{"id":39,"date":"2016-12-16T18:14:05","date_gmt":"2016-12-16T18:14:05","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/5-and\/"},"modified":"2025-10-13T17:45:44","modified_gmt":"2025-10-13T17:45:44","slug":"5-and","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/introtologic\/chapter\/5-and\/","title":{"raw":"\u201cAnd\u201d","rendered":"\u201cAnd\u201d"},"content":{"raw":"

5.1 \u00a0The conjunction<\/h2>\r\n
\r\n

Pre-Reading Questions<\/p>\r\n\r\n<\/header>\r\n

\r\n
    \r\n \t
  1. What does it mean to \u201cconjoin\u201d two statements logically?<\/li>\r\n \t
  2. Why is it valid to infer a single component from a conjunction?<\/li>\r\n \t
  3. In what types of arguments do conjunctions commonly appear?<\/li>\r\n \t
  4. How might the use of conjunctions affect the length or structure of a proof?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n
    \r\n

    Key Terms<\/p>\r\n\r\n<\/header>\r\n

    \r\n
      \r\n \t
    • Conjunction (\u2227)<\/strong> - combines two statements, creating a new statement that is true only if both original statements are true.<\/li>\r\n \t
    • Conjunction Elimination (\u2227E)<\/strong> - a rule of inference that allows you to derive a single conjunct from a conjunction.<\/li>\r\n \t
    • Conjunction Introduction (\u2227I)<\/strong> - a rule of inference that allows you to combine two or more true statements (called conjuncts) into a single, true statement called a conjunction.<\/li>\r\n \t
    • Scope of a Rule<\/strong> - defines the portion of a logical expression or statement to which it applies.<\/li>\r\n \t
    • Well-Formed Formula (WFF)<\/strong> - a string of symbols that adheres to the syntactic rules of a formal language.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nTo make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language like English. \u00a0Our translations will not be exact, but they will be close enough that: first, we will have a way to more quickly understand the language we are constructing; and, second, we will have a way to speak English more precisely when that is required of us.\r\n\r\nConsider the following expressions. \u00a0How would we translate them into our logical language?\r\n

      Anthony will go to Berlin and Paris.<\/p>\r\n

      The number a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2 and 3.<\/p>\r\n

      Malik is from Colorado but not from Denver.<\/p>\r\nWe could translate each of these using an atomic sentence. \u00a0But then we would have lost\u2014or rather we would have hidden\u2014information that is clearly there in the English sentences. \u00a0We can capture this information by introducing a new connective; one that corresponds to our \u201cand\u201d.\r\n\r\nTo see this, consider whether you will agree that these sentences above are equivalent to the following sentences.\r\n

      Anthony will go to Berlin and Anthony will go to Paris.<\/p>\r\n

      The number a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2 and the number a<\/span>\u00a0<\/span><\/em><\/strong>is evenly divisible by 3.<\/p>\r\n

      Malik is from Colorado and it is not the case that Malik is from Denver.<\/p>\r\nOnce we grant that these sentences are equivalent to those above, we see that we can treat the \u201cand\u201d in each sentence as a truth functional connective.\r\n\r\nSuppose we assume the following key.\r\n

      P<\/span><\/strong>: \u00a0Anthony will go to Berlin.<\/p>\r\n

      Q<\/span><\/strong>: \u00a0Anthony will go to Paris.<\/p>\r\n

      R<\/span><\/strong>: \u00a0a<\/span>\u00a0<\/em><\/strong>\u00a0is evenly divisible by 2.<\/p>\r\n

      S<\/span><\/strong>: \u00a0a<\/span>\u00a0<\/em><\/strong>\u00a0is evenly divisible by 3.<\/p>\r\n

      T<\/span><\/strong>: \u00a0Malik is from Colorado<\/p>\r\n

      U<\/span><\/strong>: \u00a0Malik is from Denver.<\/p>\r\nA partial translation of these sentences would then be:\r\n

      P<\/span>\u00a0<\/strong>and Q<\/span><\/strong><\/p>\r\n

      R<\/span>\u00a0<\/strong>and S<\/span><\/strong><\/p>\r\n

      T<\/span>\u00a0<\/strong>and \u00acU<\/strong><\/span><\/p>\r\nOur third sentence above might generate some controversy. \u00a0How should we understand \u201cbut\u201d? \u00a0Consider that in terms of the truth value of the connected sentences, \u201cbut\u201d is the same as \u201cand\u201d. \u00a0That is, if you say \u201cP<\/span>\u00a0<\/strong>but\u00a0Q<\/span><\/strong>\u201d you are asserting that both P<\/span>\u00a0<\/strong>and Q<\/span>\u00a0<\/strong>are true. \u00a0However, in English there is extra meaning; the English \u201cbut\u201d seems to indicate that the additional sentence is unexpected or counter-intuitive. \u00a0\u201cP<\/span>\u00a0<\/strong>but Q<\/span><\/strong>\u201d seems to say, \u201cP<\/span>\u00a0<\/strong>is true, and you will find it surprising or unexpected that Q<\/span>\u00a0<\/strong>is true also.\u201d \u00a0That extra meaning is lost in our logic. \u00a0We will not be representing surprise or expectations. \u00a0So, we can treat \u201cbut\u201d as being the same as \u201cand\u201d. \u00a0This captures the truth value of the sentence formed using \u201cbut\u201d, which is all that we require of our logic.\r\n\r\nFollowing our method up until now, we want a symbol to stand for \u201cand\u201d. \u00a0In recent years the most commonly used symbol has been \u201c^<\/span>\u201d.\r\n\r\nThe syntax for \u201c^<\/span>\u201d is simple. \u00a0If \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span>\u00a0<\/strong>are sentences, then\r\n

      (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\r\nis a sentence. \u00a0Our translations of our three example sentences should thus look like this:\r\n

      (P<\/strong>^Q<\/strong>)<\/span><\/p>\r\n

      (R<\/strong>^S<\/strong>)<\/span><\/p>\r\n

      (T<\/strong>^\u00acU<\/strong>)<\/span><\/p>\r\nEach of these is called a \"conjunction<\/strong>.\" \u00a0The two parts of a conjunction are called \"conjuncts<\/strong>.\"\r\n\r\nThe semantics of the conjunction are given by its truth table. \u00a0Most people find the conjunction\u2019s semantics obvious. \u00a0If I claim that both \u03a6 <\/span><\/strong>and\u00a0\u03a8<\/strong><\/span>\u00a0<\/strong>are true, normal usage requires that if \u03a6<\/span>\u00a0<\/strong>is false or \u03a8<\/span>\u00a0<\/strong>is false, or both are false, then I spoke falsely also.\r\n\r\nConsider an example. \u00a0Suppose your employer says, \u201cAfter one year of employment you will get a raise and two weeks vacation\u201d. \u00a0A year passes. \u00a0Suppose now that this employer gives you a raise but no vacation, or a vacation but no raise, or neither a raise nor a vacation. \u00a0In each case, the employer has broken his promise. \u00a0The sentence forming the promise turned out to be false.\r\n\r\nThus, the semantics for the conjunction are given with the following truth table. \u00a0For any sentences \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span><\/strong>:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
      \u03a6 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\n\u03a8<\/span><\/th>\r\n(\u03a6^\u03a8)<\/span><\/th>\r\n<\/tr>\r\n
      T<\/span><\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
      T<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
      F<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
      F<\/span><\/strong><\/em><\/td>\r\nF\u00a0<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
      \r\n

      Check for Understanding<\/p>\r\n\r\n<\/header>\r\n

      \r\n
        \r\n \t
      1. Critical Thinking Task:\u00a0<\/strong>Why are conjunctions good tools for avoiding information being lost in translation?<\/li>\r\n \t
      2. Critical Thinking Task:\u00a0<\/strong>Ariel states, \"The words 'and' along with 'but' can have the same meaning in logical symbols.\" Her brother, Donovan, states, \"No, they cannot!\" How can Ariel demonstrate to her brother that the word \"and\" can serve the same meanings as \"but\" in formal logic?<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

        5.2 \u00a0Alternative phrasings, and a different \u201cand\u201d<\/h2>\r\nWe have noted that in English, \u201cbut\u201d is an alternative to \u201cand\u201d, and can be translated the same way in our propositional logic. \u00a0There are other phrases that have a similar meaning: they are best translated by conjunctions, but they convey (in English) a sense of surprise or failure of expectations. \u00a0For example, consider the following sentence.\r\n

        Even though they lost the battle, they won the war.<\/p>\r\nHere \u201ceven though\u201d seems to do the same work as \u201cbut\u201d. \u00a0The implication is that it is surprising\u2014that one might expect that if they lost the battle then they lost the war. \u00a0But, as we already noted, we will not capture expectations with our logic. \u00a0So, we would take this sentence to be sufficiently equivalent to:\r\n

        They lost the battle and they won the war.<\/p>\r\nWith the exception of \u201cbut\u201d, it seems in English there is no other single word that is an alternative to \u201cand\u201d that means the same thing. However, there are many ways that one can imply a conjunction. \u00a0To see this, consider the following sentences.\r\n

        Lee, who won the race, also won the championship.<\/p>\r\n

        The star Phosphorous, that we see in the morning, is the Evening Star.<\/p>\r\n

        The Evening Star, which is called \u201cHesperus\u201d, is also the Morning Star.<\/p>\r\n

        While Mateo is tall, Steve is not.<\/p>\r\n

        Dogs are vertebrate terrestrial mammals.<\/p>\r\nDepending on what elements we take as basic in our language, these sentences all include implied conjunctions. \u00a0They are equivalent to the following sentences, for example:\r\n

        Lee won the race and Lee won the championship.<\/p>\r\n

        Phosphorous is the star that we see in the morning and Phosphorous is the Evening Star.<\/p>\r\n

        The Evening Star is called \u201cHesperus\u201d and the Evening Star is the Morning Star.<\/p>\r\n

        Mateo is tall and it is not the case that Steve is tall.<\/p>\r\n

        Dogs are vertebrates and dogs are terrestrial and dogs are mammals.<\/p>\r\nThus, we need to be sensitive to complex sentences that are conjunctions but that do not use \u201cand\u201d or \u201cbut\u201d or phrases like \u201ceven though\u201d.\r\n\r\nUnfortunately, in English there are some uses of \u201cand\u201d that are not conjunctions. \u00a0The same is true for equivalent terms in some other natural languages. \u00a0Here is an example.\r\n

        Rochester is between Buffalo and Albany.<\/p>\r\nThe \u201cand\u201d in this sentence is not a conjunction. \u00a0To see this, note that this sentence is not equivalent to the following:\r\n

        Rochester is between Buffalo and Rochester is between Albany.<\/p>\r\nThat sentence is not even semantically correct. \u00a0What is happening in the original sentence?\r\n\r\nThe issue here is that \u201cis between\u201d is what we call a \u201cpredicate\u201d. \u00a0We will learn about predicates in chapter 11, but what we can say here is that some predicates take several names in order to form a sentence. \u00a0In English, if a predicate takes more than two names, then we typically use the \u201cand\u201d to combine names that are being described by that predicate. \u00a0In contrast, the conjunction in our propositional logic only combines sentences. \u00a0So, we must say that there are some uses of the English \u201cand\u201d that are not equivalent to our conjunction.\r\n\r\nThis could be confusing because sometimes in English we put \u201cand\u201d between names and there is an implied conjunction. \u00a0Consider:\r\n

        Elijah is older than Sophia and Max.<\/p>\r\nSuperficially, this looks to have the same structure as \u201cRochester is between Buffalo and Albany\u201d. \u00a0But this sentence really is equivalent to:\r\n

        Elijah is older than Max and Elijah is older than Sophia.<\/p>\r\nThe difference, however, is that there must be three things in order for one to be between the other two. \u00a0There need only be two things for one to be older than the other. \u00a0So, in the sentence \u201cRochester is between Buffalo and Albany\u201d, we need all three names (\u201cRochester\u201d, \u201cBuffalo\u201d, and \u201cAlbany) to make a single proper atomic sentence with \u201cbetween\u201d. \u00a0This tells us that the \u201cand\u201d is just being used to combine these names, and not to combine implied sentences (since there can be no implied sentence about what is \u201cbetween\u201d, using just two or just one of these names).\r\n\r\nThat sounds complex. \u00a0Do not despair, however. \u00a0The use of \u201cand\u201d to identify names being used by predicates is less common than \u201cand\u201d being used for a conjunction. \u00a0Also, after we discuss predicates in chapter 11, and after you have practiced translating different kinds of sentences, the distinction between these uses of \u201cand\u201d will become easy to identify in almost all cases. \u00a0In the meantime, we shall pick examples that do not invite this confusion.\r\n

        \r\n

        Check for Understanding<\/p>\r\n\r\n<\/header>\r\n

        \r\n
          \r\n \t
        1. Explain the difference between how \"and\" functions in \"Elijah is older than Max and Sophia\" versus \"Rochester is between Buffalo and Albany.\"<\/li>\r\n \t
        2. Create your own example of a sentence where \"and\" is used as a logical conjunction and another example where \"and\" is not functioning as a logical conjunction.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

          5.3 \u00a0Inference rules for conjunctions<\/h2>\r\n
          \r\n

          Key Terms<\/p>\r\n\r\n<\/header>\r\n

          \r\n
            \r\n \t
          • Simplification<\/strong> - a process that allows two rules to be applied by the same name.<\/li>\r\n \t
          • Adjunction Rule\u00a0<\/strong>- an alternative way of expressing the conjunction rule to avoid confusion.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nLooking at the truth table for the conjunction should tell us two things very clearly. \u00a0First, if a conjunction is true, what else must be true? \u00a0The obvious answer is that both of the parts, the conjuncts, must be true. \u00a0We can introduce a rule to capture this insight. \u00a0In fact, we can introduce two rules and call them by the same name, since the order of conjuncts does not affect their truth value. \u00a0These rules are often called \u201csimplification\u201d.\r\n

            (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\r\n

            _____<\/p>\r\n

            \u03a6<\/span><\/strong><\/p>\r\nAnd:\r\n

            (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\r\n

            _____<\/p>\r\n

            \u03a8<\/span><\/strong><\/p>\r\nIn other words, if (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, then \u03a6<\/span>\u00a0<\/strong>must be true; and if (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, then \u03a8<\/span>\u00a0<\/strong>must be true.\r\n\r\nWe can also introduce a rule to show a conjunction, based on what we see from the truth table. \u00a0That is, it is clear that there is only one kind of condition in which (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, and that is when \u03a6<\/span>\u00a0<\/strong>is true and when \u03a8<\/span>\u00a0<\/strong>is true. \u00a0This suggests the following rule:\r\n

            \u03a6<\/span><\/strong><\/p>\r\n

            \u03a8<\/span><\/strong><\/p>\r\n

            _____<\/p>\r\n

            (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\r\nWe might call this rule \u201cconjunction\u201d, but to avoid confusion with the name of the sentences, we will call this rule \u201cadjunction\u201d.\r\n

            5.4 \u00a0Reasoning with conjunctions<\/h2>\r\nIt would be helpful to consider some examples of reasoning with conjunctions. \u00a0Let\u2019s begin with an argument in a natural language.\r\n

            River and Emma will go to London. \u00a0If Emma goes to London, then she will ride the Eye. \u00a0River will ride the Eye too, provided that they go to London. \u00a0So, both Emma and River will ride the Eye.<\/p>\r\nWe need a translation key.\r\n

            T<\/span><\/strong>: \u00a0Emma will go to London.<\/p>\r\n

            S<\/span><\/strong>: \u00a0River will go to London.<\/p>\r\n

            U<\/span><\/strong>: \u00a0Emma will ride the Eye.<\/p>\r\n

            V<\/span><\/strong>: \u00a0River will ride the Eye.<\/p>\r\nThus our argument is:\r\n

            (T<\/strong>^S<\/strong>)<\/span><\/p>\r\n

            (T<\/strong>\u2192U<\/strong>)<\/span><\/p>\r\n

            (S<\/strong>\u2192V<\/strong>)<\/span><\/p>\r\n

            _____<\/span><\/strong><\/p>\r\n

            (U<\/strong>^V<\/strong>)<\/span><\/p>\r\nOur direct proof will look like this.\r\n\r\n\"\"\r\n\r\nNow an example using just our logical language. \u00a0Consider the following argument.\r\n

            (Q<\/strong>\u2192<\/span>\u00acS<\/strong>)<\/span><\/p>\r\n

            (P<\/strong><\/span>\u2192(Q<\/strong>^R<\/strong>))<\/span><\/p>\r\n

            (T<\/strong>\u2192<\/span>\u00ac<\/span>R<\/strong>)<\/span><\/p>\r\n

            P<\/span><\/strong><\/p>\r\n

            _____<\/span><\/strong><\/p>\r\n

            (<\/span>\u00ac<\/span>S<\/strong>^<\/span>\u00ac<\/span>T<\/strong>)<\/span><\/p>\r\nHere is one possible proof.\r\n\r\n\"\"\r\n

            \r\n

            Check for Understanding<\/p>\r\n\r\n<\/header>\r\n

            \r\n
              \r\n \t
            1. Explain how simplification works as an inference rule. What does it allow us to do with conjunctions?<\/li>\r\n \t
            2. Create your own natural language argument that would have the same logical structure as the first example.<\/li>\r\n \t
            3. Identify a real-world situation where reasoning similar to the second proof might be used to draw a conclusion.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

              5.5 \u00a0Alternative symbolizations for the conjunction<\/h2>\r\nAlternative notations for the conjunction include the symbols \u201c&<\/span><\/strong>\u201d and the symbol \u201c\u2219\u201d. \u00a0Thus, the expression (P<\/strong>^Q<\/strong>)<\/span>\u00a0would be written in these different styles, as:\r\n

              (P<\/strong>&Q<\/strong>)<\/span><\/p>\r\n

              (P<\/strong>\u2219Q<\/strong>)<\/span><\/p>\r\n\r\n

              5.6 Complex sentences<\/h2>\r\n
              \r\n

              Key Terms<\/p>\r\n\r\n<\/header>\r\n

              \r\n
                \r\n \t
              • Main Connective<\/strong> - the connective that combines two parts to allow two sentences to serve as constituents with a single connective.<\/li>\r\n \t
              • Equivalent<\/strong> - two or more sentences have the same value in a proof.<\/li>\r\n \t
              • Tautology<\/strong> - a principle in symbolic logic requiring any atomic sentence to be self-identical.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\nNow that we have three different connectives, this is a convenient time to consider complex sentences. \u00a0The example that we just considered required us to symbolize complex sentences, which use several different kinds of connectives. \u00a0We want to avoid confusion by being clear about the nature of these sentences. \u00a0We also want to be able to understand when such sentences are true and when they are false. \u00a0These two goals are closely related.\r\n\r\nConsider the following sentences.\r\n

                \u00ac<\/span>(P<\/strong><\/span>\u2192Q<\/strong>)<\/span><\/p>\r\n

                (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192Q<\/strong>)<\/span><\/p>\r\n

                (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192<\/span>\u00ac<\/span>Q<\/strong>)<\/span><\/p>\r\nWe want to understand what kinds of sentences these are, and also when they are true and when they are false. \u00a0(Sometimes people wrongly assume that there is some simple distribution law for negation and conditionals, so there is some additional value to reviewing these particular examples.) \u00a0The first task is to determine what kinds of sentences these are. \u00a0If the first symbol of your expression is a negation, then you know the sentence is a negation. \u00a0The first sentence above is a negation. \u00a0If the first symbol of your expression is a parenthesis, then for our logical language we know that we are dealing with a connective that combines two sentences.\r\n\r\nThe way to proceed is to match parentheses. \u00a0Generally people are able to do this by eye, but if you are not, you can use the following rule. \u00a0Moving left to right, the last \u201c(<\/span>\u201d that you encounter always matches the first \u201c)<\/span>\u201d that you encounter. \u00a0These form a sentence that must have two parts combined with a connective. \u00a0You can identify the two parts because each will be an atomic sentence, a negation sentence, or a more complex sentence bound with parentheses on each side of the connective.\r\n\r\nIn our propositional logic, each set of paired parentheses forms a sentence of its own. \u00a0So, when we encounter a sentence that begins with a parenthesis, we find that if we match the other parentheses, we will ultimately end up with two sentences as constituents, one on each side of a single connective. \u00a0The connective that combines these two parts is called the \u201cmain connective\u201d, and it tells us what kind of sentence this is. \u00a0Thus, above we have examples of a negation, a conditional, and a conditional.\r\n\r\nHow should we understand the meaning of these sentences? \u00a0Here we can use truth tables in a new, third way (along with defining a connective and checking arguments). \u00a0Our method will be this.\r\n\r\nFirst, write out the sentence on the right, leaving plenty of room. \u00a0Identify what kind of sentence this is. \u00a0If it is a negation sentence, you should add just to the left a column for the non-negated sentence. \u00a0This is because the truth table defining negation tells us what a negated sentence means in relation to the non-negated sentence that forms the sentence. \u00a0If the sentence is a conditional, make two columns to the left, one for the antecedent and one for the consequent. \u00a0If the sentence is a conjunction, make two columns to the left, one for each conjunct. \u00a0Here again, we do this because the semantic definitions of these connectives tell us what the truth value of the sentence is, as a function of the truth value of its two parts. \u00a0Continue this process until the parts would be atomic sentences. \u00a0Then, we stipulate all possible truth values for the atomic sentences. \u00a0Once we have done this, we can fill out the truth table, working left to right.\r\n\r\nLet\u2019s try it for \u00ac(P<\/strong><\/span>\u2192Q<\/strong>)<\/span>. \u00a0We write it to the right.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                \u00a0\u00ac(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/th>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/th>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/th>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/th>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis is a negation sentence, so we write to the left the sentence being negated.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                \u00a0(P\u2192Q)<\/span><\/th>\r\n\u00ac(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/td>\r\n\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/td>\r\n\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/td>\r\n\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n
                \u00a0\u00a0\u00a0<\/span><\/td>\r\n\u00a0\u00a0\u00a0<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis sentence is a conditional. \u00a0Its two parts are atomic sentences. \u00a0We put these to the left of the dividing line, and we stipulate all possible combinations of truth values for these atomic sentences.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P<\/span><\/th>\r\nQ<\/span><\/th>\r\n\u00a0(P\u2192Q)<\/span><\/th>\r\n\u00ac(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nNow, we can fill out each column, moving left to right. \u00a0We have stipulated the values for P<\/span>\u00a0<\/strong>and Q<\/span><\/strong>, so we can identify the possible truth values of (P<\/strong>\u2192Q<\/strong>)<\/span>. \u00a0The semantic definition for \u201c\u2192<\/span>\u201d tells us how to do that, given that we know for each row the truth value of its parts.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\nQ<\/span><\/th>\r\n\u00a0(P\u2192Q)<\/span><\/th>\r\n\u00ac(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis column now allows us to fill in the last column. \u00a0The sentence in the last column is a negation of (P<\/strong>\u2192Q<\/strong>)<\/span>, so the definition of \u201c\u00ac<\/span>\u201d tell us that \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true when (P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false, and \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false when (P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P<\/span><\/th>\r\nQ<\/span><\/th>\r\n(P\u2192Q)<\/span><\/th>\r\n\u00ac(P\u2192Q)<\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis truth table tells us what \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0means in our propositional logic. \u00a0Namely, if we assert \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0we are asserting that P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false.\r\n\r\nWe can make similar truth tables for the other sentences.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P<\/span><\/th>\r\nQ<\/span><\/th>\r\n\u00ac<\/span>P<\/span><\/th>\r\n(\u00ac<\/span>P<\/span>\u2192Q)<\/span><\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F\u00a0<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F\u00a0<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHow did we make this table? \u00a0The sentence (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192Q<\/strong>)<\/span>\u00a0is a conditional with two parts, \u00ac<\/span>P<\/span>\u00a0<\/strong>and\u00a0Q<\/strong><\/span>. \u00a0Because Q<\/span>\u00a0<\/strong>is atomic, it will be on the left side. \u00a0We make a row for \u00ac<\/span>P<\/span><\/strong>. \u00a0The sentence \u00ac<\/span>P<\/span>\u00a0<\/strong>is a negation of P<\/span><\/strong>, which is atomic, so we put P<\/span>\u00a0<\/strong>also on the left. \u00a0We fill in the columns, going left to right, using our definitions of the connectives.\r\n\r\nAnd:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\r\nQ<\/span><\/th>\r\n\u00acP<\/span><\/th>\r\n\u00a0\u00acQ<\/span><\/th>\r\n(\u00acP\u2192\u00acQ)<\/span><\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nSuch a truth table is very helpful in determining when sentences are, and are not, equivalent. \u00a0We have used the concept of equivalence repeatedly, but have not yet defined it. \u00a0We can offer a semantic, and a syntactic, explanation of equivalence. \u00a0The semantic notion is relevant here: \u00a0we say two sentences \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span>\u00a0<\/strong>are \u201cequivalent\u201d or \u201clogically equivalent\u201d when they must have the same truth value. \u00a0(For the syntactic concept of equivalence, see section 9.2). \u00a0These truth tables show that these three sentences are not equivalent, because it is not the case that they must have the same truth value. \u00a0For example, if P<\/span>\u00a0<\/strong>and Q<\/span>\u00a0<\/strong>are both true, then \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false but (\u00acP<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true and (\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span>\u00a0is true. \u00a0If P<\/span>\u00a0<\/strong>is false and Q<\/span>\u00a0<\/strong>is true, then (\u00acP<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true but (\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span>\u00a0is false. \u00a0Thus, each of these sentences is true in some situation where one of the others is false. \u00a0No two of them are equivalent.\r\n\r\nWe should consider an example that uses conjunction, and which can help in some translations. \u00a0How should we translate \u201cNot both Steve and Tom will go to Berlin\u201d? \u00a0This sentence tells us that it is not the case that both Steve will go to Berlin and Tom will go to Berlin. \u00a0The sentence does allow, however, that one of them will go to Berlin. \u00a0Thus, let U<\/span>\u00a0<\/strong>mean Steve will go to Berlin<\/span>\u00a0and V<\/span>\u00a0<\/strong>mean Tom will go to Berlin.<\/span>\u00a0 Then we should translate this sentence, \u00ac(U<\/strong>^V<\/strong>)<\/span>. We should not translate the sentence (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>. \u00a0To see why, consider their truth tables.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                U<\/span><\/th>\r\nV<\/span><\/th>\r\n(U^V)<\/span><\/th>\r\n\u00ac(U^V)<\/span><\/th>\r\n\u00acU<\/span><\/span><\/th>\r\n\u00acV<\/span><\/span><\/th>\r\n(\u00acU^\u00acV)<\/span><\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe can see that \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0and (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>\u00a0are not equivalent. \u00a0Also, note the following. \u00a0Both \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0and (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>\u00a0are true if Steve does not go to Berlin and Tom does not go to Berlin. \u00a0This is captured in the last row of this truth table, and this is consistent with the meaning of the English sentence. \u00a0But, now note: \u00a0it is true that not both Steve and Tom will go to Berlin, if Steve goes and Tom does not. \u00a0This is captured in the second row of this truth table. \u00a0It is true that not both Steve and Tom will go to Berlin, if Steve does not go but Tom does. \u00a0This is captured in the third row of this truth table. \u00a0In both kinds of cases (in both rows of the truth table), \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0is true but (\u00acU<\/strong>^\u00acV<\/strong>) <\/span>is false. \u00a0Thus, we can see that \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0is the correct translation of \u201cNot both Steve and Tom will go to Berlin\u201d.\r\n\r\nLet\u2019s consider a more complex sentence that uses all of our connectives so far: \u00a0((P<\/strong>^\u00acQ<\/strong>)\u2192\u00ac(P<\/strong>\u2192Q<\/strong>))<\/span>. \u00a0This sentence is a conditional. \u00a0The antecedent is a conjunction. \u00a0The consequent is a negation. \u00a0Here is the truth table, completed.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                P<\/span><\/th>\r\nQ<\/span><\/th>\r\n\u00a0\u00acQ<\/th>\r\n(P\u2192Q)<\/th>\r\n(P^\u00acQ)<\/th>\r\n\u00a0 \u00ac(P\u2192Q)<\/span><\/th>\r\n((P^\u00acQ)\u2192\u00ac(P\u2192Q))<\/span><\/th>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n\u00a0F<\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\nF\u00a0<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                T<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\n\u00a0T<\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n\u00a0F<\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n
                F<\/span><\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\n\u00a0T<\/strong><\/em><\/td>\r\nT<\/strong><\/em><\/td>\r\nF<\/strong><\/em><\/td>\r\nF<\/span><\/strong><\/em><\/td>\r\nT<\/span><\/strong><\/em><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nThis sentence has an interesting property: \u00a0it cannot be false. \u00a0That is not surprising, once we think about what it says. \u00a0In English, the sentence says: If P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false, then it is not the case that P<\/span>\u00a0<\/strong>implies Q<\/span><\/strong>. \u00a0That must be true: \u00a0if it were the case that P <\/span><\/strong>implies Q<\/span><\/strong>, then if P<\/span>\u00a0<\/strong>is true then Q<\/span>\u00a0<\/strong>is true. \u00a0But the antecedent says P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false.\r\n\r\nSentences of the propositional logic that must be true are called \u201ctautologies\u201d. \u00a0We will discuss them at length in later chapters.\r\n\r\nFinally, note that we can combine this method for finding the truth conditions for a complex sentence with our method for determining whether an argument is valid using a truth table. \u00a0We will need to do this if any of our premises or the conclusion are complex. \u00a0Here is an example. \u00a0We\u2019ll start with an argument in English:\r\n

                If whales are mammals, then they have vestigial limbs. \u00a0If whales are mammals, then they have a quadrupedal ancestor. \u00a0Therefore, if whales are mammals then they have a quadrupedal ancestor and they have vestigial limbs.<\/p>\r\nWe need a translation key.\r\n

                P<\/span><\/strong>: \u00a0Whales are mammals.<\/p>\r\n

                Q<\/span><\/strong>: Whales have have vestigial limbs.<\/p>\r\n

                R<\/span><\/strong>: Whales have a quadrupedal ancestor.<\/p>\r\nThe argument will then be symbolized as:\r\n

                \r\n

                (P<\/strong>\u2192Q<\/strong>)<\/span><\/p>\r\n

                (P<\/strong>\u2192R<\/strong>)<\/span><\/p>\r\n

                ____<\/span><\/strong><\/p>\r\n

                (P<\/strong>\u2192(R<\/strong>^Q<\/strong>))<\/span><\/p>\r\n\r\n<\/div>\r\nHere is a semantic check of the argument.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
                <\/td>\r\n<\/td>\r\n<\/td>\r\npremise<\/td>\r\npremise<\/td>\r\n<\/td>\r\n\u00a0conclusion<\/td>\r\n<\/tr>\r\n
                P<\/span><\/td>\r\nQ<\/span><\/td>\r\nR<\/span><\/td>\r\n(P\u2192Q)<\/strong><\/td>\r\n(P\u2192R)<\/strong><\/td>\r\n(R^Q)<\/strong><\/td>\r\n(P\u2192(R^Q))<\/strong><\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nF <\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                T\u00a0<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                T<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nF <\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF <\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n
                F<\/span><\/td>\r\nF<\/span><\/td>\r\nF<\/span><\/td>\r\nT\u00a0<\/span><\/td>\r\nT<\/span><\/td>\r\nF<\/span><\/td>\r\nT<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nWe have highlighted the rows where the premises are all true. \u00a0Note that for these, the conclusion is true. \u00a0Thus, in any kind of situation in which all the premises are true, the conclusion is true. \u00a0This is equivalent, we have noted, to our definition of valid: \u00a0necessarily, if all the premises are true, the conclusion is true. \u00a0So this is a valid argument. \u00a0The third column of the analyzed sentences (the column for (R<\/strong>^Q<\/strong>)<\/span>) is there so that we can identify when the conclusion is true. \u00a0The conclusion is a conditional, and we needed to know, for each kind of situation, if its antecedent P<\/span><\/strong>, and if its consequent (R<\/strong>^Q<\/strong>)<\/span>, are true. \u00a0The third column tells us the situations in which the consequent is true. \u00a0The stipulations on the left tell us in what kind of situation the antecedent P<\/span>\u00a0<\/strong>is true.\r\n
                \r\n

                Check for Understanding<\/p>\r\n\r\n<\/header>\r\n

                \r\n
                  \r\n \t
                1. Create your own complex sentence using negation, conjunction, and conditional, and construct its truth table to determine under what conditions it is true.<\/li>\r\n \t
                2. Consider the English sentence \"Either Jacob won't go to Paris or he will go to Rome.\" Translate this into symbolic form and explain your reasoning.<\/li>\r\n \t
                3. Critical Thinking Task:\u00a0<\/strong>Generate a song, image, or poem explaining the concept of complex sentences.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n

                  5.7\u00a0 Problems<\/h2>\r\n
                    \r\n \t
                  1. Self Reflection:\u00a0<\/strong>What are some of the challenges you are facing with complex sentences? Why do you believe this is the case?<\/li>\r\n \t
                  2. Translate the following sentences into our logical language. \u00a0You will need to create your own key to do so.\r\n
                      \r\n \t
                    1. Ulysses, who is crafty, is from Ithaca.<\/li>\r\n \t
                    2. Ulysses, who isn't crafty, is from Ithaca.<\/li>\r\n \t
                    3. Ulysses, who is crafty, isn't from Ithaca.<\/li>\r\n \t
                    4. Ulysses isn't both crafty and from Ithaca.<\/li>\r\n \t
                    5. Ulysses will go home only if he's from Ithaca and not Troy.<\/li>\r\n \t
                    6. Ulysses is not both from Ithaca and Troy, though he is crafty.<\/li>\r\n \t
                    7. If Ulysses outsmarts both Circes and the Cyclops, then he can go home.<\/li>\r\n \t
                    8. If Ulysses outsmarts Circes but not the Cyclops, then he will be eaten.<\/li>\r\n \t
                    9. Though he won't outsmart Circe, Ulysses will outsmart the Cyclops, even given that he is from Ithaca.<\/li>\r\n \t
                    10. Ulysses won't outsmart both Circes and the Cyclops, but he won't be eaten and will go home even though he is from Ithaca.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                    11. Prove the following arguments are valid, using a direct derivation.\r\n
                        \r\n \t
                      1. Premise: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ \u00acQ<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00acP<\/strong><\/span>.<\/li>\r\n \t
                      2. Premises: \u00a0(((P<\/strong>\u2192Q<\/strong>) ^ (Q<\/strong>\u2192R<\/strong>))^P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0R<\/span><\/strong>.<\/li>\r\n \t
                      3. Premises: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ (R<\/strong>\u2192S<\/strong>))<\/span>, (\u00acQ<\/strong> ^ \u00acS<\/strong>)<\/span>. \u00a0Conclusion: \u00a0(\u00acP<\/strong> ^ \u00acR<\/strong>)<\/span>.<\/li>\r\n \t
                      4. Premises: \u00a0((R<\/strong> ^ S<\/strong>) <\/span>\u2192<\/span>\u00a0T<\/strong>)<\/span>, (Q<\/strong> ^ \u00acT<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00ac(R<\/strong> ^ S<\/strong>)<\/span>.<\/li>\r\n \t
                      5. Premises: \u00a0(P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span>, (Q<\/strong> ^ P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0R<\/span><\/strong>.<\/li>\r\n \t
                      6. Premises: \u00a0(P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span>, (\u00acR<\/strong> ^ P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00acQ<\/strong><\/span>.<\/li>\r\n \t
                      7. Premises: \u00a0((P<\/strong>^Q<\/strong>)\u2192(R<\/strong>\u2192S<\/strong>))<\/span>, (Q<\/strong> ^ (P<\/strong> ^ \u00acS<\/strong>))<\/span>. \u00a0Conclusion: \u00a0(\u00acR<\/strong> ^ Q<\/strong>)<\/span>.<\/li>\r\n \t
                      8. Premises: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ (R<\/strong>\u2192S<\/strong>))<\/span>, (P<\/strong> ^ R<\/strong>)<\/span>. \u00a0Conclusion: \u00a0((P<\/strong> ^ Q<\/strong>) ^ (R<\/strong> ^ S<\/strong>))<\/span>.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                      9. Make truth tables for the following complex sentences. \u00a0Identify which are tautologies.\r\n
                          \r\n \t
                        1. \u00ac(P<\/strong> ^ Q<\/strong>)<\/span><\/li>\r\n \t
                        2. \u00ac(\u00acP<\/strong> \u2192 \u00acQ<\/strong>)<\/span><\/li>\r\n \t
                        3. (P<\/strong> ^ \u00acP<\/strong>)<\/span><\/li>\r\n \t
                        4. \u00ac(P<\/strong> ^ \u00acP<\/strong>)<\/span><\/li>\r\n \t
                        5. (((P<\/strong>\u2192Q<\/strong>)^ \u00acQ<\/strong>)\u2192\u00acP<\/strong>)<\/span><\/li>\r\n \t
                        6. (((P<\/strong>\u2192Q<\/strong>)^ \u00acP<\/strong>)\u2192\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                        7. (((P<\/strong>\u2192Q<\/strong>)^ P<\/strong>)\u2192Q<\/strong>)<\/span><\/li>\r\n \t
                        8. (((P<\/strong>\u2192Q<\/strong>)^ Q<\/strong>)\u2192P<\/strong>)<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                        9. Make truth tables to show when the following sentences are true and when they are false. \u00a0State which of these sentences are equivalent. Also, can you identify if any have the same truth table as some of our connectives?\r\n
                            \r\n \t
                          1. \u00ac(P<\/strong>^Q<\/strong>)<\/span><\/li>\r\n \t
                          2. (\u00acP<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                          3. \u00ac(\u00acP<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                          4. \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span><\/li>\r\n \t
                          5. (P<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                          6. (\u00acP<\/strong>^Q<\/strong>)<\/span><\/li>\r\n \t
                          7. \u00ac(P<\/strong>\u2192\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                          8. \u00ac(\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span><\/li>\r\n \t
                          9. (P<\/strong> ^ (Q<\/strong> ^ R<\/strong>))<\/span><\/li>\r\n \t
                          10. ((P<\/strong> ^ Q<\/strong>) ^ R<\/strong>))<\/span><\/li>\r\n \t
                          11. (P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span><\/li>\r\n \t
                          12. ((P<\/strong>\u2192Q<\/strong>)\u2192R<\/strong>))<\/span><\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                          13. Write a valid argument in normal colloquial English with at least two premises, one of which is a conjunction or includes a conjunction. \u00a0Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like formal logic). \u00a0Translate the argument into propositional logic. \u00a0Prove it is valid.<\/li>\r\n \t
                          14. Write a valid argument in normal colloquial English with at least three premises, one of which is a conjunction or includes a conjunction and one of which is a conditional or includes a conditional. \u00a0Translate the argument into propositional logic. \u00a0Prove it is valid.<\/li>\r\n \t
                          15. Often in a natural language like English, there are many implicit conjunctions in descriptions and other phrases. Here are some passages from literature. Translate them into our propositional logic. You will want to make a separate key for each particular problem.\r\n
                              \r\n \t
                            1. \"But Achilles the son of Peleus again shouted at Agamemnon the son of Atreus, for he was still in a rage.\"\r\n(Homer, The Illiad<\/i>)<\/li>\r\n \t
                            2. \"Socrates is an evil-doer,\u00a0and a curious person, who searches into things under the earth and in heaven,\u00a0and he makes the worse appear the better cause.\u2026\u201d (Plato, The Apology<\/i>)<\/li>\r\n \t
                            3. \"Incensed with indignation, Satan stood\r\nUnterrified\u2026.\" (Milton, Paradise Lost<\/i>)<\/li>\r\n \t
                            4. \"Teiresias, seer who comprehends all\u2026\r\nYou know, though thy blind eyes see nothing,\r\nWhat plague infects our city Thebes.\" (Sophocles, Oedipus Rex<\/i>)<\/li>\r\n \t
                            5. \"Scrooge! a squeezing, wrenching, grasping, scraping, clutching, covetous, old sinner!\" (Charles Dickens, \"A Christmas Carrol\")<\/li>\r\n \t
                            6. \"When I wrote the following pages, or rather the bulk of them, I lived alone, in the woods, a mile from any neighbor, in a house which I had built myself, on the shore of Walden Pond, in Concord, Massachusetts, and earned my living by the labor of my hands only.\" [Here one can substitute \u201cThoreau\u201d for \u201cI\u201d in the translation, if helpful.]. (Henry David Thoreau, Walden<\/i>)<\/li>\r\n \t
                            7. \"In appearance Shatov was in complete harmony with his convictions: he was short, awkward, had a shock of flaxen hair, broad shoulders, thick lips, very thick overhanging white eyebrows, a wrinkled forehead, and a hostile, obstinately downcast, as it were shamefaced, expression in his eyes.\" (Fyodor Dostoevsky, The Possessed<\/i>)<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t
                            8. Make your own key to translate the following argument into our propositional logic. \u00a0Translate only the parts in bold. \u00a0Prove the argument is valid.<\/li>\r\n<\/ol>\r\n
                                \r\n \t
                              1. <\/li>\r\n<\/ol>\r\n
                                  \r\n \t
                                1. <\/li>\r\n<\/ol>\r\n
                                    \r\n \t
                                  1. <\/li>\r\n<\/ol>\r\n
                                      \r\n \t
                                    1. <\/li>\r\n<\/ol>\r\n
                                        \r\n \t
                                      1. <\/li>\r\n<\/ol>\r\n

                                        \u201cI suspect Dr. Kronecker of the crime of stealing Cantor\u2019s book,\u201d Inspector Tarski said. \u00a0His assistant, Mr. Carroll, waited patiently for his reasoning. \u00a0\u201cFor,\u201d Tarski said, \u201cThe thief left cigarette ashes on the table. \u00a0The thief also did not wear shoes, but slipped silently into the room. \u00a0Thus, If Dr. Kronecker smokes and is in his stocking feet, then he most likely stole Cantor\u2019s book<\/strong>.<\/span>\u201d \u00a0At this point, Tarski pointed at Kronecker\u2019s feet. \u00a0\u201cDr. Kronecker is in his stocking feet.<\/span>\u201d \u00a0Tarski reached forward and pulled from Kronecker\u2019s pocket a gold cigarette case. \u00a0\u201cAnd Kronecker smokes<\/span><\/strong>.\u201d \u00a0Mr. Carroll nodded sagely, \u201cYour conclusion is obvious: \u00a0Dr. Kronecker most likely stole Cantor\u2019s book<\/strong>.\u201d<\/span><\/p>","rendered":"

                                        5.1 \u00a0The conjunction<\/h2>\n
                                        \n
                                        \n

                                        Pre-Reading Questions<\/p>\n<\/header>\n

                                        \n
                                          \n
                                        1. What does it mean to \u201cconjoin\u201d two statements logically?<\/li>\n
                                        2. Why is it valid to infer a single component from a conjunction?<\/li>\n
                                        3. In what types of arguments do conjunctions commonly appear?<\/li>\n
                                        4. How might the use of conjunctions affect the length or structure of a proof?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n
                                          \n
                                          \n

                                          Key Terms<\/p>\n<\/header>\n

                                          \n
                                            \n
                                          • Conjunction (\u2227)<\/strong> – combines two statements, creating a new statement that is true only if both original statements are true.<\/li>\n
                                          • Conjunction Elimination (\u2227E)<\/strong> – a rule of inference that allows you to derive a single conjunct from a conjunction.<\/li>\n
                                          • Conjunction Introduction (\u2227I)<\/strong> – a rule of inference that allows you to combine two or more true statements (called conjuncts) into a single, true statement called a conjunction.<\/li>\n
                                          • Scope of a Rule<\/strong> – defines the portion of a logical expression or statement to which it applies.<\/li>\n
                                          • Well-Formed Formula (WFF)<\/strong> – a string of symbols that adheres to the syntactic rules of a formal language.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

                                            To make our logical language more easy and intuitive to use, we can now add to it elements that make it able to express the equivalents of other sentences from a natural language like English. \u00a0Our translations will not be exact, but they will be close enough that: first, we will have a way to more quickly understand the language we are constructing; and, second, we will have a way to speak English more precisely when that is required of us.<\/p>\n

                                            Consider the following expressions. \u00a0How would we translate them into our logical language?<\/p>\n

                                            Anthony will go to Berlin and Paris.<\/p>\n

                                            The number a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2 and 3.<\/p>\n

                                            Malik is from Colorado but not from Denver.<\/p>\n

                                            We could translate each of these using an atomic sentence. \u00a0But then we would have lost\u2014or rather we would have hidden\u2014information that is clearly there in the English sentences. \u00a0We can capture this information by introducing a new connective; one that corresponds to our \u201cand\u201d.<\/p>\n

                                            To see this, consider whether you will agree that these sentences above are equivalent to the following sentences.<\/p>\n

                                            Anthony will go to Berlin and Anthony will go to Paris.<\/p>\n

                                            The number a<\/span>\u00a0<\/em><\/strong>is evenly divisible by 2 and the number a<\/span>\u00a0<\/span><\/em><\/strong>is evenly divisible by 3.<\/p>\n

                                            Malik is from Colorado and it is not the case that Malik is from Denver.<\/p>\n

                                            Once we grant that these sentences are equivalent to those above, we see that we can treat the \u201cand\u201d in each sentence as a truth functional connective.<\/p>\n

                                            Suppose we assume the following key.<\/p>\n

                                            P<\/span><\/strong>: \u00a0Anthony will go to Berlin.<\/p>\n

                                            Q<\/span><\/strong>: \u00a0Anthony will go to Paris.<\/p>\n

                                            R<\/span><\/strong>: \u00a0a<\/span>\u00a0<\/em><\/strong>\u00a0is evenly divisible by 2.<\/p>\n

                                            S<\/span><\/strong>: \u00a0a<\/span>\u00a0<\/em><\/strong>\u00a0is evenly divisible by 3.<\/p>\n

                                            T<\/span><\/strong>: \u00a0Malik is from Colorado<\/p>\n

                                            U<\/span><\/strong>: \u00a0Malik is from Denver.<\/p>\n

                                            A partial translation of these sentences would then be:<\/p>\n

                                            P<\/span>\u00a0<\/strong>and Q<\/span><\/strong><\/p>\n

                                            R<\/span>\u00a0<\/strong>and S<\/span><\/strong><\/p>\n

                                            T<\/span>\u00a0<\/strong>and \u00acU<\/strong><\/span><\/p>\n

                                            Our third sentence above might generate some controversy. \u00a0How should we understand \u201cbut\u201d? \u00a0Consider that in terms of the truth value of the connected sentences, \u201cbut\u201d is the same as \u201cand\u201d. \u00a0That is, if you say \u201cP<\/span>\u00a0<\/strong>but\u00a0Q<\/span><\/strong>\u201d you are asserting that both P<\/span>\u00a0<\/strong>and Q<\/span>\u00a0<\/strong>are true. \u00a0However, in English there is extra meaning; the English \u201cbut\u201d seems to indicate that the additional sentence is unexpected or counter-intuitive. \u00a0\u201cP<\/span>\u00a0<\/strong>but Q<\/span><\/strong>\u201d seems to say, \u201cP<\/span>\u00a0<\/strong>is true, and you will find it surprising or unexpected that Q<\/span>\u00a0<\/strong>is true also.\u201d \u00a0That extra meaning is lost in our logic. \u00a0We will not be representing surprise or expectations. \u00a0So, we can treat \u201cbut\u201d as being the same as \u201cand\u201d. \u00a0This captures the truth value of the sentence formed using \u201cbut\u201d, which is all that we require of our logic.<\/p>\n

                                            Following our method up until now, we want a symbol to stand for \u201cand\u201d. \u00a0In recent years the most commonly used symbol has been \u201c^<\/span>\u201d.<\/p>\n

                                            The syntax for \u201c^<\/span>\u201d is simple. \u00a0If \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span>\u00a0<\/strong>are sentences, then<\/p>\n

                                            (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\n

                                            is a sentence. \u00a0Our translations of our three example sentences should thus look like this:<\/p>\n

                                            (P<\/strong>^Q<\/strong>)<\/span><\/p>\n

                                            (R<\/strong>^S<\/strong>)<\/span><\/p>\n

                                            (T<\/strong>^\u00acU<\/strong>)<\/span><\/p>\n

                                            Each of these is called a “conjunction<\/strong>.” \u00a0The two parts of a conjunction are called “conjuncts<\/strong>.”<\/p>\n

                                            The semantics of the conjunction are given by its truth table. \u00a0Most people find the conjunction\u2019s semantics obvious. \u00a0If I claim that both \u03a6 <\/span><\/strong>and\u00a0\u03a8<\/strong><\/span>\u00a0<\/strong>are true, normal usage requires that if \u03a6<\/span>\u00a0<\/strong>is false or \u03a8<\/span>\u00a0<\/strong>is false, or both are false, then I spoke falsely also.<\/p>\n

                                            Consider an example. \u00a0Suppose your employer says, \u201cAfter one year of employment you will get a raise and two weeks vacation\u201d. \u00a0A year passes. \u00a0Suppose now that this employer gives you a raise but no vacation, or a vacation but no raise, or neither a raise nor a vacation. \u00a0In each case, the employer has broken his promise. \u00a0The sentence forming the promise turned out to be false.<\/p>\n

                                            Thus, the semantics for the conjunction are given with the following truth table. \u00a0For any sentences \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span><\/strong>:<\/p>\n\n\n\n\n\n\n\n
                                            \u03a6 \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\n\u03a8<\/span><\/th>\n(\u03a6^\u03a8)<\/span><\/th>\n<\/tr>\n
                                            T<\/span><\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                            T<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                            F<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                            F<\/span><\/strong><\/em><\/td>\nF\u00a0<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
                                            \n
                                            \n

                                            Check for Understanding<\/p>\n<\/header>\n

                                            \n
                                              \n
                                            1. Critical Thinking Task:\u00a0<\/strong>Why are conjunctions good tools for avoiding information being lost in translation?<\/li>\n
                                            2. Critical Thinking Task:\u00a0<\/strong>Ariel states, “The words ‘and’ along with ‘but’ can have the same meaning in logical symbols.” Her brother, Donovan, states, “No, they cannot!” How can Ariel demonstrate to her brother that the word “and” can serve the same meanings as “but” in formal logic?<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                              5.2 \u00a0Alternative phrasings, and a different \u201cand\u201d<\/h2>\n

                                              We have noted that in English, \u201cbut\u201d is an alternative to \u201cand\u201d, and can be translated the same way in our propositional logic. \u00a0There are other phrases that have a similar meaning: they are best translated by conjunctions, but they convey (in English) a sense of surprise or failure of expectations. \u00a0For example, consider the following sentence.<\/p>\n

                                              Even though they lost the battle, they won the war.<\/p>\n

                                              Here \u201ceven though\u201d seems to do the same work as \u201cbut\u201d. \u00a0The implication is that it is surprising\u2014that one might expect that if they lost the battle then they lost the war. \u00a0But, as we already noted, we will not capture expectations with our logic. \u00a0So, we would take this sentence to be sufficiently equivalent to:<\/p>\n

                                              They lost the battle and they won the war.<\/p>\n

                                              With the exception of \u201cbut\u201d, it seems in English there is no other single word that is an alternative to \u201cand\u201d that means the same thing. However, there are many ways that one can imply a conjunction. \u00a0To see this, consider the following sentences.<\/p>\n

                                              Lee, who won the race, also won the championship.<\/p>\n

                                              The star Phosphorous, that we see in the morning, is the Evening Star.<\/p>\n

                                              The Evening Star, which is called \u201cHesperus\u201d, is also the Morning Star.<\/p>\n

                                              While Mateo is tall, Steve is not.<\/p>\n

                                              Dogs are vertebrate terrestrial mammals.<\/p>\n

                                              Depending on what elements we take as basic in our language, these sentences all include implied conjunctions. \u00a0They are equivalent to the following sentences, for example:<\/p>\n

                                              Lee won the race and Lee won the championship.<\/p>\n

                                              Phosphorous is the star that we see in the morning and Phosphorous is the Evening Star.<\/p>\n

                                              The Evening Star is called \u201cHesperus\u201d and the Evening Star is the Morning Star.<\/p>\n

                                              Mateo is tall and it is not the case that Steve is tall.<\/p>\n

                                              Dogs are vertebrates and dogs are terrestrial and dogs are mammals.<\/p>\n

                                              Thus, we need to be sensitive to complex sentences that are conjunctions but that do not use \u201cand\u201d or \u201cbut\u201d or phrases like \u201ceven though\u201d.<\/p>\n

                                              Unfortunately, in English there are some uses of \u201cand\u201d that are not conjunctions. \u00a0The same is true for equivalent terms in some other natural languages. \u00a0Here is an example.<\/p>\n

                                              Rochester is between Buffalo and Albany.<\/p>\n

                                              The \u201cand\u201d in this sentence is not a conjunction. \u00a0To see this, note that this sentence is not equivalent to the following:<\/p>\n

                                              Rochester is between Buffalo and Rochester is between Albany.<\/p>\n

                                              That sentence is not even semantically correct. \u00a0What is happening in the original sentence?<\/p>\n

                                              The issue here is that \u201cis between\u201d is what we call a \u201cpredicate\u201d. \u00a0We will learn about predicates in chapter 11, but what we can say here is that some predicates take several names in order to form a sentence. \u00a0In English, if a predicate takes more than two names, then we typically use the \u201cand\u201d to combine names that are being described by that predicate. \u00a0In contrast, the conjunction in our propositional logic only combines sentences. \u00a0So, we must say that there are some uses of the English \u201cand\u201d that are not equivalent to our conjunction.<\/p>\n

                                              This could be confusing because sometimes in English we put \u201cand\u201d between names and there is an implied conjunction. \u00a0Consider:<\/p>\n

                                              Elijah is older than Sophia and Max.<\/p>\n

                                              Superficially, this looks to have the same structure as \u201cRochester is between Buffalo and Albany\u201d. \u00a0But this sentence really is equivalent to:<\/p>\n

                                              Elijah is older than Max and Elijah is older than Sophia.<\/p>\n

                                              The difference, however, is that there must be three things in order for one to be between the other two. \u00a0There need only be two things for one to be older than the other. \u00a0So, in the sentence \u201cRochester is between Buffalo and Albany\u201d, we need all three names (\u201cRochester\u201d, \u201cBuffalo\u201d, and \u201cAlbany) to make a single proper atomic sentence with \u201cbetween\u201d. \u00a0This tells us that the \u201cand\u201d is just being used to combine these names, and not to combine implied sentences (since there can be no implied sentence about what is \u201cbetween\u201d, using just two or just one of these names).<\/p>\n

                                              That sounds complex. \u00a0Do not despair, however. \u00a0The use of \u201cand\u201d to identify names being used by predicates is less common than \u201cand\u201d being used for a conjunction. \u00a0Also, after we discuss predicates in chapter 11, and after you have practiced translating different kinds of sentences, the distinction between these uses of \u201cand\u201d will become easy to identify in almost all cases. \u00a0In the meantime, we shall pick examples that do not invite this confusion.<\/p>\n

                                              \n
                                              \n

                                              Check for Understanding<\/p>\n<\/header>\n

                                              \n
                                                \n
                                              1. Explain the difference between how “and” functions in “Elijah is older than Max and Sophia” versus “Rochester is between Buffalo and Albany.”<\/li>\n
                                              2. Create your own example of a sentence where “and” is used as a logical conjunction and another example where “and” is not functioning as a logical conjunction.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                5.3 \u00a0Inference rules for conjunctions<\/h2>\n
                                                \n
                                                \n

                                                Key Terms<\/p>\n<\/header>\n

                                                \n
                                                  \n
                                                • Simplification<\/strong> – a process that allows two rules to be applied by the same name.<\/li>\n
                                                • Adjunction Rule\u00a0<\/strong>– an alternative way of expressing the conjunction rule to avoid confusion.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

                                                  Looking at the truth table for the conjunction should tell us two things very clearly. \u00a0First, if a conjunction is true, what else must be true? \u00a0The obvious answer is that both of the parts, the conjuncts, must be true. \u00a0We can introduce a rule to capture this insight. \u00a0In fact, we can introduce two rules and call them by the same name, since the order of conjuncts does not affect their truth value. \u00a0These rules are often called \u201csimplification\u201d.<\/p>\n

                                                  (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\n

                                                  _____<\/p>\n

                                                  \u03a6<\/span><\/strong><\/p>\n

                                                  And:<\/p>\n

                                                  (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\n

                                                  _____<\/p>\n

                                                  \u03a8<\/span><\/strong><\/p>\n

                                                  In other words, if (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, then \u03a6<\/span>\u00a0<\/strong>must be true; and if (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, then \u03a8<\/span>\u00a0<\/strong>must be true.<\/p>\n

                                                  We can also introduce a rule to show a conjunction, based on what we see from the truth table. \u00a0That is, it is clear that there is only one kind of condition in which (\u03a6<\/strong>^\u03a8<\/strong>)<\/span>\u00a0is true, and that is when \u03a6<\/span>\u00a0<\/strong>is true and when \u03a8<\/span>\u00a0<\/strong>is true. \u00a0This suggests the following rule:<\/p>\n

                                                  \u03a6<\/span><\/strong><\/p>\n

                                                  \u03a8<\/span><\/strong><\/p>\n

                                                  _____<\/p>\n

                                                  (\u03a6<\/strong>^\u03a8<\/strong>)<\/span><\/p>\n

                                                  We might call this rule \u201cconjunction\u201d, but to avoid confusion with the name of the sentences, we will call this rule \u201cadjunction\u201d.<\/p>\n

                                                  5.4 \u00a0Reasoning with conjunctions<\/h2>\n

                                                  It would be helpful to consider some examples of reasoning with conjunctions. \u00a0Let\u2019s begin with an argument in a natural language.<\/p>\n

                                                  River and Emma will go to London. \u00a0If Emma goes to London, then she will ride the Eye. \u00a0River will ride the Eye too, provided that they go to London. \u00a0So, both Emma and River will ride the Eye.<\/p>\n

                                                  We need a translation key.<\/p>\n

                                                  T<\/span><\/strong>: \u00a0Emma will go to London.<\/p>\n

                                                  S<\/span><\/strong>: \u00a0River will go to London.<\/p>\n

                                                  U<\/span><\/strong>: \u00a0Emma will ride the Eye.<\/p>\n

                                                  V<\/span><\/strong>: \u00a0River will ride the Eye.<\/p>\n

                                                  Thus our argument is:<\/p>\n

                                                  (T<\/strong>^S<\/strong>)<\/span><\/p>\n

                                                  (T<\/strong>\u2192U<\/strong>)<\/span><\/p>\n

                                                  (S<\/strong>\u2192V<\/strong>)<\/span><\/p>\n

                                                  _____<\/span><\/strong><\/p>\n

                                                  (U<\/strong>^V<\/strong>)<\/span><\/p>\n

                                                  Our direct proof will look like this.<\/p>\n

                                                  \"\"<\/p>\n

                                                  Now an example using just our logical language. \u00a0Consider the following argument.<\/p>\n

                                                  (Q<\/strong>\u2192<\/span>\u00acS<\/strong>)<\/span><\/p>\n

                                                  (P<\/strong><\/span>\u2192(Q<\/strong>^R<\/strong>))<\/span><\/p>\n

                                                  (T<\/strong>\u2192<\/span>\u00ac<\/span>R<\/strong>)<\/span><\/p>\n

                                                  P<\/span><\/strong><\/p>\n

                                                  _____<\/span><\/strong><\/p>\n

                                                  (<\/span>\u00ac<\/span>S<\/strong>^<\/span>\u00ac<\/span>T<\/strong>)<\/span><\/p>\n

                                                  Here is one possible proof.<\/p>\n

                                                  \"\"<\/p>\n

                                                  \n
                                                  \n

                                                  Check for Understanding<\/p>\n<\/header>\n

                                                  \n
                                                    \n
                                                  1. Explain how simplification works as an inference rule. What does it allow us to do with conjunctions?<\/li>\n
                                                  2. Create your own natural language argument that would have the same logical structure as the first example.<\/li>\n
                                                  3. Identify a real-world situation where reasoning similar to the second proof might be used to draw a conclusion.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                    5.5 \u00a0Alternative symbolizations for the conjunction<\/h2>\n

                                                    Alternative notations for the conjunction include the symbols \u201c&<\/span><\/strong>\u201d and the symbol \u201c\u2219\u201d. \u00a0Thus, the expression (P<\/strong>^Q<\/strong>)<\/span>\u00a0would be written in these different styles, as:<\/p>\n

                                                    (P<\/strong>&Q<\/strong>)<\/span><\/p>\n

                                                    (P<\/strong>\u2219Q<\/strong>)<\/span><\/p>\n

                                                    5.6 Complex sentences<\/h2>\n
                                                    \n
                                                    \n

                                                    Key Terms<\/p>\n<\/header>\n

                                                    \n
                                                      \n
                                                    • Main Connective<\/strong> – the connective that combines two parts to allow two sentences to serve as constituents with a single connective.<\/li>\n
                                                    • Equivalent<\/strong> – two or more sentences have the same value in a proof.<\/li>\n
                                                    • Tautology<\/strong> – a principle in symbolic logic requiring any atomic sentence to be self-identical.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n

                                                      Now that we have three different connectives, this is a convenient time to consider complex sentences. \u00a0The example that we just considered required us to symbolize complex sentences, which use several different kinds of connectives. \u00a0We want to avoid confusion by being clear about the nature of these sentences. \u00a0We also want to be able to understand when such sentences are true and when they are false. \u00a0These two goals are closely related.<\/p>\n

                                                      Consider the following sentences.<\/p>\n

                                                      \u00ac<\/span>(P<\/strong><\/span>\u2192Q<\/strong>)<\/span><\/p>\n

                                                      (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192Q<\/strong>)<\/span><\/p>\n

                                                      (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192<\/span>\u00ac<\/span>Q<\/strong>)<\/span><\/p>\n

                                                      We want to understand what kinds of sentences these are, and also when they are true and when they are false. \u00a0(Sometimes people wrongly assume that there is some simple distribution law for negation and conditionals, so there is some additional value to reviewing these particular examples.) \u00a0The first task is to determine what kinds of sentences these are. \u00a0If the first symbol of your expression is a negation, then you know the sentence is a negation. \u00a0The first sentence above is a negation. \u00a0If the first symbol of your expression is a parenthesis, then for our logical language we know that we are dealing with a connective that combines two sentences.<\/p>\n

                                                      The way to proceed is to match parentheses. \u00a0Generally people are able to do this by eye, but if you are not, you can use the following rule. \u00a0Moving left to right, the last \u201c(<\/span>\u201d that you encounter always matches the first \u201c)<\/span>\u201d that you encounter. \u00a0These form a sentence that must have two parts combined with a connective. \u00a0You can identify the two parts because each will be an atomic sentence, a negation sentence, or a more complex sentence bound with parentheses on each side of the connective.<\/p>\n

                                                      In our propositional logic, each set of paired parentheses forms a sentence of its own. \u00a0So, when we encounter a sentence that begins with a parenthesis, we find that if we match the other parentheses, we will ultimately end up with two sentences as constituents, one on each side of a single connective. \u00a0The connective that combines these two parts is called the \u201cmain connective\u201d, and it tells us what kind of sentence this is. \u00a0Thus, above we have examples of a negation, a conditional, and a conditional.<\/p>\n

                                                      How should we understand the meaning of these sentences? \u00a0Here we can use truth tables in a new, third way (along with defining a connective and checking arguments). \u00a0Our method will be this.<\/p>\n

                                                      First, write out the sentence on the right, leaving plenty of room. \u00a0Identify what kind of sentence this is. \u00a0If it is a negation sentence, you should add just to the left a column for the non-negated sentence. \u00a0This is because the truth table defining negation tells us what a negated sentence means in relation to the non-negated sentence that forms the sentence. \u00a0If the sentence is a conditional, make two columns to the left, one for the antecedent and one for the consequent. \u00a0If the sentence is a conjunction, make two columns to the left, one for each conjunct. \u00a0Here again, we do this because the semantic definitions of these connectives tell us what the truth value of the sentence is, as a function of the truth value of its two parts. \u00a0Continue this process until the parts would be atomic sentences. \u00a0Then, we stipulate all possible truth values for the atomic sentences. \u00a0Once we have done this, we can fill out the truth table, working left to right.<\/p>\n

                                                      Let\u2019s try it for \u00ac(P<\/strong><\/span>\u2192Q<\/strong>)<\/span>. \u00a0We write it to the right.<\/p>\n\n\n\n\n\n\n\n
                                                      \u00a0\u00ac(P\u2192Q)<\/span><\/th>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/th>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      This is a negation sentence, so we write to the left the sentence being negated.<\/p>\n\n\n\n\n\n\n\n
                                                      \u00a0(P\u2192Q)<\/span><\/th>\n\u00ac(P\u2192Q)<\/span><\/th>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/td>\n\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/td>\n\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/td>\n\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n
                                                      \u00a0\u00a0\u00a0<\/span><\/td>\n\u00a0\u00a0\u00a0<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      This sentence is a conditional. \u00a0Its two parts are atomic sentences. \u00a0We put these to the left of the dividing line, and we stipulate all possible combinations of truth values for these atomic sentences.<\/p>\n\n\n\n\n\n\n\n
                                                      P<\/span><\/th>\nQ<\/span><\/th>\n\u00a0(P\u2192Q)<\/span><\/th>\n\u00ac(P\u2192Q)<\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nF<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF<\/span><\/td>\n<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      Now, we can fill out each column, moving left to right. \u00a0We have stipulated the values for P<\/span>\u00a0<\/strong>and Q<\/span><\/strong>, so we can identify the possible truth values of (P<\/strong>\u2192Q<\/strong>)<\/span>. \u00a0The semantic definition for \u201c\u2192<\/span>\u201d tells us how to do that, given that we know for each row the truth value of its parts.<\/p>\n\n\n\n\n\n\n\n
                                                      P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\nQ<\/span><\/th>\n\u00a0(P\u2192Q)<\/span><\/th>\n\u00ac(P\u2192Q)<\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\n<\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      This column now allows us to fill in the last column. \u00a0The sentence in the last column is a negation of (P<\/strong>\u2192Q<\/strong>)<\/span>, so the definition of \u201c\u00ac<\/span>\u201d tell us that \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true when (P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false, and \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false when (P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true.<\/p>\n\n\n\n\n\n\n\n
                                                      P<\/span><\/th>\nQ<\/span><\/th>\n(P\u2192Q)<\/span><\/th>\n\u00ac(P\u2192Q)<\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nF<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      This truth table tells us what \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0means in our propositional logic. \u00a0Namely, if we assert \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0we are asserting that P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false.<\/p>\n

                                                      We can make similar truth tables for the other sentences.<\/p>\n\n\n\n\n\n\n\n
                                                      P<\/span><\/th>\nQ<\/span><\/th>\n\u00ac<\/span>P<\/span><\/th>\n(\u00ac<\/span>P<\/span>\u2192Q)<\/span><\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      F\u00a0<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      F\u00a0<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      How did we make this table? \u00a0The sentence (<\/span>\u00ac<\/span>P<\/span><\/strong>\u2192Q<\/strong>)<\/span>\u00a0is a conditional with two parts, \u00ac<\/span>P<\/span>\u00a0<\/strong>and\u00a0Q<\/strong><\/span>. \u00a0Because Q<\/span>\u00a0<\/strong>is atomic, it will be on the left side. \u00a0We make a row for \u00ac<\/span>P<\/span><\/strong>. \u00a0The sentence \u00ac<\/span>P<\/span>\u00a0<\/strong>is a negation of P<\/span><\/strong>, which is atomic, so we put P<\/span>\u00a0<\/strong>also on the left. \u00a0We fill in the columns, going left to right, using our definitions of the connectives.<\/p>\n

                                                      And:<\/p>\n\n\n\n\n\n\n\n
                                                      P \u00a0 \u00a0 \u00a0 \u00a0<\/span><\/th>\nQ<\/span><\/th>\n\u00acP<\/span><\/th>\n\u00a0\u00acQ<\/span><\/th>\n(\u00acP\u2192\u00acQ)<\/span><\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      Such a truth table is very helpful in determining when sentences are, and are not, equivalent. \u00a0We have used the concept of equivalence repeatedly, but have not yet defined it. \u00a0We can offer a semantic, and a syntactic, explanation of equivalence. \u00a0The semantic notion is relevant here: \u00a0we say two sentences \u03a6<\/span>\u00a0<\/strong>and \u03a8<\/span>\u00a0<\/strong>are \u201cequivalent\u201d or \u201clogically equivalent\u201d when they must have the same truth value. \u00a0(For the syntactic concept of equivalence, see section 9.2). \u00a0These truth tables show that these three sentences are not equivalent, because it is not the case that they must have the same truth value. \u00a0For example, if P<\/span>\u00a0<\/strong>and Q<\/span>\u00a0<\/strong>are both true, then \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is false but (\u00acP<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true and (\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span>\u00a0is true. \u00a0If P<\/span>\u00a0<\/strong>is false and Q<\/span>\u00a0<\/strong>is true, then (\u00acP<\/strong>\u2192Q<\/strong>)<\/span>\u00a0is true but (\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span>\u00a0is false. \u00a0Thus, each of these sentences is true in some situation where one of the others is false. \u00a0No two of them are equivalent.<\/p>\n

                                                      We should consider an example that uses conjunction, and which can help in some translations. \u00a0How should we translate \u201cNot both Steve and Tom will go to Berlin\u201d? \u00a0This sentence tells us that it is not the case that both Steve will go to Berlin and Tom will go to Berlin. \u00a0The sentence does allow, however, that one of them will go to Berlin. \u00a0Thus, let U<\/span>\u00a0<\/strong>mean Steve will go to Berlin<\/span>\u00a0and V<\/span>\u00a0<\/strong>mean Tom will go to Berlin.<\/span>\u00a0 Then we should translate this sentence, \u00ac(U<\/strong>^V<\/strong>)<\/span>. We should not translate the sentence (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>. \u00a0To see why, consider their truth tables.<\/p>\n\n\n\n\n\n\n\n
                                                      U<\/span><\/th>\nV<\/span><\/th>\n(U^V)<\/span><\/th>\n\u00ac(U^V)<\/span><\/th>\n\u00acU<\/span><\/span><\/th>\n\u00acV<\/span><\/span><\/th>\n(\u00acU^\u00acV)<\/span><\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\n<\/tr>\n
                                                      F<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\n<\/tr>\n
                                                      F<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      We can see that \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0and (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>\u00a0are not equivalent. \u00a0Also, note the following. \u00a0Both \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0and (\u00acU<\/strong>^\u00acV<\/strong>)<\/span>\u00a0are true if Steve does not go to Berlin and Tom does not go to Berlin. \u00a0This is captured in the last row of this truth table, and this is consistent with the meaning of the English sentence. \u00a0But, now note: \u00a0it is true that not both Steve and Tom will go to Berlin, if Steve goes and Tom does not. \u00a0This is captured in the second row of this truth table. \u00a0It is true that not both Steve and Tom will go to Berlin, if Steve does not go but Tom does. \u00a0This is captured in the third row of this truth table. \u00a0In both kinds of cases (in both rows of the truth table), \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0is true but (\u00acU<\/strong>^\u00acV<\/strong>) <\/span>is false. \u00a0Thus, we can see that \u00ac(U<\/strong>^V<\/strong>)<\/span>\u00a0is the correct translation of \u201cNot both Steve and Tom will go to Berlin\u201d.<\/p>\n

                                                      Let\u2019s consider a more complex sentence that uses all of our connectives so far: \u00a0((P<\/strong>^\u00acQ<\/strong>)\u2192\u00ac(P<\/strong>\u2192Q<\/strong>))<\/span>. \u00a0This sentence is a conditional. \u00a0The antecedent is a conjunction. \u00a0The consequent is a negation. \u00a0Here is the truth table, completed.<\/p>\n\n\n\n\n\n\n\n
                                                      P<\/span><\/th>\nQ<\/span><\/th>\n\u00a0\u00acQ<\/th>\n(P\u2192Q)<\/th>\n(P^\u00acQ)<\/th>\n\u00a0 \u00ac(P\u2192Q)<\/span><\/th>\n((P^\u00acQ)\u2192\u00ac(P\u2192Q))<\/span><\/th>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n\u00a0F<\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\nF\u00a0<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      T<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\n\u00a0T<\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      F<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n\u00a0F<\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n
                                                      F<\/span><\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\n\u00a0T<\/strong><\/em><\/td>\nT<\/strong><\/em><\/td>\nF<\/strong><\/em><\/td>\nF<\/span><\/strong><\/em><\/td>\nT<\/span><\/strong><\/em><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      This sentence has an interesting property: \u00a0it cannot be false. \u00a0That is not surprising, once we think about what it says. \u00a0In English, the sentence says: If P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false, then it is not the case that P<\/span>\u00a0<\/strong>implies Q<\/span><\/strong>. \u00a0That must be true: \u00a0if it were the case that P <\/span><\/strong>implies Q<\/span><\/strong>, then if P<\/span>\u00a0<\/strong>is true then Q<\/span>\u00a0<\/strong>is true. \u00a0But the antecedent says P<\/span>\u00a0<\/strong>is true and Q<\/span>\u00a0<\/strong>is false.<\/p>\n

                                                      Sentences of the propositional logic that must be true are called \u201ctautologies\u201d. \u00a0We will discuss them at length in later chapters.<\/p>\n

                                                      Finally, note that we can combine this method for finding the truth conditions for a complex sentence with our method for determining whether an argument is valid using a truth table. \u00a0We will need to do this if any of our premises or the conclusion are complex. \u00a0Here is an example. \u00a0We\u2019ll start with an argument in English:<\/p>\n

                                                      If whales are mammals, then they have vestigial limbs. \u00a0If whales are mammals, then they have a quadrupedal ancestor. \u00a0Therefore, if whales are mammals then they have a quadrupedal ancestor and they have vestigial limbs.<\/p>\n

                                                      We need a translation key.<\/p>\n

                                                      P<\/span><\/strong>: \u00a0Whales are mammals.<\/p>\n

                                                      Q<\/span><\/strong>: Whales have have vestigial limbs.<\/p>\n

                                                      R<\/span><\/strong>: Whales have a quadrupedal ancestor.<\/p>\n

                                                      The argument will then be symbolized as:<\/p>\n

                                                      \n

                                                      (P<\/strong>\u2192Q<\/strong>)<\/span><\/p>\n

                                                      (P<\/strong>\u2192R<\/strong>)<\/span><\/p>\n

                                                      ____<\/span><\/strong><\/p>\n

                                                      (P<\/strong>\u2192(R<\/strong>^Q<\/strong>))<\/span><\/p>\n<\/div>\n

                                                      Here is a semantic check of the argument.<\/p>\n\n\n\n\n\n\n\n\n\n\n\n\n
                                                      <\/td>\n<\/td>\n<\/td>\npremise<\/td>\npremise<\/td>\n<\/td>\n\u00a0conclusion<\/td>\n<\/tr>\n
                                                      P<\/span><\/td>\nQ<\/span><\/td>\nR<\/span><\/td>\n(P\u2192Q)<\/strong><\/td>\n(P\u2192R)<\/strong><\/td>\n(R^Q)<\/strong><\/td>\n(P\u2192(R^Q))<\/strong><\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nF <\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      T\u00a0<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      T<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nF <\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF <\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n
                                                      F<\/span><\/td>\nF<\/span><\/td>\nF<\/span><\/td>\nT\u00a0<\/span><\/td>\nT<\/span><\/td>\nF<\/span><\/td>\nT<\/span><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

                                                      We have highlighted the rows where the premises are all true. \u00a0Note that for these, the conclusion is true. \u00a0Thus, in any kind of situation in which all the premises are true, the conclusion is true. \u00a0This is equivalent, we have noted, to our definition of valid: \u00a0necessarily, if all the premises are true, the conclusion is true. \u00a0So this is a valid argument. \u00a0The third column of the analyzed sentences (the column for (R<\/strong>^Q<\/strong>)<\/span>) is there so that we can identify when the conclusion is true. \u00a0The conclusion is a conditional, and we needed to know, for each kind of situation, if its antecedent P<\/span><\/strong>, and if its consequent (R<\/strong>^Q<\/strong>)<\/span>, are true. \u00a0The third column tells us the situations in which the consequent is true. \u00a0The stipulations on the left tell us in what kind of situation the antecedent P<\/span>\u00a0<\/strong>is true.<\/p>\n

                                                      \n
                                                      \n

                                                      Check for Understanding<\/p>\n<\/header>\n

                                                      \n
                                                        \n
                                                      1. Create your own complex sentence using negation, conjunction, and conditional, and construct its truth table to determine under what conditions it is true.<\/li>\n
                                                      2. Consider the English sentence “Either Jacob won’t go to Paris or he will go to Rome.” Translate this into symbolic form and explain your reasoning.<\/li>\n
                                                      3. Critical Thinking Task:\u00a0<\/strong>Generate a song, image, or poem explaining the concept of complex sentences.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n

                                                        5.7\u00a0 Problems<\/h2>\n
                                                          \n
                                                        1. Self Reflection:\u00a0<\/strong>What are some of the challenges you are facing with complex sentences? Why do you believe this is the case?<\/li>\n
                                                        2. Translate the following sentences into our logical language. \u00a0You will need to create your own key to do so.\n
                                                            \n
                                                          1. Ulysses, who is crafty, is from Ithaca.<\/li>\n
                                                          2. Ulysses, who isn’t crafty, is from Ithaca.<\/li>\n
                                                          3. Ulysses, who is crafty, isn’t from Ithaca.<\/li>\n
                                                          4. Ulysses isn’t both crafty and from Ithaca.<\/li>\n
                                                          5. Ulysses will go home only if he’s from Ithaca and not Troy.<\/li>\n
                                                          6. Ulysses is not both from Ithaca and Troy, though he is crafty.<\/li>\n
                                                          7. If Ulysses outsmarts both Circes and the Cyclops, then he can go home.<\/li>\n
                                                          8. If Ulysses outsmarts Circes but not the Cyclops, then he will be eaten.<\/li>\n
                                                          9. Though he won’t outsmart Circe, Ulysses will outsmart the Cyclops, even given that he is from Ithaca.<\/li>\n
                                                          10. Ulysses won’t outsmart both Circes and the Cyclops, but he won’t be eaten and will go home even though he is from Ithaca.<\/li>\n<\/ol>\n<\/li>\n
                                                          11. Prove the following arguments are valid, using a direct derivation.\n
                                                              \n
                                                            1. Premise: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ \u00acQ<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00acP<\/strong><\/span>.<\/li>\n
                                                            2. Premises: \u00a0(((P<\/strong>\u2192Q<\/strong>) ^ (Q<\/strong>\u2192R<\/strong>))^P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0R<\/span><\/strong>.<\/li>\n
                                                            3. Premises: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ (R<\/strong>\u2192S<\/strong>))<\/span>, (\u00acQ<\/strong> ^ \u00acS<\/strong>)<\/span>. \u00a0Conclusion: \u00a0(\u00acP<\/strong> ^ \u00acR<\/strong>)<\/span>.<\/li>\n
                                                            4. Premises: \u00a0((R<\/strong> ^ S<\/strong>) <\/span>\u2192<\/span>\u00a0T<\/strong>)<\/span>, (Q<\/strong> ^ \u00acT<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00ac(R<\/strong> ^ S<\/strong>)<\/span>.<\/li>\n
                                                            5. Premises: \u00a0(P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span>, (Q<\/strong> ^ P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0R<\/span><\/strong>.<\/li>\n
                                                            6. Premises: \u00a0(P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span>, (\u00acR<\/strong> ^ P<\/strong>)<\/span>. \u00a0Conclusion: \u00a0\u00acQ<\/strong><\/span>.<\/li>\n
                                                            7. Premises: \u00a0((P<\/strong>^Q<\/strong>)\u2192(R<\/strong>\u2192S<\/strong>))<\/span>, (Q<\/strong> ^ (P<\/strong> ^ \u00acS<\/strong>))<\/span>. \u00a0Conclusion: \u00a0(\u00acR<\/strong> ^ Q<\/strong>)<\/span>.<\/li>\n
                                                            8. Premises: \u00a0((P<\/strong>\u2192Q<\/strong>) ^ (R<\/strong>\u2192S<\/strong>))<\/span>, (P<\/strong> ^ R<\/strong>)<\/span>. \u00a0Conclusion: \u00a0((P<\/strong> ^ Q<\/strong>) ^ (R<\/strong> ^ S<\/strong>))<\/span>.<\/li>\n<\/ol>\n<\/li>\n
                                                            9. Make truth tables for the following complex sentences. \u00a0Identify which are tautologies.\n
                                                                \n
                                                              1. \u00ac(P<\/strong> ^ Q<\/strong>)<\/span><\/li>\n
                                                              2. \u00ac(\u00acP<\/strong> \u2192 \u00acQ<\/strong>)<\/span><\/li>\n
                                                              3. (P<\/strong> ^ \u00acP<\/strong>)<\/span><\/li>\n
                                                              4. \u00ac(P<\/strong> ^ \u00acP<\/strong>)<\/span><\/li>\n
                                                              5. (((P<\/strong>\u2192Q<\/strong>)^ \u00acQ<\/strong>)\u2192\u00acP<\/strong>)<\/span><\/li>\n
                                                              6. (((P<\/strong>\u2192Q<\/strong>)^ \u00acP<\/strong>)\u2192\u00acQ<\/strong>)<\/span><\/li>\n
                                                              7. (((P<\/strong>\u2192Q<\/strong>)^ P<\/strong>)\u2192Q<\/strong>)<\/span><\/li>\n
                                                              8. (((P<\/strong>\u2192Q<\/strong>)^ Q<\/strong>)\u2192P<\/strong>)<\/span><\/li>\n<\/ol>\n<\/li>\n
                                                              9. Make truth tables to show when the following sentences are true and when they are false. \u00a0State which of these sentences are equivalent. Also, can you identify if any have the same truth table as some of our connectives?\n
                                                                  \n
                                                                1. \u00ac(P<\/strong>^Q<\/strong>)<\/span><\/li>\n
                                                                2. (\u00acP<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\n
                                                                3. \u00ac(\u00acP<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\n
                                                                4. \u00ac(P<\/strong>\u2192Q<\/strong>)<\/span><\/li>\n
                                                                5. (P<\/strong>^\u00acQ<\/strong>)<\/span><\/li>\n
                                                                6. (\u00acP<\/strong>^Q<\/strong>)<\/span><\/li>\n
                                                                7. \u00ac(P<\/strong>\u2192\u00acQ<\/strong>)<\/span><\/li>\n
                                                                8. \u00ac(\u00acP<\/strong>\u2192\u00acQ<\/strong>)<\/span><\/li>\n
                                                                9. (P<\/strong> ^ (Q<\/strong> ^ R<\/strong>))<\/span><\/li>\n
                                                                10. ((P<\/strong> ^ Q<\/strong>) ^ R<\/strong>))<\/span><\/li>\n
                                                                11. (P<\/strong>\u2192(Q<\/strong>\u2192R<\/strong>))<\/span><\/li>\n
                                                                12. ((P<\/strong>\u2192Q<\/strong>)\u2192R<\/strong>))<\/span><\/li>\n<\/ol>\n<\/li>\n
                                                                13. Write a valid argument in normal colloquial English with at least two premises, one of which is a conjunction or includes a conjunction. \u00a0Your argument should just be a paragraph (not an ordered list of sentences or anything else that looks like formal logic). \u00a0Translate the argument into propositional logic. \u00a0Prove it is valid.<\/li>\n
                                                                14. Write a valid argument in normal colloquial English with at least three premises, one of which is a conjunction or includes a conjunction and one of which is a conditional or includes a conditional. \u00a0Translate the argument into propositional logic. \u00a0Prove it is valid.<\/li>\n
                                                                15. Often in a natural language like English, there are many implicit conjunctions in descriptions and other phrases. Here are some passages from literature. Translate them into our propositional logic. You will want to make a separate key for each particular problem.\n
                                                                    \n
                                                                  1. “But Achilles the son of Peleus again shouted at Agamemnon the son of Atreus, for he was still in a rage.”
                                                                    \n(Homer, The Illiad<\/i>)<\/li>\n
                                                                  2. “Socrates is an evil-doer,\u00a0and a curious person, who searches into things under the earth and in heaven,\u00a0and he makes the worse appear the better cause.\u2026\u201d (Plato, The Apology<\/i>)<\/li>\n
                                                                  3. “Incensed with indignation, Satan stood
                                                                    \nUnterrified\u2026.” (Milton, Paradise Lost<\/i>)<\/li>\n
                                                                  4. “Teiresias, seer who comprehends all\u2026
                                                                    \nYou know, though thy blind eyes see nothing,
                                                                    \nWhat plague infects our city Thebes.” (Sophocles, Oedipus Rex<\/i>)<\/li>\n
                                                                  5. “Scrooge! a squeezing, wrenching, grasping, scraping, clutching, covetous, old sinner!” (Charles Dickens, “A Christmas Carrol”)<\/li>\n
                                                                  6. “When I wrote the following pages, or rather the bulk of them, I lived alone, in the woods, a mile from any neighbor, in a house which I had built myself, on the shore of Walden Pond, in Concord, Massachusetts, and earned my living by the labor of my hands only.” [Here one can substitute \u201cThoreau\u201d for \u201cI\u201d in the translation, if helpful.]. (Henry David Thoreau, Walden<\/i>)<\/li>\n
                                                                  7. “In appearance Shatov was in complete harmony with his convictions: he was short, awkward, had a shock of flaxen hair, broad shoulders, thick lips, very thick overhanging white eyebrows, a wrinkled forehead, and a hostile, obstinately downcast, as it were shamefaced, expression in his eyes.” (Fyodor Dostoevsky, The Possessed<\/i>)<\/li>\n<\/ol>\n<\/li>\n
                                                                  8. Make your own key to translate the following argument into our propositional logic. \u00a0Translate only the parts in bold. \u00a0Prove the argument is valid.<\/li>\n<\/ol>\n
                                                                      \n
                                                                    1. <\/li>\n<\/ol>\n
                                                                        \n
                                                                      1. <\/li>\n<\/ol>\n
                                                                          \n
                                                                        1. <\/li>\n<\/ol>\n
                                                                            \n
                                                                          1. <\/li>\n<\/ol>\n
                                                                              \n
                                                                            1. <\/li>\n<\/ol>\n

                                                                              \u201cI suspect Dr. Kronecker of the crime of stealing Cantor\u2019s book,\u201d Inspector Tarski said. \u00a0His assistant, Mr. Carroll, waited patiently for his reasoning. \u00a0\u201cFor,\u201d Tarski said, \u201cThe thief left cigarette ashes on the table. \u00a0The thief also did not wear shoes, but slipped silently into the room. \u00a0Thus, If Dr. Kronecker smokes and is in his stocking feet, then he most likely stole Cantor\u2019s book<\/strong>.<\/span>\u201d \u00a0At this point, Tarski pointed at Kronecker\u2019s feet. \u00a0\u201cDr. Kronecker is in his stocking feet.<\/span>\u201d \u00a0Tarski reached forward and pulled from Kronecker\u2019s pocket a gold cigarette case. \u00a0\u201cAnd Kronecker smokes<\/span><\/strong>.\u201d \u00a0Mr. Carroll nodded sagely, \u201cYour conclusion is obvious: \u00a0Dr. Kronecker most likely stole Cantor\u2019s book<\/strong>.\u201d<\/span><\/p>\n","protected":false},"author":158,"menu_order":5,"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-39","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\/39","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":21,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions"}],"predecessor-version":[{"id":347,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapters\/39\/revisions\/347"}],"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\/39\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/media?parent=39"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/pressbooks\/v2\/chapter-type?post=39"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/contributor?post=39"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/introtologic\/wp-json\/wp\/v2\/license?post=39"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}