{"id":41,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/6-conditional-derivations\/"},"modified":"2025-10-13T17:53:35","modified_gmt":"2025-10-13T17:53:35","slug":"6-conditional-derivations","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/6-conditional-derivations\/","title":{"raw":"Conditional Derivations","rendered":"Conditional Derivations"},"content":{"raw":"<h2>6.1 \u00a0An argument from Hobbes<\/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 does the logical \u201cif\u2026then\u201d differ from how we use it in everyday language?<\/li>\r\n \t<li>What does it mean to assume something \u201cfor the sake of argument\u201d in logic?<\/li>\r\n \t<li>Why are subproofs especially important in conditional reasoning?<\/li>\r\n \t<li>Can an implication be true even if its antecedent is 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>Conditional (\u2192)<\/strong> - executing different actions based on whether certain conditions are met.<\/li>\r\n \t<li><strong>Conditional Introduction (\u2192I)<\/strong> - a rule of inference that allows you to prove a conditional statement (an \"if-then\" statement) by assuming the antecedent (the \"if\" part) and then deriving the consequent (the \"then\" part).<\/li>\r\n \t<li><strong>Conditional Elimination \/ Modus Ponens (\u2192E)<\/strong> - allows you to derive the consequent of a conditional statement if you know that the antecedent is true.<\/li>\r\n \t<li><strong>Subproof for Conditional Derivation<\/strong> - a temporary, self-contained section of a proof where an assumption is made to derive a new conclusion<\/li>\r\n \t<li><strong>Assume-for-the-sake-of-argument<\/strong> - temporarily accepting a statement as true for the purpose of exploring its implications or consequences, even if the statement is not actually believed to be true.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nIn his great work, <span class=\"em\">Leviathan,<\/span>\u00a0the philosopher Thomas Hobbes (1588-1679) gives an important argument for government. \u00a0Hobbes begins by claiming that without a common power, our condition is very poor indeed. \u00a0He calls this state without government, \u201cthe state of nature\u201d, and claims\r\n<blockquote>Hereby it is manifest that during the time men live without a common power to keep them all in awe, they are in that condition which is called war; and such a war as is of every man against every man\u2026. In such condition there is no place for industry, because the fruit thereof is uncertain: and consequently no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; no instruments of moving and removing such things as require much force; no knowledge of the face of the earth; no account of time; no arts; no letters; no society; and which is worst of all, continual fear, and danger of violent death; and the life of man, solitary, poor, nasty, brutish, and short.[footnote]Hobbes (1886: 64).[\/footnote]<\/blockquote>\r\nHobbes developed what is sometimes called \u201ccontract theory\u201d. \u00a0This is a view of government in which one views the state as the product of a rational contract. \u00a0Although we inherit our government, the idea is that in some sense we would find it rational to choose the government, were we ever in the position to do so. \u00a0So, in the passage above, Hobbes claims that in this state of nature, we have absolute freedom, but this leads to universal struggle between all people. \u00a0There can be no property, for example, if there is no power to enforce property rights. \u00a0You are free to take other people\u2019s things, but they are also free to take yours. \u00a0Only violence can discourage such theft. \u00a0But, a common power, like a king, can enforce rules, such as property rights. \u00a0To have this common power, we must give up some freedoms. \u00a0You are (or should be, if it were ever up to you) willing to give up those freedoms because of the benefits that you get from this. \u00a0For example, you are willing to give up the freedom to just seize people\u2019s goods, because you like even more that other people cannot seize your goods.\r\n\r\nWe can reconstruct Hobbes\u2019s defense of government, greatly simplified, as being something like this:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If we want to be safe, then we should have a state that can protect us.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If we should have a state that can protect us, then we should give up some freedoms.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Therefore, if we want to be safe, then we should give up some freedoms.<\/p>\r\nLet us use the following translation key.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>P<\/strong>:<\/span>\u00a0 We want to be safe.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>Q<\/strong>:<\/span>\u00a0 We should have a state that can protect us.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>R<\/strong>:<\/span>\u00a0 We should give up some freedoms.<\/p>\r\nThe argument in our logical language would then be:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong><\/span><span class=\"strong\">\u2192<strong>Q<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>Q<\/strong>\u2192<strong>R<\/strong>)<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>_____<\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span><\/p>\r\nThis is a valid argument. \u00a0Let\u2019s take the time to show this with a truth table.\r\n<table class=\"grid\" style=\"width: 125px;\">\r\n<tbody>\r\n<tr>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border\"><\/td>\r\n<td class=\"border-right\"><\/td>\r\n<td class=\"border\">premise<\/td>\r\n<td class=\"border\">\u00a0premise<\/td>\r\n<td class=\"border\">\u00a0conclusion<\/td>\r\n<\/tr>\r\n<tr class=\"border-bottom\">\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\r\n<td class=\"border\"><strong><span class=\"strong\">Q<\/span><\/strong><\/td>\r\n<td class=\"border-right\"><strong><span class=\"strong\">R<\/span><\/strong><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><strong>(P\u2192Q)<\/strong><\/td>\r\n<td class=\"border\"><strong>(Q\u2192R)<\/strong><\/td>\r\n<td class=\"border\"><strong>(P\u2192R)<\/strong><\/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\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\r\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><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<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/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<td class=\"border\"><span class=\"em strong\">F <\/span><\/td>\r\n<td class=\"border\"><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\">T<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/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\">F<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><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<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/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\u00a0<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><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<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><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><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/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<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><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><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><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><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThe rows in which all the premises are true are the first, fifth, seventh, and eighth rows. \u00a0Note that in each such row, the conclusion is true. \u00a0Thus, in any kind of situation where the premises are true, the conclusion is true. \u00a0This is our semantics for a valid argument.\r\n\r\nWhat syntactic method can we use to prove this argument is valid? \u00a0Right now, we have none. \u00a0Other than double negation, we cannot even apply any of our inference rules using these premises.\r\n\r\nSome logic systems introduce a rule to capture this inference; this rule is typically called the \u201cchain rule\u201d. \u00a0But, there is a more general principle at stake here: we need a way to show conditionals. \u00a0So we want to take another approach to showing this argument is valid.\r\n<h2>6.2 \u00a0Conditional Derivation<\/h2>\r\nAs a handy rule of thumb, we can think of the inference rules as providing a way to either show a kind of sentence, or to make use of a kind of sentence. \u00a0For example, adjunction allows us to show a conjunction. \u00a0Simplification allows us to make use of a conjunction. \u00a0But this pattern is not complete: \u00a0we have rules to make use of a <strong>conditional<\/strong> (modus ponens and modus tollens), but no rule to show a conditional.\r\n\r\nWe will want to have some means to prove a conditional, because sometimes an argument will have a conditional as a conclusion. \u00a0It is not clear what rule we should introduce, however. \u00a0The conditional is true when the antecedent is false, or if both the antecedent and the consequent are true. \u00a0That\u2019s a rather messy affair for making an inference rule.\r\n\r\nHowever, think about what the conditional asserts: \u00a0if the antecedent is true, then the consequent is true. \u00a0We can make use of this idea not with an inference rule, but rather in the very structure of a proof. \u00a0We treat the proof as embodying a conditional relationship.\r\n\r\nOur idea is this: \u00a0let us assume some sentence, <strong><span class=\"strong\">\u03a6<\/span><\/strong>. \u00a0If we can then prove another sentence <strong><span class=\"strong\">\u03a8<\/span><\/strong>, we will have proved that if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true then <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>is true. \u00a0The proof structure will thus have a shape like this:\r\n\r\n<img class=\"size-full wp-image-348 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114526.png\" alt=\"\" width=\"197\" height=\"201\" \/>\r\n\r\nThe last line of the proof is justified by the shape of the proof: \u00a0by assuming that <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true, and then using our inference rules to prove <strong><span class=\"strong\">\u03a8<\/span><\/strong>, we know that if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true then <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>is true. \u00a0And this is just what the conditional asserts.\r\n\r\nThis method is sometimes referred to as an application of the deduction theorem. Here, we shall think of this as a proof method, traditionally called \u201c<strong>conditional derivation<\/strong>.\u201d\r\n\r\nA <strong>conditional derivation<\/strong> is like a direct derivation, but with two differences. \u00a0First, along with the premises, you get a single special assumption, called \u201cthe assumption for conditional derivation\u201d. \u00a0Second, you do not aim to show your conclusion, but rather the consequent of your conclusion. \u00a0So, to show <span class=\"strong\">(<strong>\u03a6<\/strong>\u2192<strong>\u03a8<\/strong>)<\/span>\u00a0you will always assume <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and try to show <strong><span class=\"strong\">\u03a8<\/span><\/strong>. \u00a0Also, in our logical system, a conditional derivation will always be a subproof. \u00a0A <strong>subproof<\/strong> is a proof within another proof. \u00a0We always start with a direct proof, and then do the conditional proof within that direct proof.\r\n\r\nHere is how we would apply the proof method to prove the validity of Hobbes\u2019s argument, as we reconstructed it above.\r\n\r\n<img class=\" wp-image-349 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-300x93.png\" alt=\"\" width=\"371\" height=\"115\" \/>\r\n\r\nOur Fitch bars make clear what is a <strong>sub-proof<\/strong> here; they let us see this as a direct derivation with a conditional derivation embedded in it. \u00a0This is an important concept: \u00a0we can have proofs within proofs.\r\n\r\nAn important principle is that once a subproof is done, we cannot use any of the lines in the subproof. \u00a0We need this rule because conditional derivation allowed us to make a special assumption that we use only temporarily. \u00a0Above, we assumed <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0Our goal is only to show that if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, then <strong><span class=\"strong\">R<\/span>\u00a0<\/strong>is true. \u00a0But perhaps <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>isn\u2019t true. \u00a0We do not want to later make use of <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>for some other purpose. \u00a0So, we have the rule that when a subproof is complete, you cannot use the lines that occur in the subproof. \u00a0In this case, that means that we cannot use lines 3, 4, or 5 for any other purpose than to show the conditional <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0We cannot now cite those individual lines again. \u00a0We can, however, use line 6, the conclusion of the subproof.\r\n\r\nThe Fitch bars\u2014which we have used before now in our proofs only to separate the premises from the later steps\u2014now have a very beneficial use. \u00a0They allow us to set aside a conditional derivation as a subproof, and they help remind us that we cannot cite the lines in that subproof once the subproof is complete.\r\n\r\nIt might be helpful to give an example of why this is necessary. \u00a0That is, it might be helpful to give an example of an argument made invalid because it makes use of lines in a finished subproof. \u00a0Consider the following argument.\r\n<p style=\"padding-left: 120px;\">If you are Pope, then you have a home in the Vatican.<\/p>\r\n<p style=\"padding-left: 120px;\">If you have a home in the Vatican, then you hear church bells often.<\/p>\r\n<p style=\"padding-left: 120px;\">_____<\/p>\r\n<p style=\"padding-left: 120px;\">If you are Pope, then you hear church bells often.<\/p>\r\nThat is a valid argument, with the same form as the argument we adopted from Hobbes. \u00a0However, if we broke our rule about conditional derivations, we could prove that you are Pope. Let\u2019s use this key:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong>: \u00a0You are Pope.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">T<\/span><\/strong>: \u00a0You have a home in the Vatican.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">U<\/span><\/strong>: \u00a0You hear church bells often.<\/p>\r\nNow consider this \u201cproof\u201d:\r\n\r\n<img class=\" wp-image-350 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-300x103.png\" alt=\"\" width=\"411\" height=\"141\" \/>\r\n\r\nAnd, thus, we have proven that you are Pope. \u00a0But, of course, you are not the Pope. \u00a0From true premises, we ended up with a false conclusion, so the argument is obviously invalid. \u00a0What went wrong? \u00a0The problem was that after we completed the conditional derivation that occurs in lines 3 through 5, and used that conditional derivation to assert line 6, we can no longer use those lines 3 through 5. \u00a0But on line 7 we made use of line 3. \u00a0Line 3 is not something we know to be true; our reasoning from lines 3 through line 5 was to ask, if <strong><span class=\"strong\">S<\/span>\u00a0<\/strong>were true, what else would be true? \u00a0When we are done with that conditional derivation, we can use only the conditional that we derived, and not the steps used in the conditional derivation.\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>Critical Thinking Task:\u00a0<\/strong>Select an argument in modern politics that contains the need for sub-proofing. Please explain this selection.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>6.3 \u00a0Some additional examples<\/h2>\r\nHere are a few kinds of arguments that help illustrate the power of the conditional derivation.\r\n\r\nThis argument makes use of conjunctions.\r\n<p style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\r\n<p style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><span class=\"strong\">((<strong>P<\/strong>^<strong>R<\/strong>)\u2192(<strong>Q<\/strong>^<strong>S<\/strong>))<\/span><\/p>\r\nWe always begin by constructing a direct proof, using the Fitch bar to identify the premises of our argument, if any.\r\n\r\n<img class=\"size-medium wp-image-351 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-300x75.png\" alt=\"\" width=\"300\" height=\"75\" \/>\r\n\r\nBecause the conclusion is a conditional, we assume the antecedent and show the consequent.\r\n\r\n<img class=\" wp-image-352 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-300x124.png\" alt=\"\" width=\"438\" height=\"181\" \/>\r\n\r\nHere\u2019s another example. \u00a0Note that the following argument is valid.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>S<\/strong>\u2192<strong>R<\/strong>))<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>Q<\/strong>\u2192<strong>S<\/strong>))<\/span><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P\u2192(Q\u2192R))<\/span><\/strong><\/p>\r\nThe proof will require several embedded subproofs.\r\n\r\n<img class=\" wp-image-353 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-300x140.png\" alt=\"\" width=\"375\" height=\"175\" \/>\r\n<h2>6.4 \u00a0Theorems<\/h2>\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>Theorem<\/strong> - a sentence that can be proved without premises.<\/li>\r\n \t<li><strong>Tautology<\/strong> - a sentence of the propositional logic that must be true.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nConditional derivation allows us to see an important new concept. \u00a0Consider the following sentence:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">((<strong>P<\/strong>\u2192<strong>Q<\/strong>) \u2192(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>))<\/span><\/p>\r\nThis sentence is a tautology. \u00a0To check this, we can make its truth table.\r\n<table class=\"grid\" style=\"width: 150px;\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">Q<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">\u00acQ<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">\u00acP<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\"> (P\u2192Q)<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">(\u00acQ\u2192\u00acP)<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">((P\u2192Q) \u2192 (\u00acQ\u2192\u00acP))<\/span><\/th>\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\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><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<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><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<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><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><\/td>\r\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\"><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\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\"><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<\/tbody>\r\n<\/table>\r\nThis sentence is true in every kind of situation, which is what we mean by a \u201ctautology\u201d.\r\n\r\nNow reflect on our definition of \u201cvalid\u201d: \u00a0necessarily, if the premises are true, then the conclusion is true. \u00a0What about an argument in which the conclusion is a tautology? \u00a0By our definition of \u201cvalid\u201d, an argument with a conclusion that must be true must be a valid argument\u2014no matter what the premises are! \u00a0(If this confuses you, look back at the truth table for the conditional. \u00a0Our definition of valid includes the conditional: if the premises are true, then the conclusion is true. \u00a0Suppose now our conclusion must be true. \u00a0Any conditional with a true consequent is true. \u00a0So the definition of \u201cvalid\u201d must be true of any argument with a tautology as a conclusion.) \u00a0And, given that, it would seem that it is irrelevant whether we have any premises at all, since any will do. \u00a0This suggests that there can be valid arguments with no premises.\r\n\r\nConditional derivation lets us actually construct such arguments. \u00a0First, we will draw our Fitch bar for our main argument to indicate that we have no premises. \u00a0 Then, we will construct a conditional derivation. \u00a0It will start like this:\r\n\r\n<img class=\"size-medium wp-image-354 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-300x61.png\" alt=\"\" width=\"300\" height=\"61\" \/>\r\n\r\nBut what now? \u00a0Well, we have assumed the antecedent of our sentence, and we should strive now to show the consequent. \u00a0But note that the consequent is a conditional. \u00a0So, we will again do a conditional derivation.\r\n\r\n<img class=\" wp-image-355 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-300x100.png\" alt=\"\" width=\"372\" height=\"124\" \/>\r\n\r\nThis is a proof, without premises, of <span class=\"strong\">((<strong>P<\/strong>\u2192<strong>Q<\/strong>)\u2192(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>))<\/span>. \u00a0The top of the proof shows that we have no premises. \u00a0Our conclusion is a conditional, so, on line 1, we assumed the antecedent of the conditional. \u00a0We now have to show the consequent of the conditional; but the consequent of the conditional is also a conditional, so we assumed its antecedent on line 2. \u00a0 Line 4 is the result of the conditional derivation from lines 2 to 3. \u00a0Lines 1 through 4 tell us that if <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0is true, then<span class=\"strong\">\u00a0(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>)<\/span>\u00a0is true. \u00a0And that is what we conclude on line 5.\r\n\r\nWe call a sentence that can be proved without premises a \u201ctheorem\u201d. \u00a0Theorems are special because they reveal the things that follow from logic alone. \u00a0It is a very great benefit of our propositional logic that all the theorems are tautologies. \u00a0It is an equally great benefit of our propositional logic that all the tautologies are theorems. \u00a0Nonetheless, these concepts are different. \u00a0\u201cTautology\u201d refers to a semantic concept: \u00a0a tautology is a sentence that must be true. \u00a0\u201cTheorem\u201d refers to a concept of syntax and derivation: \u00a0a theorem is a sentence that can be derived without premises.\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>Critical Thinking Task:\u00a0<\/strong>Generate a short story explaining the logic connecting a tautology to a theorem.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>6.5 \u00a0Problems<\/h2>\r\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\r\n \t<li>Prove the following arguments are valid. \u00a0This will require conditional derivation.\r\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\r\n \t<li>Premise:\u00a0<span class=\"strong\">(\u00ac<strong>Q<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">\u00a0(<strong>P<\/strong> \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\r\n \t<li>Premise:\u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, <span class=\"strong\">(<strong>Q<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>.<\/li>\r\n \t<li>Premises:\u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>))<\/span>, <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>.<\/li>\r\n \t<li>Premise: <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, <span class=\"strong\">(<strong>S<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion: <span class=\"strong\">((\u00ac<strong>Q<\/strong> ^ \u00ac<strong>R<\/strong>) \u2192 (\u00ac<strong>P<\/strong> ^ \u00ac<strong>S<\/strong>))<\/span>.<\/li>\r\n \t<li>Premise: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>. \u00a0Conclusion: <span class=\"strong\">((<strong>P<\/strong> ^ <strong>R<\/strong>) \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\r\n \t<li>Premise: \u00a0<span class=\"strong\">((<strong>R<\/strong>^<strong>Q<\/strong>) \u2192 <strong>S<\/strong>)<\/span>, <span class=\"strong\">(\u00ac<strong>P<\/strong> \u2192 (<strong>R<\/strong>^<strong>Q<\/strong>))<\/span>. \u00a0Conclusion: <span class=\"strong\">(\u00ac<strong>S<\/strong> \u2192 <strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li>Premise: \u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac<strong>Q<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">\u00a0 (<strong>Q<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li>Premises:<span class=\"strong\">\u00a0 (<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>,<span class=\"strong\">\u00a0(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>Q<\/strong>^<strong>R<\/strong>)))<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>Q<\/strong> ^ <strong>S<\/strong>))<\/span>, <span class=\"strong\">(<strong>Q<\/strong> \u2192 <strong>R<\/strong>)<\/span>, <span class=\"strong\">(<strong>S<\/strong> \u2192 <strong>T<\/strong>)<\/span>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>R<\/strong> ^ <strong>T<\/strong>))<\/span>.<\/li>\r\n \t<li>Premises: \u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>))<\/span>, <span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>S<\/strong> \u2192 <strong>T<\/strong>))<\/span>,<span class=\"strong\">(<strong>Q<\/strong> ^ <strong>S<\/strong>)<\/span>.Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>R<\/strong> ^ <strong>T<\/strong>))<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Prove the following theorems.\r\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\r\n \t<li><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 ((<strong>R<\/strong> \u2192 <strong>P<\/strong>) \u2192 (<strong>R<\/strong> \u2192 <strong>Q<\/strong>)))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>)) \u2192 ((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> \u2192 <strong>R<\/strong>)))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((\u00ac<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (\u00ac<strong>Q<\/strong> \u2192 <strong>P<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) ^ (<strong>P<\/strong> \u2192 <strong>R<\/strong>)) \u2192 (<strong>P<\/strong> \u2192 (<strong>Q<\/strong>^<strong>R<\/strong>)))<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Make a truth table for each of the following complex sentences, in order to see when it is true or false. \u00a0Identify which are tautologies. \u00a0Prove the tautologies.\r\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 <strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>P<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>P<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>P<\/strong>) \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li><span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac\u00ac<strong>P<\/strong>)<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> ^ <strong>Q<\/strong>))<\/span>.<\/li>\r\n \t<li><span class=\"strong\">((<strong>P<\/strong> ^ <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>In normal colloquial English, write your own valid argument with at least two premises and with a conclusion that is a conditional. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like formal logic). \u00a0Translate it into propositional logic and prove it is valid.<\/li>\r\n \t<li>Translate the following passage into our propositional logic. Prove the argument is valid.<\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ol>\r\n<ol>\r\n \t<li style=\"list-style-type: none;\"><\/li>\r\n<\/ol>\r\n<p class=\"marg-left\">Either Beneke or Mill is the culprit who burned the Logician's Club. Also, if Beneke did it, then he bought the flares. But if Beneke bought the flares, he was at the Mariner's Shop yesterday. Thus, if Beneke was not at the Mariner's Shop yesterday, Mill did it.<\/p>\r\n\r\n<div>\r\n\r\n<hr \/>\r\n\r\n<\/div>","rendered":"<h2>6.1 \u00a0An argument from Hobbes<\/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 does the logical \u201cif\u2026then\u201d differ from how we use it in everyday language?<\/li>\n<li>What does it mean to assume something \u201cfor the sake of argument\u201d in logic?<\/li>\n<li>Why are subproofs especially important in conditional reasoning?<\/li>\n<li>Can an implication be true even if its antecedent is 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>Conditional (\u2192)<\/strong> &#8211; executing different actions based on whether certain conditions are met.<\/li>\n<li><strong>Conditional Introduction (\u2192I)<\/strong> &#8211; a rule of inference that allows you to prove a conditional statement (an &#8220;if-then&#8221; statement) by assuming the antecedent (the &#8220;if&#8221; part) and then deriving the consequent (the &#8220;then&#8221; part).<\/li>\n<li><strong>Conditional Elimination \/ Modus Ponens (\u2192E)<\/strong> &#8211; allows you to derive the consequent of a conditional statement if you know that the antecedent is true.<\/li>\n<li><strong>Subproof for Conditional Derivation<\/strong> &#8211; a temporary, self-contained section of a proof where an assumption is made to derive a new conclusion<\/li>\n<li><strong>Assume-for-the-sake-of-argument<\/strong> &#8211; temporarily accepting a statement as true for the purpose of exploring its implications or consequences, even if the statement is not actually believed to be true.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>In his great work, <span class=\"em\">Leviathan,<\/span>\u00a0the philosopher Thomas Hobbes (1588-1679) gives an important argument for government. \u00a0Hobbes begins by claiming that without a common power, our condition is very poor indeed. \u00a0He calls this state without government, \u201cthe state of nature\u201d, and claims<\/p>\n<blockquote><p>Hereby it is manifest that during the time men live without a common power to keep them all in awe, they are in that condition which is called war; and such a war as is of every man against every man\u2026. In such condition there is no place for industry, because the fruit thereof is uncertain: and consequently no culture of the earth; no navigation, nor use of the commodities that may be imported by sea; no commodious building; no instruments of moving and removing such things as require much force; no knowledge of the face of the earth; no account of time; no arts; no letters; no society; and which is worst of all, continual fear, and danger of violent death; and the life of man, solitary, poor, nasty, brutish, and short.<a class=\"footnote\" title=\"Hobbes (1886: 64).\" id=\"return-footnote-41-1\" href=\"#footnote-41-1\" aria-label=\"Footnote 1\"><sup class=\"footnote\">[1]<\/sup><\/a><\/p><\/blockquote>\n<p>Hobbes developed what is sometimes called \u201ccontract theory\u201d. \u00a0This is a view of government in which one views the state as the product of a rational contract. \u00a0Although we inherit our government, the idea is that in some sense we would find it rational to choose the government, were we ever in the position to do so. \u00a0So, in the passage above, Hobbes claims that in this state of nature, we have absolute freedom, but this leads to universal struggle between all people. \u00a0There can be no property, for example, if there is no power to enforce property rights. \u00a0You are free to take other people\u2019s things, but they are also free to take yours. \u00a0Only violence can discourage such theft. \u00a0But, a common power, like a king, can enforce rules, such as property rights. \u00a0To have this common power, we must give up some freedoms. \u00a0You are (or should be, if it were ever up to you) willing to give up those freedoms because of the benefits that you get from this. \u00a0For example, you are willing to give up the freedom to just seize people\u2019s goods, because you like even more that other people cannot seize your goods.<\/p>\n<p>We can reconstruct Hobbes\u2019s defense of government, greatly simplified, as being something like this:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If we want to be safe, then we should have a state that can protect us.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">If we should have a state that can protect us, then we should give up some freedoms.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Therefore, if we want to be safe, then we should give up some freedoms.<\/p>\n<p>Let us use the following translation key.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>P<\/strong>:<\/span>\u00a0 We want to be safe.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>Q<\/strong>:<\/span>\u00a0 We should have a state that can protect us.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\"><strong>R<\/strong>:<\/span>\u00a0 We should give up some freedoms.<\/p>\n<p>The argument in our logical language would then be:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong><\/span><span class=\"strong\">\u2192<strong>Q<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>Q<\/strong>\u2192<strong>R<\/strong>)<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong>_____<\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span><\/p>\n<p>This is a valid argument. \u00a0Let\u2019s take the time to show this with a truth table.<\/p>\n<table class=\"grid\" style=\"width: 125px;\">\n<tbody>\n<tr>\n<td class=\"border\"><\/td>\n<td class=\"border\"><\/td>\n<td class=\"border-right\"><\/td>\n<td class=\"border\">premise<\/td>\n<td class=\"border\">\u00a0premise<\/td>\n<td class=\"border\">\u00a0conclusion<\/td>\n<\/tr>\n<tr class=\"border-bottom\">\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><strong><span class=\"strong\">P<\/span><\/strong><\/td>\n<td class=\"border\"><strong><span class=\"strong\">Q<\/span><\/strong><\/td>\n<td class=\"border-right\"><strong><span class=\"strong\">R<\/span><\/strong><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><strong>(P\u2192Q)<\/strong><\/td>\n<td class=\"border\"><strong>(Q\u2192R)<\/strong><\/td>\n<td class=\"border\"><strong>(P\u2192R)<\/strong><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/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\"><span class=\"em strong\">T<\/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<td class=\"border\"><span class=\"em strong\">F <\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/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\"><span class=\"em strong\">F<\/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\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/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\"><span class=\"em strong\">F<\/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\u00a0<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><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><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/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<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><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><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><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><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"shaded\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"shaded\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>The rows in which all the premises are true are the first, fifth, seventh, and eighth rows. \u00a0Note that in each such row, the conclusion is true. \u00a0Thus, in any kind of situation where the premises are true, the conclusion is true. \u00a0This is our semantics for a valid argument.<\/p>\n<p>What syntactic method can we use to prove this argument is valid? \u00a0Right now, we have none. \u00a0Other than double negation, we cannot even apply any of our inference rules using these premises.<\/p>\n<p>Some logic systems introduce a rule to capture this inference; this rule is typically called the \u201cchain rule\u201d. \u00a0But, there is a more general principle at stake here: we need a way to show conditionals. \u00a0So we want to take another approach to showing this argument is valid.<\/p>\n<h2>6.2 \u00a0Conditional Derivation<\/h2>\n<p>As a handy rule of thumb, we can think of the inference rules as providing a way to either show a kind of sentence, or to make use of a kind of sentence. \u00a0For example, adjunction allows us to show a conjunction. \u00a0Simplification allows us to make use of a conjunction. \u00a0But this pattern is not complete: \u00a0we have rules to make use of a <strong>conditional<\/strong> (modus ponens and modus tollens), but no rule to show a conditional.<\/p>\n<p>We will want to have some means to prove a conditional, because sometimes an argument will have a conditional as a conclusion. \u00a0It is not clear what rule we should introduce, however. \u00a0The conditional is true when the antecedent is false, or if both the antecedent and the consequent are true. \u00a0That\u2019s a rather messy affair for making an inference rule.<\/p>\n<p>However, think about what the conditional asserts: \u00a0if the antecedent is true, then the consequent is true. \u00a0We can make use of this idea not with an inference rule, but rather in the very structure of a proof. \u00a0We treat the proof as embodying a conditional relationship.<\/p>\n<p>Our idea is this: \u00a0let us assume some sentence, <strong><span class=\"strong\">\u03a6<\/span><\/strong>. \u00a0If we can then prove another sentence <strong><span class=\"strong\">\u03a8<\/span><\/strong>, we will have proved that if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true then <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>is true. \u00a0The proof structure will thus have a shape like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-348 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114526.png\" alt=\"\" width=\"197\" height=\"201\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114526.png 197w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114526-65x66.png 65w\" sizes=\"auto, (max-width: 197px) 100vw, 197px\" \/><\/p>\n<p>The last line of the proof is justified by the shape of the proof: \u00a0by assuming that <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true, and then using our inference rules to prove <strong><span class=\"strong\">\u03a8<\/span><\/strong>, we know that if <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>is true then <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>is true. \u00a0And this is just what the conditional asserts.<\/p>\n<p>This method is sometimes referred to as an application of the deduction theorem. Here, we shall think of this as a proof method, traditionally called \u201c<strong>conditional derivation<\/strong>.\u201d<\/p>\n<p>A <strong>conditional derivation<\/strong> is like a direct derivation, but with two differences. \u00a0First, along with the premises, you get a single special assumption, called \u201cthe assumption for conditional derivation\u201d. \u00a0Second, you do not aim to show your conclusion, but rather the consequent of your conclusion. \u00a0So, to show <span class=\"strong\">(<strong>\u03a6<\/strong>\u2192<strong>\u03a8<\/strong>)<\/span>\u00a0you will always assume <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and try to show <strong><span class=\"strong\">\u03a8<\/span><\/strong>. \u00a0Also, in our logical system, a conditional derivation will always be a subproof. \u00a0A <strong>subproof<\/strong> is a proof within another proof. \u00a0We always start with a direct proof, and then do the conditional proof within that direct proof.<\/p>\n<p>Here is how we would apply the proof method to prove the validity of Hobbes\u2019s argument, as we reconstructed it above.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-349 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-300x93.png\" alt=\"\" width=\"371\" height=\"115\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-300x93.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-768x238.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-65x20.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-225x70.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626-350x108.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114626.png 865w\" sizes=\"auto, (max-width: 371px) 100vw, 371px\" \/><\/p>\n<p>Our Fitch bars make clear what is a <strong>sub-proof<\/strong> here; they let us see this as a direct derivation with a conditional derivation embedded in it. \u00a0This is an important concept: \u00a0we can have proofs within proofs.<\/p>\n<p>An important principle is that once a subproof is done, we cannot use any of the lines in the subproof. \u00a0We need this rule because conditional derivation allowed us to make a special assumption that we use only temporarily. \u00a0Above, we assumed <strong><span class=\"strong\">P<\/span><\/strong>. \u00a0Our goal is only to show that if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true, then <strong><span class=\"strong\">R<\/span>\u00a0<\/strong>is true. \u00a0But perhaps <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>isn\u2019t true. \u00a0We do not want to later make use of <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>for some other purpose. \u00a0So, we have the rule that when a subproof is complete, you cannot use the lines that occur in the subproof. \u00a0In this case, that means that we cannot use lines 3, 4, or 5 for any other purpose than to show the conditional <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0We cannot now cite those individual lines again. \u00a0We can, however, use line 6, the conclusion of the subproof.<\/p>\n<p>The Fitch bars\u2014which we have used before now in our proofs only to separate the premises from the later steps\u2014now have a very beneficial use. \u00a0They allow us to set aside a conditional derivation as a subproof, and they help remind us that we cannot cite the lines in that subproof once the subproof is complete.<\/p>\n<p>It might be helpful to give an example of why this is necessary. \u00a0That is, it might be helpful to give an example of an argument made invalid because it makes use of lines in a finished subproof. \u00a0Consider the following argument.<\/p>\n<p style=\"padding-left: 120px;\">If you are Pope, then you have a home in the Vatican.<\/p>\n<p style=\"padding-left: 120px;\">If you have a home in the Vatican, then you hear church bells often.<\/p>\n<p style=\"padding-left: 120px;\">_____<\/p>\n<p style=\"padding-left: 120px;\">If you are Pope, then you hear church bells often.<\/p>\n<p>That is a valid argument, with the same form as the argument we adopted from Hobbes. \u00a0However, if we broke our rule about conditional derivations, we could prove that you are Pope. Let\u2019s use this key:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">S<\/span><\/strong>: \u00a0You are Pope.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">T<\/span><\/strong>: \u00a0You have a home in the Vatican.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">U<\/span><\/strong>: \u00a0You hear church bells often.<\/p>\n<p>Now consider this \u201cproof\u201d:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-350 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-300x103.png\" alt=\"\" width=\"411\" height=\"141\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-300x103.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-768x262.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-65x22.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-225x77.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657-350x120.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114657.png 875w\" sizes=\"auto, (max-width: 411px) 100vw, 411px\" \/><\/p>\n<p>And, thus, we have proven that you are Pope. \u00a0But, of course, you are not the Pope. \u00a0From true premises, we ended up with a false conclusion, so the argument is obviously invalid. \u00a0What went wrong? \u00a0The problem was that after we completed the conditional derivation that occurs in lines 3 through 5, and used that conditional derivation to assert line 6, we can no longer use those lines 3 through 5. \u00a0But on line 7 we made use of line 3. \u00a0Line 3 is not something we know to be true; our reasoning from lines 3 through line 5 was to ask, if <strong><span class=\"strong\">S<\/span>\u00a0<\/strong>were true, what else would be true? \u00a0When we are done with that conditional derivation, we can use only the conditional that we derived, and not the steps used in the conditional derivation.<\/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>Critical Thinking Task:\u00a0<\/strong>Select an argument in modern politics that contains the need for sub-proofing. Please explain this selection.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>6.3 \u00a0Some additional examples<\/h2>\n<p>Here are a few kinds of arguments that help illustrate the power of the conditional derivation.<\/p>\n<p>This argument makes use of conjunctions.<\/p>\n<p style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span><\/p>\n<p style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>R<\/strong>\u2192<strong>S<\/strong>)<\/span><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><span class=\"strong\">((<strong>P<\/strong>^<strong>R<\/strong>)\u2192(<strong>Q<\/strong>^<strong>S<\/strong>))<\/span><\/p>\n<p>We always begin by constructing a direct proof, using the Fitch bar to identify the premises of our argument, if any.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-351 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-300x75.png\" alt=\"\" width=\"300\" height=\"75\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-300x75.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-65x16.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-225x56.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821-350x88.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114821.png 592w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Because the conclusion is a conditional, we assume the antecedent and show the consequent.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-352 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-300x124.png\" alt=\"\" width=\"438\" height=\"181\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-300x124.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-768x318.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-65x27.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-225x93.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847-350x145.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114847.png 865w\" sizes=\"auto, (max-width: 438px) 100vw, 438px\" \/><\/p>\n<p>Here\u2019s another example. \u00a0Note that the following argument is valid.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>S<\/strong>\u2192<strong>R<\/strong>))<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>Q<\/strong>\u2192<strong>S<\/strong>))<\/span><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P\u2192(Q\u2192R))<\/span><\/strong><\/p>\n<p>The proof will require several embedded subproofs.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-353 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-300x140.png\" alt=\"\" width=\"375\" height=\"175\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-300x140.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-768x359.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-65x30.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-225x105.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914-350x164.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-114914.png 881w\" sizes=\"auto, (max-width: 375px) 100vw, 375px\" \/><\/p>\n<h2>6.4 \u00a0Theorems<\/h2>\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>Theorem<\/strong> &#8211; a sentence that can be proved without premises.<\/li>\n<li><strong>Tautology<\/strong> &#8211; a sentence of the propositional logic that must be true.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>Conditional derivation allows us to see an important new concept. \u00a0Consider the following sentence:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><span class=\"strong\">((<strong>P<\/strong>\u2192<strong>Q<\/strong>) \u2192(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>))<\/span><\/p>\n<p>This sentence is a tautology. \u00a0To check this, we can make its truth table.<\/p>\n<table class=\"grid\" style=\"width: 150px;\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">Q<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">\u00acQ<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">\u00acP<\/span><\/th>\n<th class=\"border\"><span class=\"strong\"> (P\u2192Q)<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">(\u00acQ\u2192\u00acP)<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">((P\u2192Q) \u2192 (\u00acQ\u2192\u00acP))<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><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<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><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<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><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><\/td>\n<td class=\"border\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\"><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<\/tbody>\n<\/table>\n<p>This sentence is true in every kind of situation, which is what we mean by a \u201ctautology\u201d.<\/p>\n<p>Now reflect on our definition of \u201cvalid\u201d: \u00a0necessarily, if the premises are true, then the conclusion is true. \u00a0What about an argument in which the conclusion is a tautology? \u00a0By our definition of \u201cvalid\u201d, an argument with a conclusion that must be true must be a valid argument\u2014no matter what the premises are! \u00a0(If this confuses you, look back at the truth table for the conditional. \u00a0Our definition of valid includes the conditional: if the premises are true, then the conclusion is true. \u00a0Suppose now our conclusion must be true. \u00a0Any conditional with a true consequent is true. \u00a0So the definition of \u201cvalid\u201d must be true of any argument with a tautology as a conclusion.) \u00a0And, given that, it would seem that it is irrelevant whether we have any premises at all, since any will do. \u00a0This suggests that there can be valid arguments with no premises.<\/p>\n<p>Conditional derivation lets us actually construct such arguments. \u00a0First, we will draw our Fitch bar for our main argument to indicate that we have no premises. \u00a0 Then, we will construct a conditional derivation. \u00a0It will start like this:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-354 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-300x61.png\" alt=\"\" width=\"300\" height=\"61\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-300x61.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-768x155.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-65x13.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-225x45.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051-350x71.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115051.png 861w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>But what now? \u00a0Well, we have assumed the antecedent of our sentence, and we should strive now to show the consequent. \u00a0But note that the consequent is a conditional. \u00a0So, we will again do a conditional derivation.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-355 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-300x100.png\" alt=\"\" width=\"372\" height=\"124\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-300x100.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-768x257.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-65x22.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-225x75.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113-350x117.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115113.png 879w\" sizes=\"auto, (max-width: 372px) 100vw, 372px\" \/><\/p>\n<p>This is a proof, without premises, of <span class=\"strong\">((<strong>P<\/strong>\u2192<strong>Q<\/strong>)\u2192(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>))<\/span>. \u00a0The top of the proof shows that we have no premises. \u00a0Our conclusion is a conditional, so, on line 1, we assumed the antecedent of the conditional. \u00a0We now have to show the consequent of the conditional; but the consequent of the conditional is also a conditional, so we assumed its antecedent on line 2. \u00a0 Line 4 is the result of the conditional derivation from lines 2 to 3. \u00a0Lines 1 through 4 tell us that if <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>\u00a0is true, then<span class=\"strong\">\u00a0(\u00ac<strong>Q<\/strong>\u2192\u00ac<strong>P<\/strong>)<\/span>\u00a0is true. \u00a0And that is what we conclude on line 5.<\/p>\n<p>We call a sentence that can be proved without premises a \u201ctheorem\u201d. \u00a0Theorems are special because they reveal the things that follow from logic alone. \u00a0It is a very great benefit of our propositional logic that all the theorems are tautologies. \u00a0It is an equally great benefit of our propositional logic that all the tautologies are theorems. \u00a0Nonetheless, these concepts are different. \u00a0\u201cTautology\u201d refers to a semantic concept: \u00a0a tautology is a sentence that must be true. \u00a0\u201cTheorem\u201d refers to a concept of syntax and derivation: \u00a0a theorem is a sentence that can be derived without premises.<\/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>Critical Thinking Task:\u00a0<\/strong>Generate a short story explaining the logic connecting a tautology to a theorem.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>6.5 \u00a0Problems<\/h2>\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\n<li>Prove the following arguments are valid. \u00a0This will require conditional derivation.\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\n<li>Premise:\u00a0<span class=\"strong\">(\u00ac<strong>Q<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">\u00a0(<strong>P<\/strong> \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\n<li>Premise:\u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, <span class=\"strong\">(<strong>Q<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>.<\/li>\n<li>Premises:\u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>))<\/span>, <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>.<\/li>\n<li>Premise: <span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>, <span class=\"strong\">(<strong>S<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion: <span class=\"strong\">((\u00ac<strong>Q<\/strong> ^ \u00ac<strong>R<\/strong>) \u2192 (\u00ac<strong>P<\/strong> ^ \u00ac<strong>S<\/strong>))<\/span>.<\/li>\n<li>Premise: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>. \u00a0Conclusion: <span class=\"strong\">((<strong>P<\/strong> ^ <strong>R<\/strong>) \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\n<li>Premise: \u00a0<span class=\"strong\">((<strong>R<\/strong>^<strong>Q<\/strong>) \u2192 <strong>S<\/strong>)<\/span>, <span class=\"strong\">(\u00ac<strong>P<\/strong> \u2192 (<strong>R<\/strong>^<strong>Q<\/strong>))<\/span>. \u00a0Conclusion: <span class=\"strong\">(\u00ac<strong>S<\/strong> \u2192 <strong>P<\/strong>)<\/span>.<\/li>\n<li>Premise: \u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac<strong>Q<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">\u00a0 (<strong>Q<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>.<\/li>\n<li>Premises:<span class=\"strong\">\u00a0 (<strong>P<\/strong>\u2192<strong>Q<\/strong>)<\/span>,<span class=\"strong\">\u00a0(<strong>P<\/strong>\u2192<strong>R<\/strong>)<\/span>. \u00a0Conclusion:<span class=\"strong\">(<strong>P<\/strong>\u2192(<strong>Q<\/strong>^<strong>R<\/strong>)))<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>Q<\/strong> ^ <strong>S<\/strong>))<\/span>, <span class=\"strong\">(<strong>Q<\/strong> \u2192 <strong>R<\/strong>)<\/span>, <span class=\"strong\">(<strong>S<\/strong> \u2192 <strong>T<\/strong>)<\/span>. \u00a0Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>R<\/strong> ^ <strong>T<\/strong>))<\/span>.<\/li>\n<li>Premises: \u00a0<span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>))<\/span>, <span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>S<\/strong> \u2192 <strong>T<\/strong>))<\/span>,<span class=\"strong\">(<strong>Q<\/strong> ^ <strong>S<\/strong>)<\/span>.Conclusion: \u00a0<span class=\"strong\">(<strong>P<\/strong>\u2192 (<strong>R<\/strong> ^ <strong>T<\/strong>))<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>Prove the following theorems.\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\n<li><span class=\"strong\">(<strong>P<\/strong>\u2192<strong>P<\/strong>)<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 ((<strong>R<\/strong> \u2192 <strong>P<\/strong>) \u2192 (<strong>R<\/strong> \u2192 <strong>Q<\/strong>)))<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>R<\/strong>)) \u2192 ((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> \u2192 <strong>R<\/strong>)))<\/span>.<\/li>\n<li><span class=\"strong\">((\u00ac<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (\u00ac<strong>Q<\/strong> \u2192 <strong>P<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">(((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) ^ (<strong>P<\/strong> \u2192 <strong>R<\/strong>)) \u2192 (<strong>P<\/strong> \u2192 (<strong>Q<\/strong>^<strong>R<\/strong>)))<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>Make a truth table for each of the following complex sentences, in order to see when it is true or false. \u00a0Identify which are tautologies. \u00a0Prove the tautologies.\n<ol class=\"lst-kix_list_33-0 start\" start=\"1\">\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 <strong>Q<\/strong>)<\/span>.<\/li>\n<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 <strong>P<\/strong>)<\/span>.<\/li>\n<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>Q<\/strong> \u2192 <strong>P<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">(<strong>P<\/strong> \u2192 (<strong>P<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>P<\/strong>) \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac<strong>P<\/strong>)<\/span>.<\/li>\n<li><span class=\"strong\">(<strong>P<\/strong> \u2192 \u00ac\u00ac<strong>P<\/strong>)<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> \u2192 <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> ^ <strong>Q<\/strong>))<\/span>.<\/li>\n<li><span class=\"strong\">((<strong>P<\/strong> ^ <strong>Q<\/strong>) \u2192 (<strong>P<\/strong> \u2192 <strong>Q<\/strong>))<\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>In normal colloquial English, write your own valid argument with at least two premises and with a conclusion that is a conditional. Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like formal logic). \u00a0Translate it into propositional logic and prove it is valid.<\/li>\n<li>Translate the following passage into our propositional logic. Prove the argument is valid.<\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<ol>\n<li style=\"list-style-type: none;\"><\/li>\n<\/ol>\n<p class=\"marg-left\">Either Beneke or Mill is the culprit who burned the Logician&#8217;s Club. Also, if Beneke did it, then he bought the flares. But if Beneke bought the flares, he was at the Mariner&#8217;s Shop yesterday. Thus, if Beneke was not at the Mariner&#8217;s Shop yesterday, Mill did it.<\/p>\n<div>\n<hr \/>\n<\/div>\n<hr class=\"before-footnotes clear\" \/><div class=\"footnotes\"><ol><li id=\"footnote-41-1\">Hobbes (1886: 64). <a href=\"#return-footnote-41-1\" class=\"return-footnote\" aria-label=\"Return to footnote 1\">&crarr;<\/a><\/li><\/ol><\/div>","protected":false},"author":158,"menu_order":6,"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-41","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\/41","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":11,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions"}],"predecessor-version":[{"id":356,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/41\/revisions\/356"}],"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\/41\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=41"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=41"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=41"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=41"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}