Predicate Calculus

Logic consists of
• A language which tells us how to build up sentences in the language (i.e., syntax), and and what the sentences mean (i.e., semantics)
• An inference procedure which tells us which sentences are valid inferences from other sentences
Propositional logic
The symbols of propositional calculus are the propositional symbols: P, Q, R, S, … the truth symbols: true, false and connectives: ?, ?, ?, ?, ?
10
Propositional Calculus Sentences
Every propositional symbol and truth symbol is a sentence. Examples: true, P, Q, R. The negation of a sentence is a sentence. Examples: ?P, ? false. The conjunction, or and, of two sentences is a sentence. Example: P ? ?P
11
Propositional Calculus Sentences (cont’d)
The disjunction, or or, of two sentences is a sentence.
Example: P ? ?P
The implication of one sentence from another is a sentence.
Example: P ? Q
The equivalence of two sentences is a sentence.
Example: P ? Q ? R
Legal sentences are also called well-formed formulas or WFFs.
12
Propositional calculus semantics
An interpretation of a set of propositions is the assignment of a truth value, either T or F to each propositional symbol. The symbol true is always assigned T, and the symbol false is assigned F. The truth assignment of negation, ?P, where P is any propositional symbol, is F if the assignment to P is T, and is T is the assignment to P is F. The truth assignment of conjunction, ?, is T only when both conjuncts have truth value T; otherwise it is F.