Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 19Slide 20Slide 21Slide 22Slide 23Slide 24Slide 25Slide 26Slide 27Slide 28Slide 29Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Truth-functional form . . .We complicated our FOL when we introduced quantifiers. Remember that quantified expressions are not truth-functional (i.e., the truth values of quantified expressions don’t ‘build up’ from the truth values of the atomic parts of those expressions in any straightforward way)Quantification is not truth-functional:TT Tb is a cube and c is a cube. vs.?Every rich actor is a good actor.How we identified tautologies in the good old days: Replace each atomic sentence with a capital letter and see if the necessary truth of the sentence is apparent even when we’re ‘blinded’ to the meanings of the atomic sentences and predicates …Cube(a) ᴠ ¬Cube(a) A ᴠ ¬ALikewise, tautological consequence . . .Tet(c) Dodec(a) A BTet(c) ADodec(a)B#But how do quantifiers fit in now?Two cases of modus ponens?? Yet, the first argument is valid while the second is not …∀x (Cube(x) Small(x))∀x Cube(x)∀x Small(x)**Ǝx (Cube(x) Small(x)) counterexample:Ǝx Cube(x)Ǝx Small(x) large largeIn 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. ** satisfies 1st premiseƎx (Cube(x) Small(x))Ǝx Cube(x)Ǝx Small(x) large largeAs 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-quantified conditional 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.** satisfies 2nd premiseƎx (Cube(x) Small(x))Ǝx Cube(x)Ǝx Small(x) large largeActually, neither of the previous arguments is really a case of the tautology form modus ponens. The problem is that the quantified phrases are not internally truth-functional, so they are throwing off our ability to accurately determine the presence or absence of a tautology …In order to determine tautological relations among sentences that contain quantification, we will need to be ‘blind’ to the inner workings of the 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 will give us truth-functional form.The first of the above arguments put in truth-functional form:truth-functional form∀x (Cube(x) Small(x)) A∀x Cube(x) B∀x Small(x) C*Logical consequence . . . but not tautological consequenceAnd, similarly, the second argument above in truth-functional form:* truth-functional formƎx (Cube(x) Small(x)) AƎx Cube(x) BƎx Small(x) CNeither logical nor tautological consequence …Now we have a way to identify tautological relations in sentences containing quantifiers …… but we have to treat everything within the scope of a quantifier as a single ‘chunk’ the same way we did atomic sentences before, where we ‘blind’ ourselves to what’s inside the chunk.Truth-functional form: Replace . . .(1)quantified phrases (i.e., the quantifier along with everything in its scope) and (2) any atomic sentences . . . with capital letters.Ǝy (Large(y) ᴧ LeftOf(y,c)) ᴠ ¬ Ǝy (Large(y) ᴧ LeftOf(y,c))quantified phrase quantified phrase A ᴠ ¬ ATruth-functional form algorithm (p. 263)*(a) Annotate the sentences by underlining and labeling each ‘chunk’ that we must blind ourselves to in order to see the truth-functional form of the sentence. There are two kinds of ‘chunks’: (1) quantified phrases (i.e., underline beginning at the quantifier and continue underlining until you reach the end of the phrase over which the quantifier has scope), and (2) any atomic sentences that are not within quantified. Label each such underlined phrase with a capital letter, using the same capital letter only for multiple underlined phrases that are exactly alike.(b) Replace each of the entire underlined constituents from (a) above with a capital letter as guided by the letters you used to label each constituent.For example:(Ǝy (P(y) ᴠ R(y)) ∀x (P(x) ᴧ Q(x))) (¬∀x (P(x) ᴧ Q(x)) ¬Ǝy (P(y) ᴠ R(y)))First step (underline and annotate):(Ǝy (P(y) ᴠ R(y))A ∀x (P(x) ᴧ Q(x))B ) (¬∀x (P(x) ᴧ Q(x))B ¬Ǝy (P(y) ᴠ R(y))A )Second step (replace with capital letters):(A B) (¬B ¬A)It’s a tautology !Sentences on p. 264 (which are tautologies?):FO Sentence T. F. Form∀x Cube(x) ᴠ ¬ ∀x Cube(x) A ᴠ ¬A(Ǝy Tet(y) ᴧ ∀z Small(z)) ∀z Small(z) (A ᴧ B) B∀x Cube(x) ᴠ Ǝy Tet(y) A ᴠ B∀x Cube(x) Cube(a) A B∀x (Cube(x) ᴠ ¬Cube(x)) A∀x (Cube(x) Small(x)) ᴠ Ǝx Dodec(x) A ᴠ BAnswer: only the first two sentences are tautologies.T. F. Form truth?1. ∀x x≠x ___________ __________*Moving on to another set of sentences, what do you make of the sentence above? It would translate as “for every object x, x does not equal itself” or more simply,
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