Logic

Read e-book online An Introduction to Many-Valued and Fuzzy Logic: Semantics, PDF

By Merrie Bergmann

ISBN-10: 0521707579

ISBN-13: 9780521707572

ISBN-10: 0521881285

ISBN-13: 9780521881289

This quantity is an available creation to the topic of many-valued and fuzzy good judgment compatible to be used in proper complicated undergraduate and graduate classes. The textual content opens with a dialogue of the philosophical matters that supply upward thrust to fuzzy good judgment - difficulties coming up from obscure language - and returns to these matters as logical structures are provided. For historic and pedagogical purposes, three-valued logical structures are provided as precious intermediate structures for learning the foundations and thought at the back of fuzzy good judgment.

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Extra info for An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems

Example text

We’ll explain the transformation process using the formula ¬(¬(P → Q) ∨ (¬R → (S ↔ T))) First we use the Implication equivalences to eliminate all conditionals and biconditionals, producing the formula ¬(¬(¬P ∨ Q) ∨ (¬¬R ∨ ((¬S ∨ T) ∧ (¬T ∨ S)))) Next we use DeMorgan’s Laws to move negations deeper into the formula until all negations appear in front of atomic formulas. In our example we use the second DeMorgan Law first to obtain ¬¬(¬P ∨ Q) ∧ ¬(¬¬R ∨ ((¬S ∨ T) ∧ (¬T ∨ S))) and then ¬(¬¬P ∧ ¬Q) ∧ (¬¬¬R ∧ ¬((¬S ∨ T) ∧ (¬T ∨ S)) Next we can use the first DeMorgan Law twice to obtain (¬¬¬P ∨ ¬¬Q) ∧ (¬¬¬R ∧ (¬(¬S ∨ T) ∨ ¬(¬T ∨ S))) and then the second Law twice more to obtain (¬¬¬P ∨ ¬¬Q) ∧ (¬¬¬R ∧ ((¬¬S ∧ ¬T) ∨ (¬¬T ∧ ¬S))) Now Double Negation eliminates all double negations: (¬P ∨ Q) ∧ (¬R ∧ ((S ∧ ¬T) ∨ (T ∧ ¬S))).

The sole purpose of lines 3–11 of the derivation is to derive ¬M → (M → S) from ¬M → (¬S → ¬M) and (¬S → ¬M) → (M → S). Examining these formulas we see that they are instances of a general inference pattern that derives a formula of the form P → R from formulas having the form P → Q and Q → R. We may introduce this inference pattern, commonly called hypothetical syllogism, as a derived rule: HS (Hypothetical Syllogism). From P → Q and Q → R, infer P → R. 8 Claims about a derivation system are part of what is called the metatheory of logic.

3, each such conjunction is a phrase. So, for example, phrases corresponding to the four rows of the neither-nor function template are, respectively, P ∧ Q, P ∧ ¬Q, ¬P ∧ Q, and ¬P ∧ ¬Q. Note that each of these phrases is true exactly when P and Q have the truth-values in its corresponding row. Next we form a disjunction of the phrases corresponding to the rows that have T to the right of the vertical line, thus producing a formula in disjunctive normal form. In the case of the neither-nor function there is one such row, the fourth, so we form the 33 P1: RTJ 9780521881289c02 CUNY1027/Bergmann 34 978-0 521 88128 9 November 24, 2007 17:15 Review of Classical Propositional Logic “disjunction” of the single phrase for that row: ¬P ∧ ¬Q.

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An Introduction to Many-Valued and Fuzzy Logic: Semantics, Algebras, and Derivation Systems by Merrie Bergmann


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