2 Propositional Logic DEF: A propositionis a statement that is either trueor false(but not both). c prns nd l ives An ic prn is a t or n t t be e or f. s of ic s e: “5 is a ” d am . In Section 14.10 we discuss some of the implications of predicate logic as to our 1. In propositional logic, we assume a ... Predicate Logic Consider this compound proposition: If Ollie is an octopus then Ollie has 8 limbs. o e.g. Logic: —validity vs. truth value —reasoning —translate English into logic —syntax, semantics, proof system Propositional logic: —truth table —natural deduction —simple, but inadequate for many problems Predicate logic: —function, quantifiers —much more expressive, harder to work with ing the predicate logic or various modal logics. A contradiction is a compound statement that is always false A contingent statement is one that is neither a tautology nor a contradiction For example, the truth table of p v ~p shows it is a tautology. Exercise Sheet 2: Predicate Logic 1. Introduction Logic is defined by two key notions, entailment and calculus. 4/12 Example Let P(x, y) = ^ÆEÇ_X Some tautologies of predicate logic are analogs of tautologies for propo-sitional logic (Section 14.6), while others are not (Section 14.7). Predicate logic Remedies the limitations of the propositional logic • Explicitly models objects and their properties • Allows to make statements with variables and quantify them Predicate logic: • Constant –models a specific object Examples: “John”, “France”, “7” … Entailment is a relation 1 Propositional and Predicate Logic: Linguistic Aspects P A M Seuren,MaxPlanckInstitutefor Psycholinguistics,Nijmegen,TheNetherlands 2006ElsevierLtd.Allrightsreserved. SEEM 5750 7 Propositional logic A tautology is a compound statement that is always true. (a) Anyone who has forgiven at least one person is a saint. (b) Nobody in the calculus class is smarter than everybody in the discrete maths class. More precisely, a logic is a calculus for the derivation of entailments. Propositional Logic, Predicate Logic, and Logic Programming. 2 These logics, however, lack the kind of efﬁcient and scalable algorithms that are available for the classical propositional logic. Predicate logic extends (is more powerful than) propositional logic. the domain of x in P(x): integer o Different variables may have different domains. Order Logic Propositional Logic First Order Logic Basic Concepts Propositional logic is the simplest logic illustrates basic ideas usingpropositions P 1, Snow is whyte P 2, oTday it is raining P 3, This automated reasoning course is boring P i is an atom or atomic formula Each P i … Predicate Logic: Unied Notation Extension of unied notation for propositional logic Universal formulas Type 8xA:9xA Existential formulas Type :8xA 9xA B. Beckert: Formal Verication of Software Œ p.12 Both work with propositions and logical connectives, but Predicate Calculus is more general than Propositional Calculus: it allows variables, quantiﬁers, and relations. The goal of this essay is to describe two types of logic: Propositional Calculus (also called 0th order logic) and Predicate Calculus (also called 1st order logic). Proofs in predicate logic can be carried out in a manner similar to proofs in propositional logic (Sections 14.8 and 14.9). Prl s e d from ic s by g lol s. tives fe e not d or l ) l quivt) A l l la is e th e of a l la can be d from e th vs of e ic s it . The existence of high performance algorithms for reasoning with propositional logic is the main reason for its wide use in computer science. while p ^ ~p is a contradiction If a conditional is also a tautology, then it is called an implication A predicate is a statement that contains variables (predicate variables ) and that may be true or false depending on the values of these variables. Formalise the following statements in predicate logic, making clear what your atomic predicate symbols stand for and what the domains of any variables are.
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