Slides / Bullets
 Making a truthtable for a single sentence
 Build reference columns: one for each atomic sentence in the sentence you’re making a truthtable for.
 More generally: one for each sentence that isn’t built up out of other sentences using truthfunctional 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 truthtable
 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 truthtable in which P is true, and P is true in every row in their joint truthtable in which Q is true.
 In other words: the columns under the main connectives of P and Q in their joint truthtable 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:
 Read: 4.14.4, chapter 5.
 Remember: exercises 4.2 and 4.44.7 are due next Tuesday, although they’ll be counted as part of last week’s homework from the point of view of grading.
 Also do 3.23 (30%), 4.13, 4.16, 4.17, (10% each), 4.22, 4.23 (15% each).