Slides / Bullets
• Making a truth-table for a single sentence
• Build reference columns: one for each atomic sentence in the sentence you’re making a truth-table for.
• More generally: one for each sentence that isn’t built up out of other sentences using truth-functional connectives.
• Fill in the reference columns so that every possible assignment of T’s and F’s to the atomic sentences is covered.
• Standard method: like counting backwards in binary
• Search through the connectives in the sentence for one whose column you can fill in.
• You can fill in the column under a connective if the sentence(s) the connective operates on are either (i) atomic, or (ii) sentences whose main connective’s column has already been filled in.
• Has the column for the main connective of the sentence been filled in yet? If it has, stop. Otherwise, return to step 3.
• Making a joint truth-table
• The procedure for multiple sentences is the same.
• You’ll need one reference column for each atomic sentence that occurs in ANY of the sentences.
• Tautological consequence
• A sentence Q is a tautological consequence of a sentence P if and only if Q is true in every row in their joint table in which P is true.
• More generally: Q is a tautological consequencce of P1, P2... Pn if and only if Q is true in every row in their joint table in which all of P1...Pn are true.
• Tautological equivalence
• P and Q are tautologically equivalent if and only if P is a tautological consequence of Q, and Q is a tautological consequence of P.
• In other words: Q is true in every row in their joint truth-table in which P is true, and P is true in every row in their joint truth-table in which Q is true.
• In other words: the columns under the main connectives of P and Q in their joint truth-table are identical.
• Tautological and logical consequence
• If Q is a tautological consequence of P1...Pn, then Q is a logical consequence of P1...Pn.
• Hence, if P and Q are tautologically equivalent, they are logically equivalent.
• The reverse does not hold.
• For next week: