PHIL12A Section answers, 14 February 2011 - Philosophy

could abbreviate the sentence in the following way: ¬(P ∧ Q) ∨ R, where P stands for ‘all men are mortal’, Qstands for ‘Socrates is a man’, and R stands for ‘Socrates is mortal’. This sentence is not a tautology, as drawingup a truth table will show.The moral of the story is that our truth table method is not fine-grained enough to capture the meanings ofpredicates and names. There is logical structure embedded within the atomic sentences that cannot be capturedusing truth tables. (In fact, many logic textbooks teach logic in two stages: propositional logic, dealing onlywith complex sentences made up of atomic sentences connected by truth-functional connectives; and predicatelogic, which allows you to get into the structure of atomic sentences by working with predicates, names andquantifiers. Truth tables are taught in propositional logic, but drop out of the picture with predicate logic.)2. Name at least two sentences that are logically necessary but not tautological.For example:(a) All bachelors are unmarried men.(b) There are infinitely many prime numbers.3. Name at least two sentences that are TW-necessary but not logically necessary.For example:(a) Cube(a) ∨ Tet(a) ∨ Dodec(a)(b) (Large(a) ∧ Large(b)) ∧ ¬Adjoins(a,b)4. For each of the following, determine whether the sentence is a tautology, a logical necessity, or a TWnecessity.(a) a=b ∨ b=c ∨ c=cThis is not a necessity of any sort, though it is logically possible.(b) BackOf(a,b) ∨ ¬BackOf(a,b)This is a tautology, and therefore also a logical necessity and a TW necessity.(c) ¬(Cube(b) ∧ Cube(e)) ∨ Cube(b)A truth table shows that this is a tautology, and therefore also a logical necessity and a TW necessity.(d) ¬(Cube(b) ∧ Cube(e)) ∨ SameShape(b,e)This is a TW necessity, since in Tarski’s World there are one of two possibilities: either b and e are thesame kind of block, for example they are both tetrahedrons or they are both cubes, in which case thesentence is true; or they are different shapes. But if they are different shapes, then they are certainly notboth cubes, which means that the sentence is true.The sentence is not a tautology, as a truth table will show; however it is a logical necessity, as far as I cantell.4