Language Proof and Logic The Boolean Connectives Chapter

Language, Proof and Logic The Boolean Connectives Chapter 3

3. 0 Truth-functional connectives The Boolean connectives are used to form compound sentences from atomic ones. They are called truth-functional connectives because the truth value of a compound sentence is fully determined by the truth values of its components. Truth tables are the most straightforward way to define Boolean connectives. Another way is through Henkin-Hintikka games.

3. 1 Negation Truth table: P P TRUE FALSE TRUE Game: Committing to the truth of P means committing to the falsity of P. Committing to the falsity of P means committing to the truth of P. Atomic sentences and their negations are called literals. “You try it” on page 69.

3. 2 Truth table: Conjunction P Q TRUE FALSE P Q TRUE FALSE Game: If you commit to the truth, then your opponent chooses P or Q, and the game continues as if you had committed to the truth of that simpler component. If you commit to the falsity, then you have to choose either P or Q and, and the game continues as if you had committed to the falsity of that component. “You try it” on page 72.

3. 3 Truth table: Disjunction P Q TRUE FALSE P Q TRUE FALSE Game: If you commit to the truth, then have to choose either P or Q, and the game continues as if you had committed to the truth of that simpler component. If you commit to the falsity, then your opponent chooses either P or Q, and the game continues as if you had committed to the falsity of that component. “You try it” on page 76.

3. 4 Ambiguity and parentheses Parentheses must be used whenever ambiguity would result from their omission. In practice, this means that conjunctions and disjunctions should be “wrapped” in parentheses whenever combined by means of some other connective. P Q R is ambiguous, should be either P (Q R) or (P Q) R

3. 5 Equivalent ways of saying things P Q P P P Double negation: P P De. Morgan 1: (P Q) P Q De. Morgan 2: (P Q) P Q

3. 6. a Translation In order for a FOL sentence to be a good translation of an English sentence, it is sufficient that the two sentences have the same truth values in all possible circumstances. It is not sufficient that they have the same truth value in some particular world.

3. 6. b Translation 1. The English expression “and” sometimes suggest temporal ordering; the FOL expression never does. Bob went home and ate 2. The English expressions “but”, “however”, “yet”, “nonetheless”, “moreover” are all stylistic variants of “and”. Bob was sleepy but he did the homework 3. The English expressions “either” and “both” are often used like parentheses to clarify an otherwise ambiguous sentence. Either Bob is home and Jane is home or Max is happy or both Bob is home and Jane is home