ARTIFICIAL INTELLIGENCE [INTELLIGENT AGENTS PARADIGM] Professor Janis Grundspenkis Riga Technical University Faculty of Computer Science and Information Technology Institute of Applied Computer Systems Department of Systems Theory and Design SYNTAX OF PROPOSITIONAL CALCULUS

Syntax of Propositional Calculus Symbols –The propositional symbols: P, Q, R, S, T,... –Truth symbols: True, False –Connectives: , , , , 

Symbols (continued) Propositional symbols denote propositions, or statements about the world that may be either true or false. Propositions are denoted by uppercase letters. Syntax of Propositional Calculus

Sentences –Every propositional symbol and truth symbol is a sentence –The negation of a sentence is a sentence –The conjunction, or AND, of two sentences is a sentence

Sentences (continued) –The disjunction, or OR, of two sentences is a sentence –The implication of one sentence for another is a sentence –The equivalence of two sentences is a sentence Syntax of Propositional Calculus

Sentences (continued) Examples 1.False, Q, True and S are sentences 2.  False and  R are sentences 3.P   Q  S  W is a sentence conjuncts

Syntax of Propositional Calculus disjuncts premise (antecedent) conclusion (consequent) 4.P   Q  S  W is a sentence 5.P  Q is a sentence 6.P  R  W is a sentence Sentences (continued)

Syntax of Propositional Calculus Sentences (continued) Legal sentences are also called well-formed formulas (WFF). The symbols ( ) and [ ] are used to group symbols into sub-expressions and to control their order of evaluation and meaning. For example, (P  Q)  S is quite different from P  (Q  S)

Syntax of Propositional Calculus Sentences (continued) The symbols ( ) and [ ] help to take into account the binding strength   and    For example,P  Q  S means (P  Q)  S P  Q  S  R means ((P  Q)  S)  R

Solution: P, Q and R are propositions and thus sentences P  Q, the conjunction of two sentences, is a sentence Syntax of Propositional Calculus PQ PP QQ RR      Sentences (continued) Question: Is P  Q  R   P   Q  R a well-formed formula?

Syntax of Propositional Calculus Sentences (continued) P  Q  R, the implication of a sentence for another, is a sentence  P and  Q, the negations of sentences, are sentences PQ PP QQ RR       P   Q, the disjunction of two sentences, is a sentence

Syntax of Propositional Calculus  P   Q  R, the disjunction of two sentences, is a sentence P  Q  R   P   Q  R, the equivalence of two sentences, is a sentence Sentences (continued) PQ PP QQ RR      This is the original sentence, which has been constructed through a series of applications of legal rules and is therefore well formed.