Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Language, Proof and Logic Conditionals Chapter 7 7.1 Material conditional symbol: Truth table: P Q PQ T T F F T F T F T F T T Game: PQ is treated as PQ The following English constructions are all translated as PQ: If P then Q; P only if Q; Provided P, Q. The following are translated as PQ: Unless P, Q; Q unless P. Q is a logical consequence of P1,…,Pn iff (P1…Pn)Q is a logical truth. 7.2 Biconditional symbol: Truth table: P Q PQ T T F F T F T F T F F T Game: P Q is treated as (PQ)(QP) The following English construction is translated as PQ: P if and only if (iff) Q. P and Q are logically equivalent (PQ) iff PQ is a logical truth. 7.3 Conversational implicature If the assertion of a sentence carries with it a suggestion that could be cancelled (without contradiction) by further elaboration by the speaker, then the suggestion is (just) a conversational implicature, not part of the content of the original claim. The waiter says: “You can have either soup or salad”. The suggested exclusiveness is cancelled by the further elaboration: “And you can have both if you wish”. A father tells his son: “You can have dessert only if you eat all your beans”. The suggested promise of dessert is cancelled by the further elaboration: “If you eat the beans, I’ll check to see if there is any ice cream left”. 7.4.a Truth-functional completeness Do we need to introduce any more truth-functional connectives? Such as, say, Neither P nor Q P unless Q If P then Q, otherwise R … etc. Not really! Theorem: The collection {,,} is truth-functionally complete, in the sense that these connectives allow us to express any truth function. In fact, so is just {}, where PQ means “neither P nor Q”. 7.4.b Truth-functional completeness Express (P,Q,R), defined by: Using only , express: • P • PQ • PQ P Q R T T T T F F F F T T F F T T F F T F T F T F T F (P,Q,R) T T F F T F T F