Aformula in conjunctive normal form(CNF) is a conjunction of clauses. An axiom schema is sentence pattern construed as a ... Propositional Logic can be reduced to equivalent sentences with these operators by applying the following rules. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. Aliteralis either a propositional variable, or the negation of one. 0.3. From our perspective we see their work as leading to boolean algebra, set theory, propositional logic, predicate logic, as clarifying the foundations of the natural and real number Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. 1.2 The syntax of propositional logic 1. Propositional logic, studied in Sections 1.1–1.3, cannot adequately express the meaning of all statements in mathematics and in natural language. Example: (p _:q _r)^(:p _:r) Similarly, one deﬁnes formulae indisjunctive normal form(DNF) by propositional variables? Examples: p, :p. Aclauseis a disjunction of literals. SEEM 5750 7 Propositional logic A tautology is a compound statement that is always true. Write out one of the laws like (A∧B)∨C ≡ (A∨C)∧(B∨C) It will actually take two lectures to get all the way through this. esentencesof(iii)arealltrueintheL Ô-structurewhichassignsTto everysentenceletter.Todemonstratethislastclaim,noteif^andψ aretrue inanL Ô-structure,then^∧ψ,^∨ψ,^→ψ and^↔ψ arealltrueinthis Peirce, and E. Schroder. Introduction to Logic using Propositional Calculus and Proof 1.1. Show that the distributive rules of ∧ and ∨ are in fact true. Let p stand for the proposition“I bought a lottery ticket”and q for“I won the jackpot”. ó Syntax and Semantics of Propositional Logic Õä esentencesof(ii)arealltrueintheL Ô-structurewhichassignsFtoevery sentenceletter. 1. We will see how to do this in Chapter 6. P 1,2 P 2,2 P 3,1 false true false With these symbols 8 possible worlds can be enumerated automatically. Five themes: logic and proofs, discrete structures, combinatorial analysis, induction and recursion, algorithmic thinking, and applications and modeling. A third “Logic” is “the study of the principles of reasoning, especially of the structure of propositions as distinguished Propositional logic: Semantics Each world specifies true/false for each proposition symbol E.g. Example: p _:q _r. Propositional Logic • Propositional resolution • Propositional theorem proving •Unification Today we’re going to talk about resolution, which is a proof strategy. For example, suppose that we know that “Every computer connected to the university network is functioning properly.” No rules of propositional logic allow us to conclude the truth of the statement •All but the final proposition are called premises. 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. Solution: 2. n . while p ^ ~p is a contradiction If a conditional is also a tautology, then it is called an implication •The argument is valid if the premises imply the conclusion. logic is relatively recent: the 19th century pioneers were Bolzano, Boole, Cantor, Dedekind, Frege, Peano, C.S. First, we’ll look at it in the propositional case, then in the first-order case. The last statement is the conclusion. Solution: We need some rules of inference without premises to get started. In more recent times, this algebra, like many algebras, has proved useful as a design tool. Express the following as natural English sentences: (a) ¬p (b) p∨ q (c) p∧ q (d) p ⇒ q (e) ¬p ⇒ ¬q (f) ¬p∨ (p∧ q) 2. Exercise Sheet 1: Propositional Logic 1. Solution: Use a truth table. 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