics.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>1.5 Problems<\/h2>\r\n<ol class=\"lst-kix_list_19-0 start\" start=\"1\">\r\n \t<li><strong>Critical Thinking Task: <\/strong>Describe some of the situations in your life in which people often contradict themselves.<\/li>\r\n \t<li>Think of a time when a misunderstanding occurred because something wasn\u2019t said clearly. How could logic have helped?<\/li>\r\n \t<li>Describe a decision you made recently. What arguments did you consider, and were they logically sound?<\/li>\r\n \t<li>Analyze this statement: 'In some cultures, being late is considered polite.' Discuss ambiguity and truth value.<\/li>\r\n \t<li>Logic helps evaluate fake news; how do we know if a claim online is true?<\/li>\r\n \t<li>Vagueness arises when the conditions under which a sentence might be true are \u201cfuzzy\u201d. \u00a0That is, in some cases, we cannot identify if the sentence is true or false. \u00a0If we say, \u201cTom is tall\u201d, this sentence is certainly true if Tom is the tallest person in the world, but it is not clear whether it is true if Tom is 185 centimeters tall. \u00a0Identify or create five declarative sentences in English that are vague.<\/li>\r\n \t<li>Ambiguity usually arises when a word or phrase has several distinct possible interpretations. \u00a0In our example above, the word \u201cpen\u201d could mean either a writing implement or a structure to hold a child. \u00a0A sentence that includes \u201cpen\u201d could be ambiguous, in which case it might be true for one interpretation and false for another. \u00a0Identify or create five declarative sentences in English that are ambiguous. \u00a0(This will probably require you to identify a homonym, a word that has more than one meaning but sounds or is written the same. \u00a0If you are stumped, consider slang: \u00a0many slang terms are ambiguous because they redefine existing words. \u00a0For example, in the 1980s, in some communities and contexts, to say something was \u201cbad\u201d meant that it was good; this obviously can create ambiguous sentences.)<\/li>\r\n \t<li>Often we can make a vague sentence precise by defining a specific interpretation of the meaning of an adjective, term, or other element of the language. \u00a0For example, we could make the sentence \u201cTom is tall\u201d precise by specifying one person referred to by \u201cTom\u201d, and also by defining \u201c\u2026is tall\u201d as true of anyone 180 centimeters tall or taller. \u00a0For each of the five vague sentences that you identified or created for problem 1, describe how the interpretation of certain elements of the sentence could make the sentence no longer vague.<\/li>\r\n \t<li>Often we can make an ambiguous sentence precise by specifying which of the possible meanings we intend to use. \u00a0We could make the sentence, \u201cTom is by the pen\u201d unambiguous by specifying which Tom we mean, and also defining \u201cpen\u201d to mean an infant play pen. \u00a0For each of the five ambiguous sentences that you identified or created for problem 2, identify and describe how the interpretation of certain elements of the sentence could make the sentence no longer ambiguous.<\/li>\r\n \t<li>Come up with five examples of your own of English sentences that are not declarative sentences. \u00a0(Examples can include commands, exclamations, and promises.)<\/li>\r\n \t<li>Here are some sentences from literary works and other famous texts. Describe as best you can what the role of the sentence is. For example, the sentence might be a declarative sentence, which aims to describe things; or a question, which aims to solicit information; or a command, which is used to make someone do something; and so on. It is not essential that you have a name for the kind of sentence, but rather can you describe what a speaker would typically intend for such a sentence to do?<\/li>\r\n \t<li>\"Though I should die with thee, yet will I not deny thee.\" (From the King James Bible)\r\n<ol class=\"lst-kix_list_19-0 start\" start=\"1\">\r\n \t<li>\"Get thee to a nunnery.\" (William Shakespeare, <i>Hamlet<\/i>.)<\/li>\r\n \t<li>\"That on the first day of January, in the year of our Lord one thousand eight hundred and sixty-three, all persons held as slaves within any State or designated part of a State, the people whereof shall then be in rebellion against the United States, shall be then, thenceforward, and forever free.\" (From \"The Emancipation Proclamation\".)<\/li>\r\n \t<li>\"Sing, goddess, of the anger of Achilles son of Peleus, that brought countless ills upon the Achaeans.\" (Homer, <i>The Illiad<\/i>.)<\/li>\r\n \t<li>\"Since the heavens grant that you recognize me, hold your tongue, and do not say a word about who I am to any one else in the house, for if you do, and if heaven grants me to take the lives of these suitors, I will not spare you, though you are my own nurse, when I am killing the other women.\" (Homer, <i>The Odyssey<\/i>.)<\/li>\r\n \t<li>\"Tyger, tyger, burning bright,\r\nIn the forests of the night,\r\nWhat immortal hand or eye,\r\ncould frame thy fearful symmetry?\" (William Blake, <i>The Tyger<\/i>.)<\/li>\r\n \t<li>\"As wicked dew as e'er my mother brush'd\r\nWith raven's feather from unwholesome fen\r\nDrop on you both! A south-west blow on ye\r\nAnd blister you all o'er!\" (William Shakespeare, <i>The Tempest<\/i>.)<\/li>\r\n \t<li>\"For he to-day that sheds his blood with me\r\nShall be my brother.\" (William Shakespeare, <i>Henry V<\/i>.)<\/li>\r\n \t<li>\"Astonishing, Pip!\" (Charles Dickens, <i>Great Expectations<\/i>.)<\/li>\r\n \t<li>\"Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.\" (The Constitution of the United States.)<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>","rendered":"<h2>1.1 Starting with sentences<\/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 is your prior knowledge related to logic?<\/li>\n<li>How can a sentence be declared as &#8220;whole&#8221;?<\/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>Propositional Logic<\/strong> &#8211; the reasoning used to create whole sentences known as propositions.<\/li>\n<li><strong>Declarative Sentence<\/strong> &#8211; a precise set of language that can be either true or false.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>We begin the study of logic by building a precise logical language. \u00a0This will allow us to do at least two things: \u00a0first, to say some things more precisely than we otherwise would be able to do; second, to study reasoning. \u00a0We will use a natural language\u2014English\u2014as our guide, but our logical language will be far simpler, far weaker, but more rigorous than English.<\/p>\n<p>We must decide where to start. \u00a0We could pick just about any part of English to try to emulate: \u00a0names, adjectives, prepositions, general nouns, and so on. \u00a0But it is traditional, and as we will see, quite handy, to begin with whole sentences. \u00a0For this reason, the first language we will develop is called \u201cthe <strong>propositional logic<\/strong>\u201d.<span class=\"em\">\u00a0 <\/span>It is also sometimes called \u201cthe sentential logic\u201d or even \u201cthe sentential calculus\u201d. These all mean the same thing: \u00a0the logic of sentences. \u00a0In this <strong>propositional logic<\/strong>, the smallest independent parts of the language are sentences (throughout this book, I will assume that sentences and propositions are the same thing in our logic, and I will use the terms \u201csentence\u201d and \u201cproposition\u201d interchangeably).<\/p>\n<p>There are of course many kinds of sentences. \u00a0To take examples from our natural language, these include:<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">What time is it?<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Open the window.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Damn you!<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">I promise to pay you back.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">It rained in City Park on June 26, 2015.<\/p>\n<p>We could multiply such examples. \u00a0Sentences in English can be used to ask questions, give commands, curse or insult, form contracts, and express emotions. \u00a0But, the last example above is of special interest because it aims to describe the world. \u00a0Such sentences, which are sometimes called \u201cdeclarative sentences\u201d, will be our model sentences for our logical language. \u00a0We know a <strong>declarative sentence<\/strong> when we encounter it because it can be either true or false.<\/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>Please explain the logic of whole sentences.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>1.2 Precision in sentences<\/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 makes a sentence clearly true or false?<\/li>\n<li>Is it possible for a declarative sentence to be both true and false?<\/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>Truth Value <\/strong>&#8211; a clear assessment of whether a statement is true or false in reality.<\/li>\n<li><strong>Principle of Bivalence\u00a0<\/strong>&#8211; each sentence of our language must be either true or false, not both, not neither.<\/li>\n<li><strong>Principle of Non-Contradiction\u00a0<\/strong>&#8211; a claim must be asserted or denied.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>We want our logic of declarative sentences to be precise. \u00a0But what does this mean? \u00a0We can help clarify how we might pursue this by looking at sentences in a natural language that are perplexing, apparently because they are not precise. \u00a0Here are three.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Chris is kind of tall.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">When Jasmine had a baby, her mother gave her a pen.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">This sentence is false.<\/p>\n<p>We have already observed that an important feature of our declarative sentences is that they can be true or false. \u00a0We call this the \u201ctruth value\u201d of the sentence. \u00a0These three sentences are perplexing because their truth values are unclear. \u00a0The first sentence is vague, it is not clear under what conditions it would be true, and under what conditions it would be false. \u00a0If Chris is six feet tall, is he kind of tall? \u00a0There is no clear answer. \u00a0The second sentence is ambiguous<span class=\"em\">.<\/span>\u00a0 If \u201cpen\u201d means writing implement, and Jasmine\u2019s mother bought a playpen for the baby, then the sentence is false. \u00a0But until we know what \u201cpen\u201d means in this sentence, we cannot tell if the sentence is true.<\/p>\n<p>The third sentence is strange. \u00a0Many logicians have spent many years studying this sentence, which is traditionally called \u201cthe Liar\u201d. \u00a0It is related to an old paradox about a Cretan who said, \u201cAll Cretans are liars\u201d. \u00a0The strange thing about the Liar is that its truth value seems to explode. \u00a0If it is true, then it is false. \u00a0If it is false, then it is true. \u00a0Some philosophers think this sentence is, therefore, neither true nor false; some philosophers think it is both true and false. \u00a0In either case, it is confusing. \u00a0How could a sentence that looks like a declarative sentence have both or no truth value?<\/p>\n<p>Since ancient times, philosophers have believed that we will deceive ourselves, and come to believe untruths, if we do not accept a principle sometimes called \u201c<strong>the principle of <\/strong><strong>bivalence<\/strong>\u201d, or a related principle called \u201c<strong>the principle of non-contradiction<\/strong>\u201d. \u00a0Bivalence is the view that there are only two truth values (true and false) and that they exclude each other. \u00a0The principle of non-contradiction states that you have made a mistake if you both assert and deny a claim. \u00a0One or the other of these principles seems to be violated by the Liar.<\/p>\n<p>We can take these observations for our guide: \u00a0we want our language to have no vagueness and no ambiguity. \u00a0In our propositional logic, this means we want it to be the case that each sentence is either true or false. \u00a0It will not be kind of true, or partially true, or true from one perspective and not true from another. \u00a0We also want to avoid things like the Liar. \u00a0We do not need to agree on whether the Liar is both true and false, or neither true nor false. \u00a0Either would be unfortunate. \u00a0So, we will specify that our sentences have neither vice.<\/p>\n<p>We can formulate our own revised version of the principle of bivalence, which states that:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>Principle of Bivalence<\/strong>: \u00a0Each sentence of our language must be either true or false, not both, not neither.<\/p>\n<p>This requirement may sound trivial, but in fact it constrains what we do from now on in interesting and even surprising ways. \u00a0Even as we build more complex logical languages later, this principle will be fundamental.<\/p>\n<p>Some readers may be thinking: \u00a0what if I reject bivalence, or the <strong>principle of non-contradiction<\/strong>? \u00a0There is a long line of philosophers who would like to argue with you, and propose that either move would be a mistake, and perhaps even incoherent. \u00a0Set those arguments aside. \u00a0If you have doubts about bivalence, or the principle of non-contradiction, stick with logic. That is because we could develop a logic in which there were more than two truth values. \u00a0Logics have been created and studied in which we allow for three truth values, or continuous truth values, or stranger possibilities. \u00a0The issue for us is that we must start somewhere, and the principle of bivalence is an intuitive way and\u2014it would seem\u2014the simplest way to start with respect to truth values. \u00a0Learn basic logic first, and then you can explore these alternatives.<\/p>\n<p>This points us to an important feature, and perhaps a mystery, of logic. \u00a0In part, what a logical language shows us is the consequences of our assumptions. \u00a0That might sound trivial, but, in fact, it is anything but. \u00a0From very simple assumptions, we will discover new, and ultimately shocking, facts. \u00a0So, if someone wants to study a logical language where we reject the principle of bivalence, they can do so. The difference between what they are doing, and what we will do in the following chapters, is that they will discover the consequences of rejecting the principle of bivalence, whereas we will discover the consequences of adhering to it. \u00a0In either case, it would be wise to learn traditional logic first, before attempting to study or develop an alternative logic.<\/p>\n<p>We should note at this point that we are not going to try to explain what \u201ctrue\u201d and \u201cfalse\u201d mean, other than saying that \u201cfalse\u201d means <span class=\"em\">not true.<\/span>\u00a0 When we add something to our language without explaining its meaning, we call it a \u201cprimitive\u201d<span class=\"em\">.<\/span>\u00a0 Philosophers have done much to try to understand what truth is, but it remains quite difficult to define truth in any way that is not controversial. Fortunately, taking <span class=\"em\">true<\/span>\u00a0as a primitive will not get us into trouble, and it appears unlikely to make logic mysterious. \u00a0We all have some grasp of what \u201ctrue\u201d means, and this grasp will be sufficient for our development of the propositional logic.<\/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>Why is the statement &#8220;This sentence is false&#8221; a unique statement in logic?<\/li>\n<li>Does logic allow for more than two truth values? Why or why not?<\/li>\n<li>What is the idea behind a &#8220;primitive&#8221; statement?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>1.3 Atomic sentences<\/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 some examples of sentences that are clearly true or false?<\/li>\n<li>Why are some sentences more fundamental than others?<\/li>\n<li>Please explain the logic of the word &#8220;atom.&#8221;<\/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>Atomic Sentence\u00a0<\/strong>&#8211; a sentence that can have no parts that are sentences.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Our language will be concerned with declarative sentences, sentences that are either true or false, never both, and never neither. \u00a0Here are some example sentences.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">2+2=4.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little<strong>*<\/strong> is tall.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If Obama wins the election, then Obama will be President.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If it rains tomorrow, then the picnic will be canceled.<\/p>\n<p>These are all declarative sentences. \u00a0These all appear to satisfy our principle of bivalence. \u00a0But they differ in important ways. \u00a0The first two sentences do not have sentences as parts.\u00a0 For example, try to break up the first sentence. \u00a0\u201c2+2\u201d is a function. \u00a0\u201c4\u201d is a name. \u00a0\u201c=4\u201d is a meaningless fragment, as is \u201c2+\u201d. \u00a0Only the whole expression, \u201c2+2=4\u201d, is a sentence with a truth value. \u00a0The second sentence is similar in this regard. \u00a0\u201cMalcolm Little\u201d is a name. \u00a0\u201cis tall\u201d is an adjective phrase (we will discover later that logicians call this a \u201cpredicate\u201d). \u00a0\u201cMalcolm Little is\u201d or \u201cis tall\u201d are fragments, they have no truth value.<a class=\"footnote\" title=\"There is a complex issue here that we will discuss later. \u00a0But, in brief: \u00a0\u201cis\u201d is ambiguous; it has several meanings. \u00a0\u201cMalcolm Little is\u201d is a sentence if it is meant to assert the existence of Malcolm Little. \u00a0The \u201cis\u201d that appears in the sentence, \u201cMalcolm Little is tall\u201d, however, is what we call the \u201c\u2018is\u2019 of predication\u201d.\u00a0 In that sentence, \u201cis\u201d is used to assert that a property is had by Malcolm Little (the property of being tall); and here \u201cis tall\u201d is what we are calling a \u201cpredicate\u201d.\u00a0 So, the \u201cis\u201d of predication has no clear meaning when appearing without the rest of the predicate; it does not assert existence.\" id=\"return-footnote-29-1\" href=\"#footnote-29-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><sup class=\"super\">\u00a0<\/sup>Only \u201cMalcolm Little is tall\u201d is a complete sentence.<\/p>\n<p>The first two example sentences above are of a kind we call \u201c<strong>atomic sentences<\/strong>\u201d. \u00a0The word \u201catom\u201d comes from the ancient Greek word \u201catomos\u201d, meaning <span class=\"em\">cannot be cut.<\/span>\u00a0 When the ancient Greeks reasoned about matter, for example, some of them believed that if you took some substance, say a rock, and cut it into pieces, then cut the pieces into pieces, and so on, eventually you would get to something that could not be cut. \u00a0This would be the smallest possible thing. \u00a0(The fact that we now talk of having \u201csplit the atom\u201d just goes to show that we changed the meaning of the word \u201catom\u201d. \u00a0We came to use it as a name for a particular kind of thing, which then turned out to have parts, such as electrons, protons, and neutrons.) In logic, the idea of an atomic sentence is of a sentence that can have no parts that are sentences. In other words, atomic sentences are like LEGO bricks\u2014they build bigger ideas but stand alone.<\/p>\n<p>In reasoning about these atomic sentences, we could continue to use English. \u00a0But for reasons that become clear as we proceed, there are many advantages to coming up with our own way of writing our sentences. \u00a0It is traditional in logic to use upper case letters from <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>on (<strong><span class=\"strong\">P<\/span><\/strong>, <strong><span class=\"strong\">Q<\/span><\/strong>, <strong><span class=\"strong\">R<\/span><\/strong>, <strong><span class=\"strong\">S<\/span><\/strong>\u2026.) to stand for atomic sentences. \u00a0Thus, instead of writing<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little is tall.<\/p>\n<p>We could write<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\n<p>If we want to know how to translate <span class=\"strong\">P<\/span>\u00a0to English, we can provide a translation key. \u00a0Similarly, instead of writing<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Malcolm Little is a great orator.<\/p>\n<p>We could write<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>And so on. \u00a0Of course, written in this way, all we can see about such a sentence is that it is a sentence, and that perhaps <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are different sentences. \u00a0But for now, these will be sufficient.<\/p>\n<p>Note that not all sentences are atomic. \u00a0The third sentence in our four examples above contains parts that are sentences. \u00a0It contains the atomic sentence, \u201cObama wins the election\u201d and also the atomic sentence, \u201cObama will be President\u201d. \u00a0We could represent this whole sentence with a single letter. \u00a0That is, we could let<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">If Obama wins the election, Obama will be president.<\/p>\n<p>be represented in our logical language by<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong><\/p>\n<p>However, this would have the disadvantage that it would hide some of the sentences that are inside this sentence, and also it would hide their relationship. \u00a0Our language would tell us more if we could capture the relation between the parts of this sentence, instead of hiding them. \u00a0We will do this in chapter 2.<\/p>\n<p><strong>*<\/strong>Malcolm Little, later known as Malcolm X, was a civil rights leader\u2014his story adds depth to discussions of truth and belief.<\/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>Why are some sentences atomic and some not?<\/li>\n<li>What do upper case letters (<strong>P<\/strong>, <strong>Q<\/strong>,\u00a0<strong>R<\/strong>, <strong>S<\/strong>, &#8230;) represent in the language of logic?<\/li>\n<li>Is it logical to use upper case letters to represent whole sentences? Please explain.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>1.4 Syntax and semantics<\/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 do we examine the shape of a sentence?<\/li>\n<li>What makes a sentence clearly true or false?<\/li>\n<li>How do we communicate the veracity of a propositional statement?<\/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>Syntax<\/strong> &#8211; the &#8220;shape&#8221; of an expression in a language.<\/li>\n<li><strong>Semantics\u00a0<\/strong>&#8211; the meaning of an expression in a language.<\/li>\n<li><strong>Truth Table\u00a0<\/strong>&#8211; describes the conditions in which a sentence is true or false in a table format.<\/li>\n<li><strong>Metalangauge<\/strong> &#8211; literally the &#8220;after language,&#8221; but which we take to mean our language about our language.<\/li>\n<li><strong>Object Language\u00a0<\/strong>&#8211; the particular propositional logic that we create.<\/li>\n<li><strong>Contingent Sentence\u00a0<\/strong>&#8211; a sentence that can be either true or false.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\"><\/div>\n<p>An important and useful principle for understanding a language is the difference between syntax and semantics. \u00a0\u201c<strong>Syntax<\/strong>\u201d refers to the \u201cshape\u201d of an expression in our language. \u00a0It does not concern itself with what the elements of the language mean, but just specifies how they can be written out.<\/p>\n<p>We can make a similar distinction (though not exactly the same) in a natural language. \u00a0This expression in English has an uncertain meaning, but it has the right \u201cshape\u201d to be a sentence:<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Colorless green ideas sleep furiously.<\/p>\n<p>In other words, in English, this sentence is syntactically correct, although it may express some kind of meaning error.<\/p>\n<p>An expression made with the parts of our language must have correct syntax in order for it to be a sentence. \u00a0Sometimes, we also call an expression with the right syntactic form a \u201cwell-formed formula\u201d.<\/p>\n<p>We contrast syntax with semantics. \u00a0\u201c<strong>Semantics<\/strong>\u201d refers to the meaning of an expression of our language. \u00a0Semantics depends upon the relation of that element of the language to something else. \u00a0For example, the truth value of the sentence, \u201cThe Earth has one moon\u201d depends not upon the English language, but upon something exterior to the language. \u00a0Since the self-standing elements of our propositional logic are sentences, and the most important property of these is their truth value, the only semantic feature of sentences that will concern us in our propositional logic is their truth value.<\/p>\n<p>Whenever we introduce a new element into the propositional logic, we will specify its syntax and its semantics. In the propositional logic, the syntax is generally trivial, but the <strong>semantics<\/strong> is less so.\u00a0 We have so far introduced atomic sentences. \u00a0The syntax for an atomic sentence is trivial. \u00a0If <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is an atomic sentence, then it is syntactically correct to write down<\/p>\n<p class=\"marg-left indent\" style=\"text-align: left; padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong><\/p>\n<p>By saying that this is syntactically correct, we are not saying that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0Rather, we are saying that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is a sentence.<\/p>\n<p>If semantics in the propositional logic concerns only truth value, then we know that there are only two possible semantic values for <strong><span class=\"strong\">P<\/span><\/strong>; it can be either true or false. \u00a0We have a way of writing this that will later prove helpful. \u00a0It is called a \u201c<strong>truth table<\/strong>\u201d. \u00a0For an atomic sentence, the truth table is trivial, but when we look at other kinds of sentences their truth tables will be more complex.<\/p>\n<p>The idea of a truth table is to describe the conditions in which a sentence is true or false. \u00a0We do this by identifying all the atomic sentences that compose that sentence. \u00a0Then, on the left side, we stipulate all the possible truth values of these atomic sentences and write these out. \u00a0On the right side, we then identify under what conditions the sentence (that is composed of the other atomic sentences) is true or false.<\/p>\n<p>The idea is that the sentence on the right is dependent on the sentence(s) on the left. \u00a0So the truth table is filled in like this:<\/p>\n<table class=\"grid aligncenter\" style=\"height: 102px; width: 400px; width: 563px;\">\n<tbody>\n<tr style=\"height: 31px;\">\n<td class=\"border\" style=\"height: 31px; width: 320.633px;\" colspan=\"1\" rowspan=\"1\">Atomic sentence(s) that compose the dependent sentence on the right<\/td>\n<td class=\"border\" style=\"height: 31px; width: 339.933px;\" colspan=\"1\" rowspan=\"1\">Dependent sentence composed of the atomic sentences on the left<\/td>\n<\/tr>\n<tr style=\"height: 46px;\">\n<td class=\"border\" style=\"height: 46px; width: 320.633px;\" colspan=\"1\" rowspan=\"1\">All possible combinations of truth values of the composing atomic sentences<\/td>\n<td class=\"border\" style=\"height: 46px; width: 339.933px;\" colspan=\"1\" rowspan=\"1\">Resulting truth values for each possible combination of truth values of the composing atomic sentences<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We stipulate all the possible truth values on the bottom left because the propositional logic alone will not determine whether an atomic sentence is true or false; thus, we will simply have to consider both possibilities. \u00a0Note that there are many ways that an atomic sentence can be true, and there are many ways that it can be false. \u00a0For example, the sentence, \u201cJeremiah is American\u201d might be true if Jeremiah was born in New York, in Texas, in Ohio, and so on. \u00a0The sentence might be false because Jeremiah was born to Italian parents in Italy, to French parents in France, and so on. \u00a0So, we group all these cases together into two kinds of cases.<\/p>\n<p>These are two rows of the truth table for an atomic sentence. \u00a0Each row of the truth table represents a kind of way that the world could be. \u00a0So here is the left side of a truth table with only a single atomic sentence, <strong><span class=\"strong\">P<\/span>. \u00a0<\/strong>We will write \u201c<strong><span class=\"em strong\">T<\/span><\/strong>\u201d for <span class=\"em\">true<\/span>\u00a0and \u201c<strong><span class=\"em strong\">F<\/span><\/strong>\u201d for <span class=\"em\">false.<\/span><\/p>\n<table class=\"grid\" style=\"height: 45px; width: 365px; width: 75px;\">\n<thead>\n<tr class=\"border\" style=\"height: 15px;\">\n<td class=\"border\" style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\n<td style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<\/thead>\n<tbody>\n<tr style=\"height: 15px;\">\n<td style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\n<td class=\"border\" style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"width: 305.267px; height: 15px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\n<td class=\"border\" style=\"width: 355.3px; height: 15px;\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>There are only two relevant kinds of ways that the world can be, when we are considering the semantics of an atomic sentence. \u00a0The world can be one of the many conditions such that <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, or it can be one of the many conditions such that\u00a0<strong>P<\/strong> is false.<\/p>\n<p>To complete the truth table, we place the dependent sentence on the top right side, and describe its truth value in relation to the truth value of its parts. \u00a0We want to identify the semantics of <strong><span class=\"strong\">P<\/span><\/strong>, which has only one part, <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0The truth table thus has the final form:<\/p>\n<table class=\"grid\" style=\"height: 63px; width: 75px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><strong><br \/>\n<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This truth table tells us the meaning of <strong><span class=\"strong\">P<\/span><\/strong>, as far as our propositional logic can tell us about it. \u00a0Thus, it gives us the complete semantics for <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0(As we will see later, truth tables have three uses: \u00a0to provide the semantics for a kind of sentence; to determine under what conditions a complex sentence is true or false; and to determine if an argument is good. \u00a0Here we are describing only this first use.)<\/p>\n<p>In this truth table, the first row combined together all the kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0In the second column we see that for all of these kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, unsurprisingly, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true. \u00a0The second row combines together all the kinds of ways the world could be in which <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0In those, <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0As we noted above, in the case of an atomic sentence, the truth table is trivial. Nonetheless, the basic concept is very useful, as we will begin to see in the next chapter.<\/p>\n<p>One last tool will be helpful to us. \u00a0Strictly speaking, what we have done above is give the syntax and semantics for a particular atomic sentence, <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0We need a way to make general claims about all the sentences of our language, and then give the syntax and semantics for any atomic sentences. \u00a0We do this using variables, and here we will use Greek letters for those variables, such as <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span><\/strong>. \u00a0Things said using these variables is called our \u201c<strong>metalanguage<\/strong>\u201d<span class=\"em\">,<\/span>\u00a0which means literally the <span class=\"em\">after language<\/span>, but which we take to mean, <span class=\"em\">our language about our language.<\/span>\u00a0 The particular propositional logic that we create is called our \u201c<strong>object<\/strong> <strong>language<\/strong>\u201d. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>are sentences of our object language. \u00a0<strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are elements of our metalanguage. \u00a0To specify now the syntax of atomic sentences (that is, of all atomic sentences) we can say: \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is an atomic sentence, then<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\n<p>is a sentence. \u00a0This tells us that simply writing <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>down (whatever atomic sentence it may be), as we have just done, is to write down something that is syntactically correct.<\/p>\n<p>To specify now the semantics of atomic sentences (that is, of all atomic sentences) we can say: \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is an atomic sentence, then the semantics of <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is given by<\/p>\n<table class=\"grid\" style=\"height: 58px; width: 75px;\">\n<tbody>\n<tr class=\"border-bottom\" style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">T<\/span><\/strong><\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td class=\"border-right\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\n<td class=\"border\" style=\"height: 15px; width: 234.833px; text-align: center;\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"em strong\">F<\/span><\/strong><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note an important and subtle point. \u00a0The atomic sentences of our propositional logic will be what we call \u201c<strong>contingent<\/strong>\u201d <strong>sentences.<\/strong> \u00a0A contingent sentence can be either true or false. \u00a0We will see later that some complex sentences of our propositional logic must be true, and some complex sentences of our propositional logic must be false. \u00a0But for the propositional logic, every atomic sentence is (as far as we can tell using the propositional logic alone) contingent. \u00a0This observation matters because it greatly helps to clarify where logic begins, and where the methods of another discipline ends. \u00a0For example, suppose we have an atomic sentence like:<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Force is equal to mass times acceleration.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Igneous rocks formed under pressure.<\/p>\n<p class=\"marg-left\" style=\"text-align: left; padding-left: 120px;\">Germany inflated its currency in 1923 in order to reduce its reparations debt.<\/p>\n<p>Logic cannot tell us whether these sentences are true or false. \u00a0We will turn to physicists, and use their methods, to evaluate the first claim. \u00a0We will turn to geologists, and use their methods, to evaluate the second claim. \u00a0We will turn to historians, and use their methods, to evaluate the third claim. \u00a0But the logician can tell the physicist, geologist, and historian what follows from their claims.<\/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>Why is correct syntax fundamental for whole sentences?<\/li>\n<li>What makes semantics different than syntax?<\/li>\n<li>What is the only semantical feature of sentences of importance in logic?<\/li>\n<li>How can a statement be eligible for a truth table?<\/li>\n<li>What makes the truth table &#8220;trivial&#8221; for an atomic sentence?<\/li>\n<li>Why is the issue of contingency important for every atomic sentence?<\/li>\n<li><strong>Critical Thinking Task: <\/strong>Use a book, song, or movie to symbolize your understanding of the difference between syntax and semantics.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>1.5 Problems<\/h2>\n<ol class=\"lst-kix_list_19-0 start\" start=\"1\">\n<li><strong>Critical Thinking Task: <\/strong>Describe some of the situations in your life in which people often contradict themselves.<\/li>\n<li>Think of a time when a misunderstanding occurred because something wasn\u2019t said clearly. How could logic have helped?<\/li>\n<li>Describe a decision you made recently. What arguments did you consider, and were they logically sound?<\/li>\n<li>Analyze this statement: &#8216;In some cultures, being late is considered polite.&#8217; Discuss ambiguity and truth value.<\/li>\n<li>Logic helps evaluate fake news; how do we know if a claim online is true?<\/li>\n<li>Vagueness arises when the conditions under which a sentence might be true are \u201cfuzzy\u201d. \u00a0That is, in some cases, we cannot identify if the sentence is true or false. \u00a0If we say, \u201cTom is tall\u201d, this sentence is certainly true if Tom is the tallest person in the world, but it is not clear whether it is true if Tom is 185 centimeters tall. \u00a0Identify or create five declarative sentences in English that are vague.<\/li>\n<li>Ambiguity usually arises when a word or phrase has several distinct possible interpretations. \u00a0In our example above, the word \u201cpen\u201d could mean either a writing implement or a structure to hold a child. \u00a0A sentence that includes \u201cpen\u201d could be ambiguous, in which case it might be true for one interpretation and false for another. \u00a0Identify or create five declarative sentences in English that are ambiguous. \u00a0(This will probably require you to identify a homonym, a word that has more than one meaning but sounds or is written the same. \u00a0If you are stumped, consider slang: \u00a0many slang terms are ambiguous because they redefine existing words. \u00a0For example, in the 1980s, in some communities and contexts, to say something was \u201cbad\u201d meant that it was good; this obviously can create ambiguous sentences.)<\/li>\n<li>Often we can make a vague sentence precise by defining a specific interpretation of the meaning of an adjective, term, or other element of the language. \u00a0For example, we could make the sentence \u201cTom is tall\u201d precise by specifying one person referred to by \u201cTom\u201d, and also by defining \u201c\u2026is tall\u201d as true of anyone 180 centimeters tall or taller. \u00a0For each of the five vague sentences that you identified or created for problem 1, describe how the interpretation of certain elements of the sentence could make the sentence no longer vague.<\/li>\n<li>Often we can make an ambiguous sentence precise by specifying which of the possible meanings we intend to use. \u00a0We could make the sentence, \u201cTom is by the pen\u201d unambiguous by specifying which Tom we mean, and also defining \u201cpen\u201d to mean an infant play pen. \u00a0For each of the five ambiguous sentences that you identified or created for problem 2, identify and describe how the interpretation of certain elements of the sentence could make the sentence no longer ambiguous.<\/li>\n<li>Come up with five examples of your own of English sentences that are not declarative sentences. \u00a0(Examples can include commands, exclamations, and promises.)<\/li>\n<li>Here are some sentences from literary works and other famous texts. Describe as best you can what the role of the sentence is. For example, the sentence might be a declarative sentence, which aims to describe things; or a question, which aims to solicit information; or a command, which is used to make someone do something; and so on. It is not essential that you have a name for the kind of sentence, but rather can you describe what a speaker would typically intend for such a sentence to do?<\/li>\n<li>&#8220;Though I should die with thee, yet will I not deny thee.&#8221; (From the King James Bible)\n<ol class=\"lst-kix_list_19-0 start\" start=\"1\">\n<li>&#8220;Get thee to a nunnery.&#8221; (William Shakespeare, <i>Hamlet<\/i>.)<\/li>\n<li>&#8220;That on the first day of January, in the year of our Lord one thousand eight hundred and sixty-three, all persons held as slaves within any State or designated part of a State, the people whereof shall then be in rebellion against the United States, shall be then, thenceforward, and forever free.&#8221; (From &#8220;The Emancipation Proclamation&#8221;.)<\/li>\n<li>&#8220;Sing, goddess, of the anger of Achilles son of Peleus, that brought countless ills upon the Achaeans.&#8221; (Homer, <i>The Illiad<\/i>.)<\/li>\n<li>&#8220;Since the heavens grant that you recognize me, hold your tongue, and do not say a word about who I am to any one else in the house, for if you do, and if heaven grants me to take the lives of these suitors, I will not spare you, though you are my own nurse, when I am killing the other women.&#8221; (Homer, <i>The Odyssey<\/i>.)<\/li>\n<li>&#8220;Tyger, tyger, burning bright,<br \/>\nIn the forests of the night,<br \/>\nWhat immortal hand or eye,<br \/>\ncould frame thy fearful symmetry?&#8221; (William Blake, <i>The Tyger<\/i>.)<\/li>\n<li>&#8220;As wicked dew as e&#8217;er my mother brush&#8217;d<br \/>\nWith raven&#8217;s feather from unwholesome fen<br \/>\nDrop on you both! A south-west blow on ye<br \/>\nAnd blister you all o&#8217;er!&#8221; (William Shakespeare, <i>The Tempest<\/i>.)<\/li>\n<li>&#8220;For he to-day that sheds his blood with me<br \/>\nShall be my brother.&#8221; (William Shakespeare, <i>Henry V<\/i>.)<\/li>\n<li>&#8220;Astonishing, Pip!&#8221; (Charles Dickens, <i>Great Expectations<\/i>.)<\/li>\n<li>&#8220;Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the government for a redress of grievances.&#8221; (The Constitution of the United States.)<\/li>\n<\/ol>\n<\/li>\n<\/ol>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-29-1\">There is a complex issue here that we will discuss later. \u00a0But, in brief: \u00a0\u201cis\u201d is ambiguous; it has several meanings. \u00a0\u201cMalcolm Little is\u201d is a sentence if it is meant to assert the existence of Malcolm Little. \u00a0The \u201cis\u201d that appears in the sentence, \u201cMalcolm Little is tall\u201d, however, is what we call the \u201c\u2018is\u2019 of predication\u201d.\u00a0 In that sentence, \u201cis\u201d is used to assert that a property is had by Malcolm Little (the property of being tall); and here \u201cis tall\u201d is what we are calling a \u201cpredicate\u201d.\u00a0 So, the \u201cis\u201d of predication has no clear meaning when appearing without the rest of the predicate; it does not assert existence. <a href=\"#return-footnote-29-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":1,"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-29","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\/29","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":38,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/29\/revisions"}],"predecessor-version":[{"id":332,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/29\/revisions\/332"}],"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\/29\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=29"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=29"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=29"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=29"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}