Truth table

Truth tables are a type of mathematical table used in logic to determine whether an expression (mathematics) is true or validity. (Expressions may be arguments; i.e., a conjunction of expressions, each conjunct of which is a premise with the last being the conclusion.) Truth tables derive from the work of Gottlob Frege, Charles Peirce and others from about the 1880s. They came to their present form in 1922 through the work of Emil Post and Ludwig Wittgenstein. Wittgenstein's Tractatus Logico-Philosophicus uses them to place truth functions in a series. The wide influence of this work led to the spread of the use of truth tables. Truth tables are used to compute the values of truth-functional expressions (i.e., it is a decision procedure). 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. $\backslash land$ (logical conjunction), $\backslash lnot$ (Negation) — e.g. Fx & Gx). 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. The rows show each possible valuation of T or F assignments to (i) and (ii). In other words, each row is a distinct interpretation of (i) and (ii). 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). For example, take two propositional variables, $A$ and $B$, and the logical operator "AND" ($\backslash land$), signifying the conjunction "A and B" or $A\; \backslash land\; B$. In common English, if both A and B are true, then the conjunction "$A\; \backslash land\; B$" is true; under all other possible assignments of truth values to $A\; \backslash land\; B$, the conjunction is false. This relationship is defined as follows: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $A$ || $B$ || $A\; \backslash land\; B$ |- align=center | F || F || F |- align=center | F || T || F |- align=center | T || F || F |- align=center | T || T || T |} In a boolean logic system, all the operators can be explicitly defined this way. For example, the Negation ($\backslash lnot$) relationship is defined as follows: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $A$ || $\backslash lnot\; A$ |- align=center | F || T |- align=center | T || F |} The Logical disjunction ($\backslash lor$) relationship is defined as follows: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $A$ || $B$ || $A\; \backslash lor\; B$ |- align=center | F || F || F |- align=center | F || T || T |- align=center | T || F || T |- align=center | T || T || T |} Compound expressions can be constructed, using parenthesis to denote precedence. The negation of conjunction $\backslash lnot\; (\; A\; \backslash land\; B\; )$, is depicted as follows: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $A$ || $B$ || $A\; \backslash land\; B$ || $\backslash lnot\; (\; A\; \backslash land\; B\; )$ |- align=center | F || F || F || T |- align=center | F || T || F || T |- align=center | T || F || F || T |- align=center | T || T || T || F |} Truth tables can be used to prove logical equivalence. The truth table for the disjunction of $\backslash lnot\; A\; \backslash lor\; \backslash lnot\; B$ is: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $A$ || $B$ || $\backslash lnot\; A$ || $\backslash lnot\; B$ || $\backslash lnot\; A\; \backslash lor\; \backslash lnot\; B$ |- align=center | F || F || T || T || T |- align=center | F || T || T || F || T |- align=center | T || F || F || T || T |- align=center | T || T || F || F || F |} 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 $\backslash lnot\; (A\; \backslash land\; B)$ and $\backslash lnot\; A\; \backslash lor\; \backslash lnot\; B$, the two are logically equivalent, and may be substituted for each other. This equivalence is one of DeMorgans Laws. Here is a truth table giving definitions of the most commonly used 5 of Tractatus Logico-Philosophicus#Propositions 4.*-5.*: {| border="1" cellspacing="1" cellpadding="5" align="center" |- | $P$ || $Q$ || $P\; \backslash land\; Q$ || $P\; \backslash lor\; Q$ || $P\; \backslash oplus\; Q$ || $P\; \backslash rightarrow\; Q$ || $P\; \backslash leftrightarrow\; Q$ |- align=center | F || F || F || F || F || T || T |- align=center | F || T || F || T || T || T || F |- align=center | T || F || F || T || T || F || F |- align=center | T || T || T || T || F || T || T |} Key: :T = true, F = false :$\backslash land$ = logical conjunction (logical conjunction) :$\backslash lor$ = logical disjunction (logical disjunction) :$\backslash oplus$ = Exclusive disjunction (exclusive disjunction) :$\backslash rightarrow$ = logical conditional :$\backslash leftrightarrow$ = iff Johnston diagrams, similar to Venn diagrams and Venn diagram, provide a way of visualizing truth tables. An interactive Johnston diagram illustrating truth tables is at [http://logictutorial.com LogicTutorial.com] == See also == * Connective * Propositional calculus * List of Boolean algebra topics Logic

Truth table

The comments about "finite mathematics" are silly. "Finite mathematics" is not a field within mathematics, but rather a collection of diverse topics in elementary mathematics that the curriculum brings togetther in a single undergraduate course for business students. Truth tables are not different in "finite mathematics" than in other disciplines. -- Mike Hardy ---- The "arrow" connective, it is to be understood as a truth-functional operator, should not be described as "implication." Call it "conditional" or "if-then." Doing otherwise involves confusing the use-mention distinction that Quine first noticed and spent his whole career trying to enforce (Perhaps hopelessly: quantified modal logic is deeply infected with use-mention confusions.) See his ''Mathematical Logic'', Section 5. In any case, "if...then" is not the same as "implies." "Implies" is a relation between sentences: a two-place predicate that takes sentences as the values of its variables and produces a sentence from them: it is a function from ''names'' of sentences--''terms''--to a sentence. By constrast "if then" is a not a predicate but a connective; it is a funtion from sentences to a sentence. It does not take anything as values because it does not contain variables. "Implies" talks about--mentions--two sentences, and can only be used in a meta-language. "If...then" ''uses'' two sentences; it ''mentions'' whatever the sentences mention, and is itself a term within the object language. Shortly: If A then B. If the light goes out then the monsters will come. but "A" implies "B". "The light goes out" implies "The monsters will come". Sorry for the rant. If anyone sees this mistake elsewhere, please correct it. ---- (Added) Same goes for equivalence. Sentences are equivalent to one another; but the things they say are related as "...if and only if ..." Also, variables don't generally have truth-values. ''That's'' too confused to explain at all. P, Q, and R here aren't being used as variables, (though if they were, they'd have sentences, not truth-values, as their values). They're ''schemata''; their standing in for sentences, but there's no assumption that you can quantify over them. The best way to explain this stuff is using Quine-corners and his Greek-letter metavariables. But, alas, no one cares about being rigorous anymore. Sigh. ---- ==Peirce== Can anyone provide a reference for Charles Peirce�s development of truth tables? I would like to check the form that they took. It appears from a quick bit of research that the tables he developed were substantially different in form to those presented here (see http://plato.stanford.edu/entries/peirce-logic/ ) Those sources that support the claim appear to derive from the Wikipedia. Was it Wittgenstein who developed the form that is now used? User:Banno 22:59, May 15, 2004 (UTC) :Discussed in depth in this thread: http://sunsite.utk.edu/math_archives/.http/hypermail/historia/apr00/0117.html An excellent summary of the issue. Thanks for pointing it out to me. The conclusion appears to be that Frege, Peirce, and Schr�der all played a part in the development of the truth table, and so the attribution of them to Peirce alone should be altered. Wittgenstein perhaps had the role of popularising their use. User:Banno 00:04, May 16, 2004 (UTC)

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