PHIL 012 2/14/2001 Logical Equivalence & Translation

Outline Announcements Logical Equivalence Association, Idempotence, & Commutativity Double Negation DeMorgan’s Theorems Translation Hand Back Tests

Announcements Reminder about new homework policy. Check schedule to see what homework is due via disk & hardcopy on Monday. Test solutions are online. UTS is mailing scanner results. Test forms handed back after class.

Logical Equivalence Two statements are said to be logically equivalents of one another IFF they have the same truth value in every world. The symbol for logical equivalence is , though some texts use . Note that  is not a logical connective.

Rules: Association Suppose that P, Q, & R are statements... (P ^ Q ^ R)  (P ^ Q) ^ R  P ^ (Q ^ R) It is not the case, however, that (P ^ Q) v R  P ^ (Q v R) Association works only if the connectives are all the same. You will not need to cite association.

Rules: Idempotence If a conjunction has a repeated conjunct, then the repeated conjunct may be removed without changing truth value. Idempotence of ^: P ^ Q ^ P  P ^ Q and (P ^ Q) ^ (R ^ S) ^ (P ^ S)  P ^ Q ^ R ^ S

Rules: Idempotence If a disjunction has a repeated disjunct, then the repeated conjunct may be removed without changing truth value. Idempotence of v: P v Q v P  P v Q and (P v Q) v (R v S) v (P v S)  P v Q v R v S

Commutativity ^ and v are Commutative. This means that the arrangement of a sequence of conjuncts or disjuncts does not affect truth value. So, P ^ Q ^ R  P ^ R ^ Q and P v Q v R  P v R v Q

Double Negation  P  P and, in general, an even number of  preserves truth value whereas an odd number of  flips truth value. So, if P is TRUE,  P will also be TRUE whereas  P will be FALSE.

DeMorgan’s Theorems  is not distributive. So, it is not the case that  (P ^ Q)   P ^  Q However, DeMorgan’s Theorems state that  (P ^ Q)   P v  Q and  (P v Q)   P ^  Q

DeMorgan’s Theorems So, DeMorgan’s Theorems allow us to –switch back and forth between ^ and v –reduce an expression to a series of literals A literal is either P or  P, in contrast with, for example,  (P ^ Q). That is, a literal is an atomic sentence or its negation.

Simplification Examples

Translation In general, any translation that has the same truth value in all possible worlds is a good translation. However, some translations are better than others in that they preserve word order.

Translation Either Mary is not Home or Susan is not Happy.  Happy(Susan) v  Home(Mary) = good  (Home(Mary) ^ Happy(Susan)) = good  Home(Mary) v  Happy(Susan) = best All preserve truth value, but the last is best because it preserves word order.

Questions

Hand Back Tests