Truth table

Truth tables are a type of mathematical table used in logic to determine whether an expression is true or valid.

Related Topics:
Mathematical table - Logic - Expression - Valid

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(Expressions may be arguments; i.e., a conjunction of expressions, each conjunct of which is a premise with the last being the conclusion.)

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Truth tables derive from the work of Gottlob Frege, Charles Peirce and others from about the 1880s.

Related Topics:
Gottlob Frege - Charles Peirce - 1880s

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They came to their present form in 1922 through the work of Emil Post and Ludwig Wittgenstein.

Related Topics:
1922 - Emil Post - Ludwig Wittgenstein

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Wittgenstein's Tractatus Logico-Philosophicus uses them to place truth functions in a series.

Related Topics:
Tractatus Logico-Philosophicus - Truth function

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The wide influence of this work led to the spread of the use of truth tables.

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Truth tables are used to compute the values of truth-functional expressions (i.e., it is a decision procedure).

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A truth-functional expression is either atomic (i.e., a propositional variable (or placeholder) or a propositional function — e.g. Px) or built up from atomic formulas from logical operators (i.e. land (AND), lnot (NOT) — e.g. Fx & Gx).

Related Topics:
AND - NOT

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The column headings on a truth table show (i) the propositional functions and/or variables, and (ii) the truth-functional expression built up from those propositional functions or variables and operators.

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The rows show each possible valuation of T or F assignments to (i) and (ii).

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In other words, each row is a distinct interpretation of (i) and (ii).

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Truth tables for classical (i.e., bivalent) logic are limited to Boolean logic systems where only two truth values are possible, true or false, usually denoted simply T and F in the tables (as remarked above).

Related Topics:
Boolean logic - Truth value

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For example, take two propositional variables, A and B, and the logical operator "AND" (land), signifying the conjunction "A and B" or A land B.

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In common English, if both A and B are true, then the conjunction "A land B" is true; under all other possible assignments of truth values to A land B, the conjunction is false.

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This relationship is defined as follows:

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In a boolean logic system, all the operators can be explicitly defined this way.

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For example, the NOT (lnot) relationship is defined as follows:

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The OR (lor) relationship is defined as follows:

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Compound expressions can be constructed, using parenthesis to denote precedence.

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The negation of conjunction lnot ( A land B ), is depicted as follows:

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Truth tables can be used to prove logical equivalence.

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The truth table for the disjunction of lnot A lor lnot B is:

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Comparing the above two truth tables, since the enumeration of all possible truth-values for A and B yields the same truth-value under both lnot (A land B) and lnot A lor lnot B, the two are logically equivalent, and may be substituted for each other.

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This equivalence is one of DeMorgan's Laws.

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Here is a truth table giving definitions of the most commonly used 5 of the 16 possible truth functions of 2 binary variables (P,Q are thus boolean variables):

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Key:

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:T = true, F = false

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:land = AND (logical conjunction)

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:lor = OR (logical disjunction)

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:oplus = XOR (exclusive disjunction)

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: ightarrow = conditional or "if-then"

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:leftrightarrow = biconditional or "if-and-only-if"

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Johnston diagrams, similar to Venn diagrams and Euler diagrams, provide a way of visualizing truth tables. An interactive Johnston diagram illustrating truth tables is at LogicTutorial.com

Related Topics:
Johnston diagram - Venn diagram

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