first-order logic
The language describing the truth of mathematical formulas. Formulas describe properties of terms and have a truth value. The following are atomic formulas: True False p(t1,..tn) where t1,..,tn are terms and p is a predicate. If F1, F2 and F3 are formulas and v is a variable then the following are compound formulas: F1 ^ F2 conjunction - true if both F1 and F2 are true, F1 V F2 disjunction - true if either or both are true, F1 => F2 implication - true if F1 is false or F2 is true, F1 is the antecedent, F2 is the consequent (sometimes written with a thin arrow), F1 <= F2 true if F1 is true or F2 is false, F1 == F2 true if F1 and F2 are both true or both false (normally written with a three line equivalence symbol) ~F1 negation - true if f1 is false (normally written as a dash ' -' with a shorter vertical line hanging from its right hand end). For all v . F universal quantification - true if F is true for all values of v (normally written with an inverted A). Exists v . F existential quantification - true if there exists some value of v for which F is true. (Normally written with a reversed E). The operators ^ V => <= == ~ are called connectives. "For all" and "Exists" are quantifiers whose scope is F. A term is a mathematical expression involving numbers, operators, functions and variables. The "order" of a logic specifies what entities "For all" and "Exists" may quantify over. First-order logic can only quantify over sets of atomic propositions. (E.g. For all p . p => p). Second-order logic can quantify over functions on propositions, and higher-order logic can quantify over any type of entity. The sets over which quantifiers operate are usually implicit but can be deduced from well-formedness constraints. In first-order logic quantifiers always range over ALL the elements of the domain of discourse. By contrast, second-order logic allows one to quantify over subsets of M. ["The Realm of First-Order Logic", Jon Barwise, Handbook of Mathematical Logic (Barwise, ed., North Holland, NYC, 1977)]. (1995-05-02) In addition suitable contents:
[ 2 ] [ = ] [ ag ] [ ai ] [ AL ] [ al ] [ am ] [ an ] [ ar ] [ arc ] [ as ] [ ash ] [ at ] [ atomic ] [ av ] [ B ] [ b ] [ be ] [ bi ] [ bo ] [ bot ] [ bs ] [ C ] [ ca ] [ cat ] [ ch ] [ ci ] [ co ] [ com ] [ con ] [ conjunction ] [ connect ] [ connective ] [ cons ] [ constraint ] [ cr ] [ de ] [ disc ] [ do ] [ domain ] [ du ] [ E ] [ ec ] [ ed ] [ edu ] [ ee ] [ eg ] [ element ] [ er ] [ era ] [ es ] [ et ] [ expression ] [ fi ] [ file ] [ first-order ] [ fo ] [ for ] [ formula ] [ fr ] [ function ] [ ga ] [ ge ] [ gh ] [ gi ] [ gu ] [ h ] [ hang ] [ hat ] [ hose ] [ hr ] [ ht ] [ id ] [ ie ] [ il ] [ in ] [ int ] [ io ] [ ir ] [ is ] [ it ] [ J ] [ la ] [ language ] [ Lex ] [ li ] [ line ] [ LL ] [ ls ] [ lu ] [ lv ] [ ly ] [ M ] [ ma ] [ mall ] [ Mathematica ] [ mo ] [ mod ] [ module ] [ mp ] [ ms ] [ mu ] [ N ] [ na ] [ nc ] [ ne ] [ ng ] [ ni ] [ nl ] [ nn ] [ no ] [ norm ] [ ns ] [ nu ] [ numbers ] [ O ] [ om ] [ op ] [ operator ] [ pe ] [ ph ] [ pl ] [ pound ] [ pr ] [ quantifier ] [ query ] [ range ] [ rc ] [ re ] [ ro ] [ row ] [ ru ] [ rw ] [ S ] [ sa ] [ sc ] [ scope ] [ se ] [ set ] [ sh ] [ si ] [ sit ] [ sj ] [ so ] [ spec ] [ st ] [ su ] [ sy ] [ T ] [ ] [ th ] [ tn ] [ to ] [ tr ] [ tt ] [ type ] [ ua ] [ um ] [ us ] [ V ] [ va ] [ value ] [ var ] [ variable ] [ ve ] [ vi ] [ win ] [ ws ] [ Y ] [ ~ ]
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