{"id":44,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/7-or\/"},"modified":"2025-10-13T18:00:01","modified_gmt":"2025-10-13T18:00:01","slug":"7-or","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/7-or\/","title":{"raw":"\u201cOr\u201d","rendered":"\u201cOr\u201d"},"content":{"raw":"<h2>7.1 \u00a0A historical example: \u00a0The Euthyphro argument<\/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 a logical \u201cor\u201d differ from an exclusive \u201cor\u201d in natural language?<\/li>\r\n \t<li>Why might adding a disjunction to a proof be helpful?<\/li>\r\n \t<li>What is the value of proving something by eliminating all other possibilities?<\/li>\r\n \t<li>When using disjunction elimination, what must be shown about each disjunct?<\/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>Disjunction (\u2228)<\/strong> - a compound statement formed by combining two or more simpler statements with the logical connective \"or.\"<\/li>\r\n \t<li><strong>Disjunction Introduction (\u2228I)<\/strong> - a rule of inference that allows you to add a disjunction (using the \"or\" operator, symbolized as \"\u2228\") to a true statement.<\/li>\r\n \t<li><strong>Disjunction Elimination (\u2228E)<\/strong> - a rule of inference that allows you to deduce a conclusion from a disjunction (an \"or\" statement) if you can prove that the conclusion follows from each of the disjuncts separately.<\/li>\r\n \t<li><strong>Exhaustive Possibilities<\/strong> - refer to a set of potential outcomes or conditions that collectively cover all conceivable scenarios within a given context.<\/li>\r\n \t<li><strong>Branching Subproofs<\/strong> - smaller, temporary proofs nested within a larger proof. They are used to explore the consequences of an assumption.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nThe philosopher Plato (who lived from approximately 427 BC to 347 BC) wrote a series of great philosophical texts. \u00a0Plato was the first philosopher to deploy argument in a vigorous and consistent way, and in so doing he showed how philosophy takes logic as its essential method. \u00a0We think of Plato as the principal\u00a0founder of Western philosophy. \u00a0The American philosopher Alfred Whitehead (1861-1947) in fact once famously quipped that philosophy is a \u201cseries of footnotes to Plato\u201d.\r\n\r\nPlato\u2019s teacher was Socrates (c. 469-399 B.C.), a gadfly of ancient Athens who made many enemies by showing people how little they knew. \u00a0Socrates did not write anything, but most of Plato\u2019s writings are dialogues, which are like small plays, in which Socrates is the protagonist of the philosophical drama that ensues. \u00a0Several of the dialogues are named after the person who will be seen arguing with Socrates. \u00a0In the dialogue <span class=\"em\">Euthyphro,<\/span>\u00a0Socrates is standing in line, awaiting his trial. \u00a0He has been accused of corrupting the youth of Athens. \u00a0A trial in ancient Athens was essentially a debate before the assembled citizen men of the city. \u00a0Before Socrates in line is a young man, Euthyphro. \u00a0Socrates asks Euthyphro what his business is that day, and Euthyphro proudly proclaims he is there to charge his own father with murder. \u00a0Socrates is shocked. \u00a0In ancient Athens, respect for one\u2019s father was highly valued and expected. \u00a0Socrates, with characteristic sarcasm, tells Euthyphro that he must be very wise to be so confident. \u00a0Here are two profound and conflicting duties: \u00a0to respect one\u2019s father, and to punish murder. \u00a0Euthyphro seems to find it very easy to decide which is the greater duty. \u00a0Euthyphro is not bothered. \u00a0To him, these ethical matters are simple: \u00a0one should be pious. \u00a0When Socrates demands a definition of piety that applies to all pious acts, Euthyphro says,\r\n<p id=\"h.30j0zll\" class=\"marg-left\" style=\"padding-left: 120px;\">Piety is that which is loved by the gods and impiety is that which is not loved by them.<\/p>\r\nSocrates observes that this is ambiguous. \u00a0It could mean, an act is good because the gods love that act. \u00a0Or it could mean, the gods love an act because it is good. \u00a0We have, then, an \u201cor\u201d statement, which logicians call a \u201c<strong>disjunction<\/strong>\u201d:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\r\nMight the former be true? \u00a0This view\u2014that an act is good because the gods love it\u2014is now called \u201cdivine command theory\u201d, and theists have disagreed since Socrates\u2019s time about whether it is true. \u00a0But, Socrates finds it absurd. \u00a0For, if tomorrow the gods love, say, murder, then, tomorrow murder would be good.\r\n\r\nEuthyphro comes to agree that it cannot be that an act is good because the gods love that act. \u00a0Our argument so far has this form:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that an act is good because the gods love it.<\/p>\r\nSocrates concludes that the gods love an act because it is good.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that an act is good because the gods love it.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">_____<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The gods love an act because it is good.<\/p>\r\nThis argument is one of the most important arguments in philosophy. \u00a0Most philosophers consider some version of this argument both valid and sound. \u00a0Some who disagree with it bite the bullet and claim that if tomorrow God (most theistic philosophers alive today are monotheists) loved puppy torture, adultery, random acts of cruelty, pollution, and lying, these would all be good things. \u00a0(If you are inclined to say, \u201cThat is not fair, God would never love those things\u201d, then you have already agreed with Socrates. \u00a0For, the reason you believe that God would never love these kinds of acts is because these kinds of acts are bad. \u00a0But then, being bad or good is something independent of the love of God.) \u00a0But most philosophers agree with Socrates: \u00a0they find it absurd to believe that random acts of cruelty and other such acts could be good. \u00a0There is something inherently bad to these acts, they believe. \u00a0The importance of the Euthyphro argument is not that it helps illustrate that divine command theory is an enormously strange and costly position to hold (though that is an important outcome), but rather that the argument shows ethics can be studied independently of theology. \u00a0For, if there is something about acts that makes them good or bad independently of a god\u2019s will, then we do not have to study a god\u2019s will to study what makes those acts good or bad.\r\n\r\nOf course, many philosophers are atheists so they already believed this, but for most of philosophy\u2019s history, one was obliged to be a theist. \u00a0Even today, lay people tend to think of ethics as an extension of religion. \u00a0Philosophers believe instead that ethics is its own field of study. \u00a0The Euthyphro argument explains why, even if you are a theist, you can study ethics independently of studying theology.\r\n\r\nBut is Socrates\u2019s argument valid? \u00a0Is it sound?\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> Create your own mini-lesson explaining the Euthyphro argument.<\/li>\r\n \t<li><strong>Critical Thinking Task:\u00a0<\/strong>Do you agree or disagree with the \"divine command\" theory? Please select an image, song, or movie clip to support your argument.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>7.2 \u00a0The disjunction<\/h2>\r\nWe want to extend our language so that it can represent sentences that contain an \u201cor\u201d. \u00a0Sentences like\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Anthony will go to Berlin or Paris.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">We have coffee or tea.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">This web page contains the phrase \u201cMark Twain\u201d or \u201cSamuel Clemens.\u201d<\/p>\r\nLogicians call these kinds of sentences \u201cdisjunctions\u201d. \u00a0Each of the two parts of a disjunction is called a \u201cdisjunct\u201d. \u00a0The idea is that these are really equivalent to the following sentences:\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Anthony will go to Berlin or Anthony will go to Paris.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">We have coffee or we have tea.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">This web page contains the phrase \u201cMark Twain\u201d or this web page contains the phrase \u201cSamuel Clemens.\u201d<\/p>\r\nWe can, therefore, see that (at least in many sentences) the \u201cor\u201d operates as a connective between two sentences.\r\n\r\nIt is traditional to use the symbol \u201c<strong><span class=\"strong\">v<\/span><\/strong>\u201d for \u201cor\u201d. \u00a0This comes from the Latin \u201cvel,\u201d meaning (in some contexts) <span class=\"em\">or.<\/span>\r\n\r\nThe syntax for the disjunction is very basic. \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\r\nis a sentence.\r\n\r\nThe semantics is a little more controversial. \u00a0This much of the defining truth table, most people find obvious:\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6\u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6v\u03a8)<\/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-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><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\">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-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nConsider: if I promise that I will bring you roses or lilacs, then it seems that I told the truth either if I have brought you roses but not lilacs, or if I brought you lilacs but not roses. \u00a0Similarly, the last row should be intuitive, also. \u00a0If I promise I will bring you roses or lilacs, and I bring you nothing, then I spoke falsely.\r\n\r\nWhat about the first row? \u00a0Many people who are not logicians want it to be the case that we define this condition as false. \u00a0The resulting meaning would correspond to what is sometimes called the \u201cexclusive \u2018or\u2019\u201d. \u00a0Logicians disagree. \u00a0They favor the definition where a disjunction is true if its two parts are true; this is sometimes called the \u201cinclusive \u2018or\u2019\u201d. \u00a0Of course, all that matters is that we pick a definition and stick with it, but we can offer some reasons why the \u201cinclusive \u2018or\u2019\u201d, as we call it, is more general than the \u201cexclusive \u2018or\u2019\u201d.\r\n\r\nConsider the first two sentences above. \u00a0It seems that the first sentence\u2014\u201cAnthony will go to Berlin or Paris\u201d\u2014should be true if Anthony goes to both. \u00a0Or consider the second sentence, \u201cWe have coffee or tea.\u201d \u00a0In most restaurants, this means they have both coffee and they have tea, but they expect that you will order only one of these. \u00a0After all, it would be strange to be told that they have coffee or tea, and then be told that it is false that they have both coffee and tea. \u00a0Or, similarly, suppose the waiter said, \u201cWe have coffee or tea\u201d, and then you said \u201cI\u2019ll have both\u201d, and the waiter replied \u201cWe don\u2019t have both\u201d. \u00a0This would seem strange. \u00a0But if you find it strange, then you implicitly agree that the disjunction should be interpreted as the inclusive \u201cor\u201d.\r\n\r\nExamples like these suggest to logicians that the inclusive \u201cor\u201d (where the first row of the table is true) is the default case, and that the context of our speech tells us when not both disjuncts are true. \u00a0For example, when a restaurant has a fixed price menu\u2014where you pay one fee and then get either steak or lobster\u2014it is understood by the context that this means you can have one or the other but not both. \u00a0But that is not logic, that is social custom. \u00a0One must know about restaurants to determine this.\r\n\r\nThus, it is customary to define the semantics of the disjunction as\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6\u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6v\u03a8)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><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\">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-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe haven\u2019t lost the ability to express the exclusive \u201cor\u201d. \u00a0We can say, \u201cone or the other but not both\u201d, which is expressed by the formula \u201c<strong><span class=\"strong\">((\u03a6 v \u03a8) ^ \u00ac(\u03a6 ^ \u03a8))<\/span><\/strong>\u201d. \u00a0To check, we can make the truth table for this complex expression:\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6 ^ \u03a8)<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">\u00ac(\u03a6 ^ \u03a8)<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">((\u03a6 v \u03a8) ^ \u00ac(\u03a6 ^ \u03a8))<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T\u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\r\n<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\"><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-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\r\n<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\"><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-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0<\/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\"><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-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\">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<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that this formula is equivalent to the exclusive \u201cor\u201d (it is true when <span class=\"strong\">\u03a6<\/span>\u00a0is true or <span class=\"strong\">\u03a8 <\/span>is true, but not when both are true or both are false). \u00a0So, if we need to say something like the exclusive \u201cor\u201d, we can do so.\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:<\/strong> Write a short story explaining the difference between social customs and logic in the concept of disjunction (v).<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>7.3 \u00a0Alternative forms<\/h2>\r\nThere do not seem to be many alternative expressions in English equivalent to the \u201cor\u201d. \u00a0We have\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>or <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>or <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\r\nThese are both expressed in our logic with <strong><span class=\"strong\">(P v Q)<\/span><\/strong>.\r\n\r\nOne expression that does arise in English is \u201cneither\u2026nor\u2026\u201d. \u00a0This expression seems best captured by simply making it into \u201cnot either\u2026 or\u2026\u201d. \u00a0Let\u2019s test this proposal. \u00a0Consider the sentence\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Neither Smith nor Jones will go to London.<\/p>\r\nThis sentence expresses the idea that Smith will not go to London, and that Jones will not go to London. \u00a0So, it would surely be a mistake to express it as\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either Smith will not go to London or Jones will not go to London.<\/p>\r\nWhy? \u00a0Because this latter sentence would be true if one of them went to London and one of them did not. \u00a0Consider the truth table for this expression to see this. \u00a0Use the following translation key.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0Smith will go to London.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0Jones will go to London.<\/p>\r\nThen suppose we did (wrongly) translate \u201cNeither Smith nor Jones will go to London\u201d with\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u00acP v \u00acQ)<\/span><\/strong><\/p>\r\nHere is the truth table for this expression.\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P\u00a0<\/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\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u00acP<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">(\u00acPv\u00acQ)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T\u00a0<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\r\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F\u00a0<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T\u00a0<\/span><\/strong><\/em><\/td>\r\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNote that this sentence is true if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is false, or if <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0In other words, it is true if one of the two goes to London. \u00a0That\u2019s not what we mean in English by that sentence claiming that neither of them will go to London.\r\n\r\nThe better translation is <strong><span class=\"strong\">\u00ac(PvQ)<\/span>.<\/strong>\r\n<table class=\"grid\">\r\n<tbody>\r\n<tr class=\"border-bottom\">\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border-right\"><span class=\"strong\">Q<\/span><\/th>\r\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(PvQ) \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n<th class=\"border\"><span class=\"strong\">\u00ac(PvQ)<\/span><\/th>\r\n<\/tr>\r\n<tr>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\r\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\r\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis captures the idea well: \u00a0it is only true if each does not go to London. \u00a0So, we can simply translate \u201cneither\u2026nor\u2026\u201d as \u201cIt is not the case that either\u2026 or\u2026\u201d.\r\n<h2>7.4 \u00a0Reasoning with disjunctions<\/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>Addition<\/strong> <strong>Rule<\/strong> -\u00a0 the principle that a sentence is true whether is it composed of the first or second disjunct.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nHow shall we reason with the disjunction? \u00a0Looking at the truth table that defines the disjunction, we find that we do not know much if we are told that, say, <strong><span class=\"strong\">(P v Q)<\/span><\/strong>. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>could be true, or it could be false. \u00a0The same is so for <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0All we know is that they cannot both be false.\r\n\r\nThis does suggest a reasonable and useful kind of inference rule. \u00a0If we have a disjunction, and we discover that half of it is false, then we know that the other half must be true. \u00a0This is true for either disjunct. \u00a0This means we have two rules, but we can group together both rules with a single name and treat them as one rule:\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00ac\u03a6<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a8<\/span><\/strong><\/p>\r\nand\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00ac\u03a8<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\r\nThis rule is traditionally called \u201cmodus tollendo ponens\u201d.\r\n\r\nWhat if we are required to show a disjunction? \u00a0One insight we can use is that if some sentence is true, then any disjunction that contains it is true. \u00a0This is so whether the sentence makes up the first or second disjunct. \u00a0Again, then, we would have two rules, which we can group together under one name:\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\r\nand\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a8<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\r\nThis rule is often called \u201caddition\u201d.\r\n\r\nThe addition rule often confuses students. \u00a0It seems to be a cheat, as if we are getting away with something for free. \u00a0But a moment of reflection will help clarify that just the opposite is true. \u00a0We lose information when we use the addition rule. \u00a0If you ask me where John is, and I say, \u201cJohn is in New York\u201d, I told you more than if I answered you, \u201cJohn is either in New York or in New Jersey\u201d. \u00a0Just so, when we go from some sentence <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>to <strong><span class=\"strong\">(PvQ)<\/span><\/strong>, we did not get something for free.\r\n\r\nThis rule does have the seemingly odd consequence that from, say, 2+2=4 you can derive that either 2+2=4 or 7=0. \u00a0But that only seems odd because in normal speech, we have a number of implicit rules. \u00a0The philosopher Paul Grice (1913-1988) described some of these rules, and we sometimes call the rules he described \u201cGrice\u2019s Maxims\u201d.<sup class=\"super\"><a id=\"ftnt_ref9\" href=\"#ftnt9\">[9]<\/a><\/sup>\u00a0 He observed that in conversation we expect people to give all the information required but not more; to try to be truthful; to say things that are relevant; and to be clear and brief and orderly. \u00a0So, in normal English conversations, if someone says, \u201cChen is in New York or New Jersey,\u201d they would be breaking the rule to give enough information, and to say what is relevant, if they knew that Chen was in New York. \u00a0This also means that we expect people to use a disjunction when they have reason to believe that either or both disjuncts could be true. \u00a0But our logical language is designed only to be precise, and we have been making the language precise by specifying when a sentence is true or false, and by specifying the relations between sentences in terms of their truth values. \u00a0We are thus not representing, and not putting into our language, Grice\u2019s maxims of conversation. \u00a0It remains true that if you knew Chen is in New York, but answered my question \u201cWhere is Chen?\u201d by saying \u201cChen is in New York or New Jersey\u201d, then you have wasted my time. \u00a0But you did not say something false.\r\n\r\nWe are now in a position to test Socrates\u2019s argument. \u00a0Using the following translation key, we can translate the argument into symbolic form.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0An act is good because the gods love that act.<\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0The gods love an act because it is good.<\/p>\r\nEuthyphro had argued\r\n\r\n<img class=\"size-medium wp-image-357 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-300x55.png\" alt=\"\" width=\"300\" height=\"55\" \/>\r\n\r\nSocrates had got Euthryphro to admit that\r\n\r\n<img class=\"size-medium wp-image-358 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-300x69.png\" alt=\"\" width=\"300\" height=\"69\" \/>\r\n\r\nAnd so we have a simple direct derivation:\r\n\r\n<img class=\" wp-image-359 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-300x53.png\" alt=\"\" width=\"538\" height=\"95\" \/>\r\n\r\nSocrates\u2019s argument is valid. \u00a0I will leave it up to you to determine whether Socrates\u2019s argument is sound.\r\n\r\nAnother example might be helpful. \u00a0Here is an argument in our logical language.\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P v Q)<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00acP<\/span><\/strong><\/p>\r\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00a0(\u00acP <\/span><span class=\"strong\">\u2192 (Q \u2192 R))<\/span><\/strong><\/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\">(R v S)<\/span><\/strong><\/p>\r\nThis will make use of the addition rule, and so is useful to illustrating that rule\u2019s application. \u00a0Here is one possible proof.\r\n\r\n<img class=\" wp-image-360 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-300x90.png\" alt=\"\" width=\"427\" height=\"128\" \/>\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>Design a short lesson explaining the \"modus tollendo ponens\" rule.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<h2>7.5 \u00a0Alternative symbolizations of disjunction<\/h2>\r\nWe are fortunate that there have been no popular alternatives to the use of \u201c<strong><span class=\"strong\">v<\/span><\/strong>\u201d as a symbol for disjunction. \u00a0Perhaps the second most widely used alternative symbol was \u201c<strong><span class=\"strong\">||<\/span><\/strong>\u201d, such that <strong><span class=\"strong\">(P v Q)<\/span><\/strong>\u00a0would be symbolized:\r\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P || Q)<\/span><\/strong><\/p>\r\n\r\n<h2>7.6 \u00a0Problems<\/h2>\r\n<ol class=\"lst-kix_list_17-0\" start=\"1\">\r\n \t<li>Prove the following using a derivation.\r\n<ol class=\"lst-kix_list_17-0\" start=\"1\">\r\n \t<li>Premises: <strong><span class=\"strong\">(PvQ)<\/span>, <span class=\"strong\">(Q \u2192 S)<\/span>, <span class=\"strong\">(\u00acS^T)<\/span><\/strong>. Conclusion: \u00a0<strong><span class=\"strong\">(T^P)<\/span><\/strong>.<\/li>\r\n \t<li>Premises: <strong><span class=\"strong\">((P \u2192 \u00acQ) ^ (R \u2192 S))<\/span>, <span class=\"strong\">(Q v R)<\/span><\/strong>. \u00a0Conclusion: \u00a0<strong><span class=\"strong\">(P \u2192 S)<\/span><\/strong>.<\/li>\r\n \t<li>Premises: <strong><span class=\"strong\">((P^Q) v R)<\/span>, <span class=\"strong\">((P^Q) \u2192 S)<\/span>, <span class=\"strong\">\u00acS<\/span><\/strong>. \u00a0Conclusion: \u00a0<strong><span class=\"strong\">R<\/span><\/strong>.<\/li>\r\n \t<li>Premises: <strong><span class=\"strong\">(RvS)<\/span>, <span class=\"strong\">((S \u2192 T) ^ V)<\/span>, <span class=\"strong\">\u00acT<\/span>, <span class=\"strong\">((R^V) \u2192 P)<\/span><\/strong>. \u00a0Conclusion: <strong>\u00a0<span class=\"strong\">(PvQ)<\/span><\/strong>.<\/li>\r\n \t<li>Premises: <strong><span class=\"strong\">((P \u2192 Q) v (\u00acR \u2192 S))<\/span>, <span class=\"strong\">((P \u2192 Q) \u2192 T)<\/span>, <span class=\"strong\">(\u00acT ^ \u00acS)<\/span><\/strong>. \u00a0Conclusion: <strong><span class=\"strong\">(R v V)<\/span><\/strong>.<\/li>\r\n \t<li>Premises: <strong><span class=\"strong\">(P v S)<\/span>, <span class=\"strong\">(T \u2192 \u00acS)<\/span>, <span class=\"strong\">T<\/span><\/strong>. \u00a0Conclusion: <strong><span class=\"strong\">((P v Q) v R)<\/span><\/strong>.<\/li>\r\n \t<li>Conclusion:<span class=\"strong\">\u00a0<strong>(P \u2192 (PvQ))<\/strong><\/span>.<\/li>\r\n \t<li>Conclusion:<span class=\"strong\">\u00a0<strong>((PvQ) \u2192 (\u00acP \u2192 Q))<\/strong><\/span>.<\/li>\r\n \t<li>Conclusion:<strong><span class=\"strong\">\u00a0((PvQ) \u2192 (\u00acQ \u2192 P))<\/span><\/strong>.<\/li>\r\n \t<li>Conclusion:<span class=\"strong\">\u00a0<strong>(((PvQ) ^ (\u00acQ v \u00acR)) \u2192 (R \u2192 P))<\/strong><\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Consider the following four cards in figure 7.1. \u00a0Each card has a letter on one side, and a shape on the other side.<\/li>\r\n<\/ol>\r\n[caption id=\"attachment_130\" align=\"aligncenter\" width=\"719\"]<img class=\"wp-image-130 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-star-triangle-q-diamond-e1467752349488.png\" alt=\"card1: star card 2: triangle card 3: q card 4: diamond\" width=\"719\" height=\"329\" \/> Figure 7.1[\/caption]\r\n\r\nFor each of the following claims, determine (1) the minimum number of cards you must turn over to check the claim, and (2) what those cards are, in order to determine if the claim is true of all four cards.\r\n<ol class=\"lower-alpha\">\r\n \t<li>If there is a <strong>P<\/strong> or <strong>Q<\/strong> on the letter side of the card, then there is a diamond on the shape side of the card.<\/li>\r\n \t<li>If there is a <strong>Q<\/strong> on the letter side of the card, then there is either a diamond or a star on the shape side of the card.<\/li>\r\n<\/ol>\r\n<ol class=\"lst-kix_list_17-0\" start=\"3\">\r\n \t<li>In normal colloquial English, write your own valid argument with at least two premises, at least one of which is a disjunction. 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<\/ol>\r\n<ol class=\"lst-kix_list_17-0 start\" start=\"4\">\r\n \t<li>Translate the following passage into our propositional logic. \u00a0Prove the argument is valid.<\/li>\r\n<\/ol>\r\n<p class=\"marg-left\">Either Dr. Kronecker or Bishop Berkeley killed Colonel Cardinality. \u00a0If Dr. Kronecker killed Colonel Cardinality, then Dr. Kronecker was in the kitchen. If Bishop Berkeley killed Colonel Cardinality, then he was in the drawing room. If Bishop Berkeley was in the drawing room, then he was wearing boots. But Bishop Berkeley was not wearing boots. So, Dr. Kronecker killed the Colonel.<\/p>\r\n\r\n<ol class=\"lst-kix_list_17-0\" start=\"5\">\r\n \t<li>Translate the following passage into our propositional logic. \u00a0Prove the argument is valid.<\/li>\r\n<\/ol>\r\n<p class=\"marg-left\">Either Wittgenstein or Meinong stole the diamonds. If Meinong stole the diamonds, then he was in the billiards room. But if Meinong was in the library, then he was not in the billiards room. Therefore, if Meinong was in the library, Wittgenstein stole the diamonds.<\/p>\r\n\r\n<div>\r\n\r\n<hr \/>\r\n\r\n<a id=\"ftnt9\" href=\"#ftnt_ref9\">[9]<\/a>\u00a0Grice (1975).\r\n\r\n<\/div>","rendered":"<h2>7.1 \u00a0A historical example: \u00a0The Euthyphro argument<\/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 a logical \u201cor\u201d differ from an exclusive \u201cor\u201d in natural language?<\/li>\n<li>Why might adding a disjunction to a proof be helpful?<\/li>\n<li>What is the value of proving something by eliminating all other possibilities?<\/li>\n<li>When using disjunction elimination, what must be shown about each disjunct?<\/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>Disjunction (\u2228)<\/strong> &#8211; a compound statement formed by combining two or more simpler statements with the logical connective &#8220;or.&#8221;<\/li>\n<li><strong>Disjunction Introduction (\u2228I)<\/strong> &#8211; a rule of inference that allows you to add a disjunction (using the &#8220;or&#8221; operator, symbolized as &#8220;\u2228&#8221;) to a true statement.<\/li>\n<li><strong>Disjunction Elimination (\u2228E)<\/strong> &#8211; a rule of inference that allows you to deduce a conclusion from a disjunction (an &#8220;or&#8221; statement) if you can prove that the conclusion follows from each of the disjuncts separately.<\/li>\n<li><strong>Exhaustive Possibilities<\/strong> &#8211; refer to a set of potential outcomes or conditions that collectively cover all conceivable scenarios within a given context.<\/li>\n<li><strong>Branching Subproofs<\/strong> &#8211; smaller, temporary proofs nested within a larger proof. They are used to explore the consequences of an assumption.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>The philosopher Plato (who lived from approximately 427 BC to 347 BC) wrote a series of great philosophical texts. \u00a0Plato was the first philosopher to deploy argument in a vigorous and consistent way, and in so doing he showed how philosophy takes logic as its essential method. \u00a0We think of Plato as the principal\u00a0founder of Western philosophy. \u00a0The American philosopher Alfred Whitehead (1861-1947) in fact once famously quipped that philosophy is a \u201cseries of footnotes to Plato\u201d.<\/p>\n<p>Plato\u2019s teacher was Socrates (c. 469-399 B.C.), a gadfly of ancient Athens who made many enemies by showing people how little they knew. \u00a0Socrates did not write anything, but most of Plato\u2019s writings are dialogues, which are like small plays, in which Socrates is the protagonist of the philosophical drama that ensues. \u00a0Several of the dialogues are named after the person who will be seen arguing with Socrates. \u00a0In the dialogue <span class=\"em\">Euthyphro,<\/span>\u00a0Socrates is standing in line, awaiting his trial. \u00a0He has been accused of corrupting the youth of Athens. \u00a0A trial in ancient Athens was essentially a debate before the assembled citizen men of the city. \u00a0Before Socrates in line is a young man, Euthyphro. \u00a0Socrates asks Euthyphro what his business is that day, and Euthyphro proudly proclaims he is there to charge his own father with murder. \u00a0Socrates is shocked. \u00a0In ancient Athens, respect for one\u2019s father was highly valued and expected. \u00a0Socrates, with characteristic sarcasm, tells Euthyphro that he must be very wise to be so confident. \u00a0Here are two profound and conflicting duties: \u00a0to respect one\u2019s father, and to punish murder. \u00a0Euthyphro seems to find it very easy to decide which is the greater duty. \u00a0Euthyphro is not bothered. \u00a0To him, these ethical matters are simple: \u00a0one should be pious. \u00a0When Socrates demands a definition of piety that applies to all pious acts, Euthyphro says,<\/p>\n<p id=\"h.30j0zll\" class=\"marg-left\" style=\"padding-left: 120px;\">Piety is that which is loved by the gods and impiety is that which is not loved by them.<\/p>\n<p>Socrates observes that this is ambiguous. \u00a0It could mean, an act is good because the gods love that act. \u00a0Or it could mean, the gods love an act because it is good. \u00a0We have, then, an \u201cor\u201d statement, which logicians call a \u201c<strong>disjunction<\/strong>\u201d:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\n<p>Might the former be true? \u00a0This view\u2014that an act is good because the gods love it\u2014is now called \u201cdivine command theory\u201d, and theists have disagreed since Socrates\u2019s time about whether it is true. \u00a0But, Socrates finds it absurd. \u00a0For, if tomorrow the gods love, say, murder, then, tomorrow murder would be good.<\/p>\n<p>Euthyphro comes to agree that it cannot be that an act is good because the gods love that act. \u00a0Our argument so far has this form:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that an act is good because the gods love it.<\/p>\n<p>Socrates concludes that the gods love an act because it is good.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either an act is good because the gods love that act, or the gods love an act because it is good.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">It is not the case that an act is good because the gods love it.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">_____<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">The gods love an act because it is good.<\/p>\n<p>This argument is one of the most important arguments in philosophy. \u00a0Most philosophers consider some version of this argument both valid and sound. \u00a0Some who disagree with it bite the bullet and claim that if tomorrow God (most theistic philosophers alive today are monotheists) loved puppy torture, adultery, random acts of cruelty, pollution, and lying, these would all be good things. \u00a0(If you are inclined to say, \u201cThat is not fair, God would never love those things\u201d, then you have already agreed with Socrates. \u00a0For, the reason you believe that God would never love these kinds of acts is because these kinds of acts are bad. \u00a0But then, being bad or good is something independent of the love of God.) \u00a0But most philosophers agree with Socrates: \u00a0they find it absurd to believe that random acts of cruelty and other such acts could be good. \u00a0There is something inherently bad to these acts, they believe. \u00a0The importance of the Euthyphro argument is not that it helps illustrate that divine command theory is an enormously strange and costly position to hold (though that is an important outcome), but rather that the argument shows ethics can be studied independently of theology. \u00a0For, if there is something about acts that makes them good or bad independently of a god\u2019s will, then we do not have to study a god\u2019s will to study what makes those acts good or bad.<\/p>\n<p>Of course, many philosophers are atheists so they already believed this, but for most of philosophy\u2019s history, one was obliged to be a theist. \u00a0Even today, lay people tend to think of ethics as an extension of religion. \u00a0Philosophers believe instead that ethics is its own field of study. \u00a0The Euthyphro argument explains why, even if you are a theist, you can study ethics independently of studying theology.<\/p>\n<p>But is Socrates\u2019s argument valid? \u00a0Is it sound?<\/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> Create your own mini-lesson explaining the Euthyphro argument.<\/li>\n<li><strong>Critical Thinking Task:\u00a0<\/strong>Do you agree or disagree with the &#8220;divine command&#8221; theory? Please select an image, song, or movie clip to support your argument.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>7.2 \u00a0The disjunction<\/h2>\n<p>We want to extend our language so that it can represent sentences that contain an \u201cor\u201d. \u00a0Sentences like<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Anthony will go to Berlin or Paris.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">We have coffee or tea.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">This web page contains the phrase \u201cMark Twain\u201d or \u201cSamuel Clemens.\u201d<\/p>\n<p>Logicians call these kinds of sentences \u201cdisjunctions\u201d. \u00a0Each of the two parts of a disjunction is called a \u201cdisjunct\u201d. \u00a0The idea is that these are really equivalent to the following sentences:<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Anthony will go to Berlin or Anthony will go to Paris.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">We have coffee or we have tea.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">This web page contains the phrase \u201cMark Twain\u201d or this web page contains the phrase \u201cSamuel Clemens.\u201d<\/p>\n<p>We can, therefore, see that (at least in many sentences) the \u201cor\u201d operates as a connective between two sentences.<\/p>\n<p>It is traditional to use the symbol \u201c<strong><span class=\"strong\">v<\/span><\/strong>\u201d for \u201cor\u201d. \u00a0This comes from the Latin \u201cvel,\u201d meaning (in some contexts) <span class=\"em\">or.<\/span><\/p>\n<p>The syntax for the disjunction is very basic. \u00a0If <strong><span class=\"strong\">\u03a6<\/span>\u00a0<\/strong>and <strong><span class=\"strong\">\u03a8<\/span>\u00a0<\/strong>are sentences, then<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\n<p>is a sentence.<\/p>\n<p>The semantics is a little more controversial. \u00a0This much of the defining truth table, most people find obvious:<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6\u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6v\u03a8)<\/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-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><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\">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-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Consider: if I promise that I will bring you roses or lilacs, then it seems that I told the truth either if I have brought you roses but not lilacs, or if I brought you lilacs but not roses. \u00a0Similarly, the last row should be intuitive, also. \u00a0If I promise I will bring you roses or lilacs, and I bring you nothing, then I spoke falsely.<\/p>\n<p>What about the first row? \u00a0Many people who are not logicians want it to be the case that we define this condition as false. \u00a0The resulting meaning would correspond to what is sometimes called the \u201cexclusive \u2018or\u2019\u201d. \u00a0Logicians disagree. \u00a0They favor the definition where a disjunction is true if its two parts are true; this is sometimes called the \u201cinclusive \u2018or\u2019\u201d. \u00a0Of course, all that matters is that we pick a definition and stick with it, but we can offer some reasons why the \u201cinclusive \u2018or\u2019\u201d, as we call it, is more general than the \u201cexclusive \u2018or\u2019\u201d.<\/p>\n<p>Consider the first two sentences above. \u00a0It seems that the first sentence\u2014\u201cAnthony will go to Berlin or Paris\u201d\u2014should be true if Anthony goes to both. \u00a0Or consider the second sentence, \u201cWe have coffee or tea.\u201d \u00a0In most restaurants, this means they have both coffee and they have tea, but they expect that you will order only one of these. \u00a0After all, it would be strange to be told that they have coffee or tea, and then be told that it is false that they have both coffee and tea. \u00a0Or, similarly, suppose the waiter said, \u201cWe have coffee or tea\u201d, and then you said \u201cI\u2019ll have both\u201d, and the waiter replied \u201cWe don\u2019t have both\u201d. \u00a0This would seem strange. \u00a0But if you find it strange, then you implicitly agree that the disjunction should be interpreted as the inclusive \u201cor\u201d.<\/p>\n<p>Examples like these suggest to logicians that the inclusive \u201cor\u201d (where the first row of the table is true) is the default case, and that the context of our speech tells us when not both disjuncts are true. \u00a0For example, when a restaurant has a fixed price menu\u2014where you pay one fee and then get either steak or lobster\u2014it is understood by the context that this means you can have one or the other but not both. \u00a0But that is not logic, that is social custom. \u00a0One must know about restaurants to determine this.<\/p>\n<p>Thus, it is customary to define the semantics of the disjunction as<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6\u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6v\u03a8)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><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\">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-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>We haven\u2019t lost the ability to express the exclusive \u201cor\u201d. \u00a0We can say, \u201cone or the other but not both\u201d, which is expressed by the formula \u201c<strong><span class=\"strong\">((\u03a6 v \u03a8) ^ \u00ac(\u03a6 ^ \u03a8))<\/span><\/strong>\u201d. \u00a0To check, we can make the truth table for this complex expression:<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u03a6 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">\u03a8<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(\u03a6 ^ \u03a8)<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">\u00ac(\u03a6 ^ \u03a8)<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">((\u03a6 v \u03a8) ^ \u00ac(\u03a6 ^ \u03a8))<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T\u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T<\/span><\/td>\n<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\"><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-right\"><span class=\"em strong\">F<\/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<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-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0<\/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\"><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-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\">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<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that this formula is equivalent to the exclusive \u201cor\u201d (it is true when <span class=\"strong\">\u03a6<\/span>\u00a0is true or <span class=\"strong\">\u03a8 <\/span>is true, but not when both are true or both are false). \u00a0So, if we need to say something like the exclusive \u201cor\u201d, we can do so.<\/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:<\/strong> Write a short story explaining the difference between social customs and logic in the concept of disjunction (v).<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>7.3 \u00a0Alternative forms<\/h2>\n<p>There do not seem to be many alternative expressions in English equivalent to the \u201cor\u201d. \u00a0We have<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span>\u00a0<\/strong>or <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>or <strong><span class=\"strong\">Q<\/span><\/strong><\/p>\n<p>These are both expressed in our logic with <strong><span class=\"strong\">(P v Q)<\/span><\/strong>.<\/p>\n<p>One expression that does arise in English is \u201cneither\u2026nor\u2026\u201d. \u00a0This expression seems best captured by simply making it into \u201cnot either\u2026 or\u2026\u201d. \u00a0Let\u2019s test this proposal. \u00a0Consider the sentence<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Neither Smith nor Jones will go to London.<\/p>\n<p>This sentence expresses the idea that Smith will not go to London, and that Jones will not go to London. \u00a0So, it would surely be a mistake to express it as<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\">Either Smith will not go to London or Jones will not go to London.<\/p>\n<p>Why? \u00a0Because this latter sentence would be true if one of them went to London and one of them did not. \u00a0Consider the truth table for this expression to see this. \u00a0Use the following translation key.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0Smith will go to London.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0Jones will go to London.<\/p>\n<p>Then suppose we did (wrongly) translate \u201cNeither Smith nor Jones will go to London\u201d with<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u00acP v \u00acQ)<\/span><\/strong><\/p>\n<p>Here is the truth table for this expression.<\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P\u00a0<\/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\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">\u00acP<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">(\u00acPv\u00acQ)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T\u00a0<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>T<\/strong><\/em><\/td>\n<td class=\"border\"><em><strong>F<\/strong><\/em><\/td>\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F\u00a0<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">F<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<td class=\"border-right\" colspan=\"1\" rowspan=\"1\"><em><strong><span class=\"em strong\">T\u00a0<\/span><\/strong><\/em><\/td>\n<td class=\"border\"><em><strong><span class=\"em strong\">T<\/span><\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Note that this sentence is true if <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is false, or if <strong><span class=\"strong\">Q<\/span>\u00a0<\/strong>is true and <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>is false. \u00a0In other words, it is true if one of the two goes to London. \u00a0That\u2019s not what we mean in English by that sentence claiming that neither of them will go to London.<\/p>\n<p>The better translation is <strong><span class=\"strong\">\u00ac(PvQ)<\/span>.<\/strong><\/p>\n<table class=\"grid\">\n<tbody>\n<tr class=\"border-bottom\">\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border-right\"><span class=\"strong\">Q<\/span><\/th>\n<th class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"strong\">(PvQ) \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n<th class=\"border\"><span class=\"strong\">\u00ac(PvQ)<\/span><\/th>\n<\/tr>\n<tr>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">T<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">T \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0\u00a0<\/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 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border-right\"><span class=\"em strong\">F<\/span><\/td>\n<td class=\"border\" colspan=\"1\" rowspan=\"1\"><span class=\"em strong\">F \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/td>\n<td class=\"border\"><span class=\"em strong\">T<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>This captures the idea well: \u00a0it is only true if each does not go to London. \u00a0So, we can simply translate \u201cneither\u2026nor\u2026\u201d as \u201cIt is not the case that either\u2026 or\u2026\u201d.<\/p>\n<h2>7.4 \u00a0Reasoning with disjunctions<\/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>Addition<\/strong> <strong>Rule<\/strong> &#8211;\u00a0 the principle that a sentence is true whether is it composed of the first or second disjunct.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<p>How shall we reason with the disjunction? \u00a0Looking at the truth table that defines the disjunction, we find that we do not know much if we are told that, say, <strong><span class=\"strong\">(P v Q)<\/span><\/strong>. \u00a0<strong><span class=\"strong\">P<\/span>\u00a0<\/strong>could be true, or it could be false. \u00a0The same is so for <strong><span class=\"strong\">Q<\/span><\/strong>. \u00a0All we know is that they cannot both be false.<\/p>\n<p>This does suggest a reasonable and useful kind of inference rule. \u00a0If we have a disjunction, and we discover that half of it is false, then we know that the other half must be true. \u00a0This is true for either disjunct. \u00a0This means we have two rules, but we can group together both rules with a single name and treat them as one rule:<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00ac\u03a6<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a8<\/span><\/strong><\/p>\n<p>and<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00ac\u03a8<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\n<p>This rule is traditionally called \u201cmodus tollendo ponens\u201d.<\/p>\n<p>What if we are required to show a disjunction? \u00a0One insight we can use is that if some sentence is true, then any disjunction that contains it is true. \u00a0This is so whether the sentence makes up the first or second disjunct. \u00a0Again, then, we would have two rules, which we can group together under one name:<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a6<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\n<p>and<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u03a8<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">_____<\/span><\/strong><\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(\u03a6 v \u03a8)<\/span><\/strong><\/p>\n<p>This rule is often called \u201caddition\u201d.<\/p>\n<p>The addition rule often confuses students. \u00a0It seems to be a cheat, as if we are getting away with something for free. \u00a0But a moment of reflection will help clarify that just the opposite is true. \u00a0We lose information when we use the addition rule. \u00a0If you ask me where John is, and I say, \u201cJohn is in New York\u201d, I told you more than if I answered you, \u201cJohn is either in New York or in New Jersey\u201d. \u00a0Just so, when we go from some sentence <strong><span class=\"strong\">P<\/span>\u00a0<\/strong>to <strong><span class=\"strong\">(PvQ)<\/span><\/strong>, we did not get something for free.<\/p>\n<p>This rule does have the seemingly odd consequence that from, say, 2+2=4 you can derive that either 2+2=4 or 7=0. \u00a0But that only seems odd because in normal speech, we have a number of implicit rules. \u00a0The philosopher Paul Grice (1913-1988) described some of these rules, and we sometimes call the rules he described \u201cGrice\u2019s Maxims\u201d.<sup class=\"super\"><a id=\"ftnt_ref9\" href=\"#ftnt9\">[9]<\/a><\/sup>\u00a0 He observed that in conversation we expect people to give all the information required but not more; to try to be truthful; to say things that are relevant; and to be clear and brief and orderly. \u00a0So, in normal English conversations, if someone says, \u201cChen is in New York or New Jersey,\u201d they would be breaking the rule to give enough information, and to say what is relevant, if they knew that Chen was in New York. \u00a0This also means that we expect people to use a disjunction when they have reason to believe that either or both disjuncts could be true. \u00a0But our logical language is designed only to be precise, and we have been making the language precise by specifying when a sentence is true or false, and by specifying the relations between sentences in terms of their truth values. \u00a0We are thus not representing, and not putting into our language, Grice\u2019s maxims of conversation. \u00a0It remains true that if you knew Chen is in New York, but answered my question \u201cWhere is Chen?\u201d by saying \u201cChen is in New York or New Jersey\u201d, then you have wasted my time. \u00a0But you did not say something false.<\/p>\n<p>We are now in a position to test Socrates\u2019s argument. \u00a0Using the following translation key, we can translate the argument into symbolic form.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">P<\/span><\/strong>: \u00a0An act is good because the gods love that act.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">Q<\/span><\/strong>: \u00a0The gods love an act because it is good.<\/p>\n<p>Euthyphro had argued<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-357 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-300x55.png\" alt=\"\" width=\"300\" height=\"55\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-300x55.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-65x12.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-225x41.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653-350x64.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115653.png 569w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>Socrates had got Euthryphro to admit that<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-358 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-300x69.png\" alt=\"\" width=\"300\" height=\"69\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-300x69.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-65x15.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-225x52.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713-350x81.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115713.png 591w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><\/p>\n<p>And so we have a simple direct derivation:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-359 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-300x53.png\" alt=\"\" width=\"538\" height=\"95\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-300x53.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-768x135.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-65x11.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-225x40.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735-350x61.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115735.png 774w\" sizes=\"auto, (max-width: 538px) 100vw, 538px\" \/><\/p>\n<p>Socrates\u2019s argument is valid. \u00a0I will leave it up to you to determine whether Socrates\u2019s argument is sound.<\/p>\n<p>Another example might be helpful. \u00a0Here is an argument in our logical language.<\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P v Q)<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00acP<\/span><\/strong><\/p>\n<p class=\"marg-left\" style=\"padding-left: 120px;\"><strong><span class=\"strong\">\u00a0(\u00acP <\/span><span class=\"strong\">\u2192 (Q \u2192 R))<\/span><\/strong><\/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\">(R v S)<\/span><\/strong><\/p>\n<p>This will make use of the addition rule, and so is useful to illustrating that rule\u2019s application. \u00a0Here is one possible proof.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-360 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-300x90.png\" alt=\"\" width=\"427\" height=\"128\" srcset=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-300x90.png 300w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-768x231.png 768w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-65x20.png 65w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-225x68.png 225w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758-350x105.png 350w, https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/Screenshot-2025-10-13-115758.png 794w\" sizes=\"auto, (max-width: 427px) 100vw, 427px\" \/><\/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>Design a short lesson explaining the &#8220;modus tollendo ponens&#8221; rule.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<h2>7.5 \u00a0Alternative symbolizations of disjunction<\/h2>\n<p>We are fortunate that there have been no popular alternatives to the use of \u201c<strong><span class=\"strong\">v<\/span><\/strong>\u201d as a symbol for disjunction. \u00a0Perhaps the second most widely used alternative symbol was \u201c<strong><span class=\"strong\">||<\/span><\/strong>\u201d, such that <strong><span class=\"strong\">(P v Q)<\/span><\/strong>\u00a0would be symbolized:<\/p>\n<p style=\"padding-left: 120px;\"><strong><span class=\"strong\">(P || Q)<\/span><\/strong><\/p>\n<h2>7.6 \u00a0Problems<\/h2>\n<ol class=\"lst-kix_list_17-0\" start=\"1\">\n<li>Prove the following using a derivation.\n<ol class=\"lst-kix_list_17-0\" start=\"1\">\n<li>Premises: <strong><span class=\"strong\">(PvQ)<\/span>, <span class=\"strong\">(Q \u2192 S)<\/span>, <span class=\"strong\">(\u00acS^T)<\/span><\/strong>. Conclusion: \u00a0<strong><span class=\"strong\">(T^P)<\/span><\/strong>.<\/li>\n<li>Premises: <strong><span class=\"strong\">((P \u2192 \u00acQ) ^ (R \u2192 S))<\/span>, <span class=\"strong\">(Q v R)<\/span><\/strong>. \u00a0Conclusion: \u00a0<strong><span class=\"strong\">(P \u2192 S)<\/span><\/strong>.<\/li>\n<li>Premises: <strong><span class=\"strong\">((P^Q) v R)<\/span>, <span class=\"strong\">((P^Q) \u2192 S)<\/span>, <span class=\"strong\">\u00acS<\/span><\/strong>. \u00a0Conclusion: \u00a0<strong><span class=\"strong\">R<\/span><\/strong>.<\/li>\n<li>Premises: <strong><span class=\"strong\">(RvS)<\/span>, <span class=\"strong\">((S \u2192 T) ^ V)<\/span>, <span class=\"strong\">\u00acT<\/span>, <span class=\"strong\">((R^V) \u2192 P)<\/span><\/strong>. \u00a0Conclusion: <strong>\u00a0<span class=\"strong\">(PvQ)<\/span><\/strong>.<\/li>\n<li>Premises: <strong><span class=\"strong\">((P \u2192 Q) v (\u00acR \u2192 S))<\/span>, <span class=\"strong\">((P \u2192 Q) \u2192 T)<\/span>, <span class=\"strong\">(\u00acT ^ \u00acS)<\/span><\/strong>. \u00a0Conclusion: <strong><span class=\"strong\">(R v V)<\/span><\/strong>.<\/li>\n<li>Premises: <strong><span class=\"strong\">(P v S)<\/span>, <span class=\"strong\">(T \u2192 \u00acS)<\/span>, <span class=\"strong\">T<\/span><\/strong>. \u00a0Conclusion: <strong><span class=\"strong\">((P v Q) v R)<\/span><\/strong>.<\/li>\n<li>Conclusion:<span class=\"strong\">\u00a0<strong>(P \u2192 (PvQ))<\/strong><\/span>.<\/li>\n<li>Conclusion:<span class=\"strong\">\u00a0<strong>((PvQ) \u2192 (\u00acP \u2192 Q))<\/strong><\/span>.<\/li>\n<li>Conclusion:<strong><span class=\"strong\">\u00a0((PvQ) \u2192 (\u00acQ \u2192 P))<\/span><\/strong>.<\/li>\n<li>Conclusion:<span class=\"strong\">\u00a0<strong>(((PvQ) ^ (\u00acQ v \u00acR)) \u2192 (R \u2192 P))<\/strong><\/span>.<\/li>\n<\/ol>\n<\/li>\n<li>Consider the following four cards in figure 7.1. \u00a0Each card has a letter on one side, and a shape on the other side.<\/li>\n<\/ol>\n<figure id=\"attachment_130\" aria-describedby=\"caption-attachment-130\" style=\"width: 719px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-130 size-full\" src=\"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-content\/uploads\/sites\/247\/2016\/12\/cards-star-triangle-q-diamond-e1467752349488.png\" alt=\"card1: star card 2: triangle card 3: q card 4: diamond\" width=\"719\" height=\"329\" \/><figcaption id=\"caption-attachment-130\" class=\"wp-caption-text\">Figure 7.1<\/figcaption><\/figure>\n<p>For each of the following claims, determine (1) the minimum number of cards you must turn over to check the claim, and (2) what those cards are, in order to determine if the claim is true of all four cards.<\/p>\n<ol class=\"lower-alpha\">\n<li>If there is a <strong>P<\/strong> or <strong>Q<\/strong> on the letter side of the card, then there is a diamond on the shape side of the card.<\/li>\n<li>If there is a <strong>Q<\/strong> on the letter side of the card, then there is either a diamond or a star on the shape side of the card.<\/li>\n<\/ol>\n<ol class=\"lst-kix_list_17-0\" start=\"3\">\n<li>In normal colloquial English, write your own valid argument with at least two premises, at least one of which is a disjunction. 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<\/ol>\n<ol class=\"lst-kix_list_17-0 start\" start=\"4\">\n<li>Translate the following passage into our propositional logic. \u00a0Prove the argument is valid.<\/li>\n<\/ol>\n<p class=\"marg-left\">Either Dr. Kronecker or Bishop Berkeley killed Colonel Cardinality. \u00a0If Dr. Kronecker killed Colonel Cardinality, then Dr. Kronecker was in the kitchen. If Bishop Berkeley killed Colonel Cardinality, then he was in the drawing room. If Bishop Berkeley was in the drawing room, then he was wearing boots. But Bishop Berkeley was not wearing boots. So, Dr. Kronecker killed the Colonel.<\/p>\n<ol class=\"lst-kix_list_17-0\" start=\"5\">\n<li>Translate the following passage into our propositional logic. \u00a0Prove the argument is valid.<\/li>\n<\/ol>\n<p class=\"marg-left\">Either Wittgenstein or Meinong stole the diamonds. If Meinong stole the diamonds, then he was in the billiards room. But if Meinong was in the library, then he was not in the billiards room. Therefore, if Meinong was in the library, Wittgenstein stole the diamonds.<\/p>\n<div>\n<hr \/>\n<p><a id=\"ftnt9\" href=\"#ftnt_ref9\">[9]<\/a>\u00a0Grice (1975).<\/p>\n<\/div>\n","protected":false},"author":158,"menu_order":7,"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-44","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\/44","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":14,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions"}],"predecessor-version":[{"id":361,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/44\/revisions\/361"}],"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\/44\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=44"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=44"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=44"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=44"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}