Propositional Logic 2. –First-Order logic •Godel’s completeness theorem showed that a proof Transcribing English sentences into wffs is sometimes a non-trivial task. x∈A x∈B ----- ∃y∈Ay∈B ?? Proof procedure is exponential in n, the number of symbols. b) Translate your set-theoretic notation into the notation of predicate logic. Two parts: ! The patterns which proofs follow are complicated, and there are a lot of them. ! •In practice, can be much faster… •Polynomial-time inference procedure exists when KB is expressed as Horn clauses: where the P i and Q are non-negated atoms. Inference Rules 3. Still have two truth values for statements (T and F) ! Proof. Variables (x,y) can take arbitrary values from some domain. A predicate P describes a relation or property. In Section 14.10 we discuss some of the implications of predicate logic as to our • A predicate is a property that is affirmed or denied about the subject (in logic, we say ‘variable’ or ‘argument’) of a statement • Consider the statement : ‘x is greater than 3’ – ‘x’ is the subject – ‘is greater than 3’ is the predicate Inference rules for propositional logic plus additional ... some of the example proofs to follow. In this course we are concerned with the transcription using given predicate symbols and the universe. Illinois Institute of Technology Class 2 … predicate logic. Predicate Logic 4. Example: Using (p → q) 㱻 (¬p ∨ q) for r, p for p, and ¬¬p for q, by substitution, we know (¬¬p → q) 㱻 (¬p ∨ q). Proofs in predicate logic can be carried out in a manner similar to proofs in propositional logic (Sections 14.8 and 14.9). ! More Answers for Practice in Logic and HW 1.doc Ling 310 Feb 27, 2006 1 More Answers for Practice in Logic and HW 1 This is an expanded version showing additional right and wrong answers. Some tautologies of predicate logic are analogs of tautologies for propo-sitional logic (Section 14.6), while others are not (Section 14.7). The use of symbolic logic also makes reasoning formal and mechanical, contributing to the simplification of the reasoning and making it less prone to errors. When we assign values to x and y, then P has a truth value. Practice in 1st-order predicate logic – with answers. Mary loves … E. Sample Proofs • Proofs in predicate logic give step-by-step reasoning for why we think the truth of one proposition is related to another. c) Give a proof that the syllogism is valid, similar to that given in the lecture 5 videos. We will learn You can't expect to do proofs by following rules, memorizing formulas, or looking at a few examples in a book. Consequence There is no algorithm that decides whether a first-order predi-cate logic sentence is a tautology. • end of proof CS 441 Discrete mathematics for CS M. Hauskrecht Informal proofs Proving theorems in practice: • The steps of the proofs are not expressed in any formal language as e.g. Such an alogithm could be used to decide satisfiable of first-order pred-icate logic sentences. A first-order predicate logic sentence G is satisfiable if, and only if, :G is not a tautology. I. The rules of inference are the essential building block in the construction of valid arguments. Statements in Predicate Logic P(x,y) ! For this reason, I'll start by discussing logic proofs. 1. 1. •Use the definition of entailment directly. Since they are more highly patterned than most proofs, they are a good place to start.
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