DAY 28: TAUTOLOGIES AND QUANTIFICATIONAssigned reading sec. 10.1POWERPOINT SLIDE #1The main new thing to learn in today’s reading is the concept of truth-functional form. The reason we need truth-functional form is that we complicated our FOL when we introduced quantifiers. POWERPOINT SLIDE #2As we discussed in lesson 20, quantification is not truth-functional …POWERPOINT SLIDE #3Before, we had a neat way to identify tautologies—we just replaced each atomic sentence with a capital letter so that we would be ‘blind’ to the meanings of the predicates involved. This allowed us to focus on how the truth-functional connectives build up complex sentences from atomic sentences, and in ways that the truth value of the larger sentence could be computed (using a truth table) from the truth-values of its parts. For example, we could see that the sentence Cube(a) ᴠ ¬Cube(a) was a tautology when we replaced the atomic sentences in it with capital letters: A ᴠ ¬A (i.e., we can still see that this complex sentence couldnever be false, even without knowing what ‘A’ means. Whatever ‘A’ is, it must be either true or not true …)POWERPOINT SLIDE #4Likewise, using the same method we could see that the conclusion of an argument like the one below was a tautological consequence of its premises:Tet(c) Dodec(a) A BTet(c) ADodec(a) BBut how do quantifiers fit into that? We already said that they are not truth-functional connectives.POWERPOINT SLIDE #5Notice, for example, that though the first argument below is valid, the second is NOT, even though they may both look like an example of the same tautologically valid modus ponens argument as above:∀x (Cube(x) Small(x))∀x Cube(x)∀x Small(x)Ǝx (Cube(x) Small(x))Ǝx Cube(x)Ǝx Small(x)The premises of the second argument above could both be true in a world with a large cube and a large dodec, in which case the conclusion of the argument would be false (i.e., nothing being small). This counterexample would can be understood in the following way . . .POWERPOINT SLIDE #6In this world, the ‘some object’ of the first premise (which, if it’s a cube, will then be small) refers only to the dodec. The dodec is not a cube, of course, so it doesn’t follow that it has to be small (and it’s not; it’s large). Remember: a conditional with a false antecedent is, as a whole conditional statement, true. POWERPOINT SLIDE #7As for the second premise of the argument (i.e., “there is a cube”), it would be satisfied by the large cube in this world. Now, you might think that this large cube would in turn falsify the first premise. But it doesn’t have to, and here’s the really weird reason: We can simply take it as an arbitrary fact about this world that the large cube in question is not the ‘some’ object described by the first premise (i.e., the object which, if it’s a cube, will be small). Keep in mind that the conditional statement in the first premise is embedded within an existential statement. An existentially-quantified conditional of this sort does not have to refer to every cube that we might come across in the world in the same way that a universally-quantifiedconditional would require. The first premise is an existential statement; therefore, it can arbitrarily refer to one object but not to another. We can, therefore, simply stipulate that the existential statement in the first premise of the argument does not refer to the large cube in this world, so the conditional statement embedded within that existential statement doesn’t get triggered by the cube in question. Weird, huh? But logical.POWERPOINT SLIDE #8What the above example shows us is that in order to determine tautological relations, we will need to be ‘blind’ to the inner workings of quantified phrases (i.e., the quantifier and everything it scopes over) just as we had to be blind before to the inner meanings of atomic sentences with their predicates. This method gives us truth-functional form …POWERPOINT SLIDE #9If we force ourselves to be ‘blind’ in this way, then when we substitute capital letters for the entire quantified phrases in the last two arguments above, we won’t end mistaking these arguments for tautologies. That is, the whole expression ∀x (Cube(x) Small(x)) from the first argument above would be replaced by ‘A’, the whole second line by a different capital letter, andthe conclusion by yet a different letter, yielding:ABC… which clearly does not expresses a tautological consequence (though the original argument still expresses a case of logical consequence).POWERPOINT SLIDE #10Similarly, the second argument above can be put in truth-functional form. The conclusion of this argument is not a logical consequence of its premises (i.e., we already saw that it is an invalid argument),and from the truth-functional form we see that the conclusion is also not a tautological consequence of the premises.POWERPOINT SLIDE #11Once we recognize that we must ‘chunk’ quantified phrases in this way, we find out that we now have a way to spot tautological relations in sentences containing quantifiers (though the preceding cases were NOT examples of this!). And even with quantifiers we can build up complex sentences in a truth-functional way from their parts. The trick, again, is that we have to treat everything within the scope of a quantifier as a single ‘chunk’ the same we that we did atomic sentences before.POWERPOINT SLIDE #12Replacing quantified phrases (i.e., a quantifier with everything in its scope) and atomic sentences with capital letters gives us the truth-functional form of a sentence, and from the truth-functional form we can identity tautological relations. For example, the following sentence, when converted to its truth-functional form, turns out to be a case of the law of excluded middle, even though the sentence containsquantified phrases:Ǝy (Large(y) ᴧ LeftOf(y,c)) ᴠ ¬ Ǝy (Large(y) ᴧ LeftOf(y,c)) A ᴠ ¬AWe see this by replacing each of the quantified phrases with a capital letter—in this case the same capital letter because the two quantified phrases are exactly the same in this sentence. And we leave the two truth-functional connectives that do NOT fall inside a quantified phrase intact (in this case, the ᴠand the ¬ ).POWERPOINT SLIDE #13The book gives the formal procedure for transforming a sentence into truth-functional form on p. 263. Basically:a)
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