while p ^ ~p is a contradiction If a conditional is also a tautology, then it is called an implication We will see how to do this in Chapter 6. •The argument is valid if the premises imply the conclusion. 1. Aliteralis either a propositional variable, or the negation of one. P 1,2 P 2,2 P 3,1 false true false With these symbols 8 possible worlds can be enumerated automatically. For example, Chapter 13 shows how propositional logic can be used in computer circuit design. Example: p _:q _r. Show that the distributive rules of ∧ and ∨ are in fact true. Five themes: logic and proofs, discrete structures, combinatorial analysis, induction and recursion, algorithmic thinking, and applications and modeling. Propositional Logic • Propositional resolution • Propositional theorem proving •Unification Today we’re going to talk about resolution, which is a proof strategy. Aformula in conjunctive normal form(CNF) is a conjunction of clauses. 1.2 The syntax of propositional logic 1. Write out one of the laws like (A∧B)∨C ≡ (A∨C)∧(B∨C) Examples: p, :p. Aclauseis a disjunction of literals. esentencesof(iii)arealltrueintheL Ô-structurewhichassignsTto everysentenceletter.Todemonstratethislastclaim,noteif^andψ aretrue inanL Ô-structure,then^∧ψ,^∨ψ,^→ψ and^↔ψ arealltrueinthis Peirce, and E. Schroder. It will actually take two lectures to get all the way through this. propositional variables? Propositional Logic In this chapter, we introduce propositional logic, an algebra whose original purpose, dating back to Aristotle, was to model reasoning. 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. Solution: Use a truth table. Propositional logic, studied in Sections 1.1–1.3, cannot adequately express the meaning of all statements in mathematics and in natural language. 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. SEEM 5750 7 Propositional logic A tautology is a compound statement that is always true. Solution: We need some rules of inference without premises to get started. 0.3. 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 In more recent times, this algebra, like many algebras, has proved useful as a design tool. Example: (p _:q _r)^(:p _:r) Similarly, one defines formulae indisjunctive normal form(DNF) by First, we’ll look at it in the propositional case, then in the first-order case. Propositional logic: Semantics Each world specifies true/false for each proposition symbol E.g. 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 1: Propositional Logic 1. logic is relatively recent: the 19th century pioneers were Bolzano, Boole, Cantor, Dedekind, Frege, Peano, C.S. The last statement is the conclusion. ó Syntax and Semantics of Propositional Logic Õä esentencesof(ii)arealltrueintheL Ô-structurewhichassignsFtoevery sentenceletter. “Logic” is “the study of the principles of reasoning, especially of the structure of propositions as distinguished Introduction to Logic using Propositional Calculus and Proof 1.1. Solution: 2. n . Let p stand for the proposition“I bought a lottery ticket”and q for“I won the jackpot”. 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