Lecture Notes - Chapter 4
by Brian W. Carver
A sentence is TT-possible iff at least one row on its truth table reads T in the column under its main connective.
A sentence is a tautology iff every row on its truth table reads T in the column under its main connective.
A sentence is TW-possible iff it is true in at least one world in Tarski's World.
A sentence is TW-necessary iff it is true in every world in Tarski's World (in which it has a truth value).
(Note: We add "in which it has a truth value" to exclude worlds in which objects named in the sentence do not exist.)
Two sentences are tautologically equivalent iff every row of their joint truth table assigns the same values to each.
The truth-table method of checking validity: We can define a notion of tautological consequence that explains when one sentence follows from some other(s). Do a joint truth table for the sentences. If you cannot find a row where all the premises are true, and the conclusion is false, then you know the argument is valid, and that the conclusion is a tautological consequence of the premises. We just fill out the table according to the rules. If we can find a row that goes TTT F (for example) then the argument is invalid, and the conclusion is not a tautological consequence of the premises. (The preceeding example assumes our conclusion is on the far right, but we could consider whether any sentence, no matter where it was written, was a tautological consequence of any others in a similar way. For example, if we listed the conclusion first, on the far left, and all the premises to the right, then we would instead be looking for a F TTT pattern, for instance.)If we constructed a joint truth table of an argument having just two premises, P1 and P2, and a conclusion, C, where P1, P2, and C are sentences of FOL built up from atomic sentences by means of truth-functional connectives alone, then we would have:
C is a tautological consequence of P1 and P2 iff every row that assigns T to P1 and P2, also assigns T to C.