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