2slides - Chapter 2: Introduction to Propositional Logic...

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Chapter 2: Introduction to Propositional Logic PART ONE: History and Motivation Origins: Stoic school of philosophy (3rd cen- tury B.C.), with the most eminent repre- sentative was Chryssipus. Modern Origins: Mid-19th century - English mathematician G. Boole , who is some- times regarded as the founder of mathe- matical logic. First Axiomatic System: 1879 by German lo- gician G. Frege. 1
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The first assumption underlying the formal- ization of classical propositional logic (cal- culus) is the following. We assume that sentences are always evalu- ated as true or false . Such sentences are called logical sentences or propositions . Hence the name propositional logic . 2
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A statement: 2+2 = 4 is a proposition (true). A statement: 2 + 2 = 5 is also a proposition (false). A statement: I am pretty is modeled as a log- ical sentence (proposition). We assume that it is false, or true. A statement: 2+ n = 5 is not a proposition; it might be true for some n, for example n=3, false for other n, for example n= 2, and moreover, we don’t know what n is. Sentences of this kind are called proposi- tional functions . We model propositional functions within propo- sitional logic by treating propositional func- tions as propositions. 3
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The classical logic reflects the black and white qualities of mathematics. We expect from mathematical theorems to be always either true or false and the rea- sonings leading to them should guarantee this without any ambiguity. 4
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Formulas We combine logical sentences to form more complicated sentences, called formulas . We combine them using the following words or phrases: not ; and ; or ; if . .., then ; if and only if . We use only symbols do denote both logical sentences and the phrases: not ; and ; or ; if ..., then ; if and only if . Hence the name symbolic logic . 5
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Logical sentences are denoted by symbols a,b,c,p,r,q,. . Symbols for logical connectives are: ¬ for ” not ”, for ” and ”, for ” or ”, for ”if . .., then” , and for ”if and only if” . 6
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Translate a natural language sentence: The fact that it is not true that at the same time 2+2 = 4 and 2+2 = 5 implies that 2 + 2 = 4 into its propositional symbolic logic for- mula. First we write it in a form: If not (2 + 2 = 4 and 2 + 2 = 5) then 2 + 2 = 4 Second we write it in a symbolic formula: ( ¬ ( a b ) a ). 7
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Translate a natural language sentence: The fact that it is not true that at the same time 2 + n = 4 and some numbers are pretty implies that 2 + n = 4 into its propositional symbolic logic for- mula. First
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2slides - Chapter 2: Introduction to Propositional Logic...

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