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) Style="border-width:thin; border-color:#333333; border-Style:daShed; padding:5px;" align="left">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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