first-order logic
The language describing the truth of mathematical ref="module.php?name=Lexikon&file=search&eid=1&query=formula">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 ref="module.php?name=Lexikon&file=search&eid=1&query=quantifier">quantifiers whose ref="module.php?name=Lexikon&file=search&eid=1&query=scope">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 ref="module.php?name=Lexikon&file=search&eid=1&query=atomic">atomic ref="module.php?name=Lexikon&file=search&eid=1&query=proposition">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:
[ ref="module.php?name=Lexikon&op=content&tid=31">2 ] [ ref="module.php?name=Lexikon&op=content&tid=134">= ] [ ref="module.php?name=Lexikon&op=content&tid=396">ag ] [ ref="module.php?name=Lexikon&op=content&tid=411">ai ] [ ref="module.php?name=Lexikon&op=content&tid=432">AL ] [ ref="module.php?name=Lexikon&op=content&tid=433">al ] [ ref="module.php?name=Lexikon&op=content&tid=544">am ] [ ref="module.php?name=Lexikon&op=content&tid=592">an ] [ ref="module.php?name=Lexikon&op=content&tid=740">ar ] [ ref="module.php?name=Lexikon&op=content&tid=743">arc ] [ ref="module.php?name=Lexikon&op=content&tid=800">as ] [ ref="module.php?name=Lexikon&op=content&tid=822">ash ] [ ref="module.php?name=Lexikon&op=content&tid=894">at ] [ ref="module.php?name=Lexikon&op=content&tid=920">atomic ] [ ref="module.php?name=Lexikon&op=content&tid=996">av ] [ ref="module.php?name=Lexikon&op=content&tid=1025">B ] [ ref="module.php?name=Lexikon&op=content&tid=1026">b ] [ ref="module.php?name=Lexikon&op=content&tid=1181">be ] [ ref="module.php?name=Lexikon&op=content&tid=1269">bi ] [ ref="module.php?name=Lexikon&op=content&tid=1444">bo ] [ ref="module.php?name=Lexikon&op=content&tid=1501">bot ] [ ref="module.php?name=Lexikon&op=content&tid=1606">bs ] [ ref="module.php?name=Lexikon&op=content&tid=1708">C ] [ ref="module.php?name=Lexikon&op=content&tid=1724">ca ] [ ref="module.php?name=Lexikon&op=content&tid=1863">cat ] [ ref="module.php?name=Lexikon&op=content&tid=2001">ch ] [ ref="module.php?name=Lexikon&op=content&tid=2099">ci ] [ ref="module.php?name=Lexikon&op=content&tid=2247">co ] [ ref="module.php?name=Lexikon&op=content&tid=2330">com ] [ ref="module.php?name=Lexikon&op=content&tid=2545">con ] [ ref="module.php?name=Lexikon&op=content&tid=2592">conjunction ] [ ref="module.php?name=Lexikon&op=content&tid=2594">connect ] [ ref="module.php?name=Lexikon&op=content&tid=2601">connective ] [ ref="module.php?name=Lexikon&op=content&tid=2606">cons ] [ ref="module.php?name=Lexikon&op=content&tid=2619">constraint ] [ ref="module.php?name=Lexikon&op=content&tid=2791">cr ] [ ref="module.php?name=Lexikon&op=content&tid=3151">de ] [ ref="module.php?name=Lexikon&op=content&tid=3470">disc ] [ ref="module.php?name=Lexikon&op=content&tid=3565">do ] [ ref="module.php?name=Lexikon&op=content&tid=3595">domain ] [ ref="module.php?name=Lexikon&op=content&tid=3752">du ] [ ref="module.php?name=Lexikon&op=content&tid=3834">E ] [ ref="module.php?name=Lexikon&op=content&tid=3865">ec ] [ ref="module.php?name=Lexikon&op=content&tid=3896">ed ] [ ref="module.php?name=Lexikon&op=content&tid=3923">edu ] [ ref="module.php?name=Lexikon&op=content&tid=3929">ee ] [ ref="module.php?name=Lexikon&op=content&tid=3946">eg ] [ ref="module.php?name=Lexikon&op=content&tid=4008">element ] [ ref="module.php?name=Lexikon&op=content&tid=4148">er ] [ ref="module.php?name=Lexikon&op=content&tid=4150">era ] [ ref="module.php?name=Lexikon&op=content&tid=4171">es ] [ ref="module.php?name=Lexikon&op=content&tid=4199">et ] [ ref="module.php?name=Lexikon&op=content&tid=4317">expression ] [ ref="module.php?name=Lexikon&op=content&tid=4497">fi ] [ ref="module.php?name=Lexikon&op=content&tid=4520">file ] [ ref="module.php?name=Lexikon&op=content&tid=4587">first-order ] [ ref="module.php?name=Lexikon&op=content&tid=4700">fo ] [ ref="module.php?name=Lexikon&op=content&tid=4727">for ] [ ref="module.php?name=Lexikon&op=content&tid=4757">formula ] [ ref="module.php?name=Lexikon&op=content&tid=4828">fr ] [ ref="module.php?name=Lexikon&op=content&tid=4940">function ] [ ref="module.php?name=Lexikon&op=content&tid=4989">ga ] [ ref="module.php?name=Lexikon&op=content&tid=5057">ge ] [ ref="module.php?name=Lexikon&op=content&tid=5134">gh ] [ ref="module.php?name=Lexikon&op=content&tid=5141">gi ] [ ref="module.php?name=Lexikon&op=content&tid=5403">gu ] [ ref="module.php?name=Lexikon&op=content&tid=5434">h ] [ ref="module.php?name=Lexikon&op=content&tid=5493">hang ] [ ref="module.php?name=Lexikon&op=content&tid=5540">hat ] [ ref="module.php?name=Lexikon&op=content&tid=5722">hose ] [ ref="module.php?name=Lexikon&op=content&tid=5768">hr ] [ ref="module.php?name=Lexikon&op=content&tid=5779">ht ] [ ref="module.php?name=Lexikon&op=content&tid=5931">id ] [ ref="module.php?name=Lexikon&op=content&tid=5956">ie ] [ ref="module.php?name=Lexikon&op=content&tid=6013">il ] [ ref="module.php?name=Lexikon&op=content&tid=6064">in ] [ ref="module.php?name=Lexikon&op=content&tid=6194">int ] [ ref="module.php?name=Lexikon&op=content&tid=6413">io ] [ ref="module.php?name=Lexikon&op=content&tid=6449">ir ] [ ref="module.php?name=Lexikon&op=content&tid=6482">is ] [ ref="module.php?name=Lexikon&op=content&tid=6558">it ] [ ref="module.php?name=Lexikon&op=content&tid=6589">J ] [ ref="module.php?name=Lexikon&op=content&tid=6918">la ] [ ref="module.php?name=Lexikon&op=content&tid=6950">language ] [ ref="module.php?name=Lexikon&op=content&tid=7091">Lex ] [ ref="module.php?name=Lexikon&op=content&tid=7107">li ] [ ref="module.php?name=Lexikon&op=content&tid=7151">line ] [ ref="module.php?name=Lexikon&op=content&tid=7245">LL ] [ ref="module.php?name=Lexikon&op=content&tid=7399">ls ] [ ref="module.php?name=Lexikon&op=content&tid=7415">lu ] [ ref="module.php?name=Lexikon&op=content&tid=7437">lv ] [ ref="module.php?name=Lexikon&op=content&tid=7441">ly ] [ ref="module.php?name=Lexikon&op=content&tid=7457">M ] [ ref="module.php?name=Lexikon&op=content&tid=7463">ma ] [ ref="module.php?name=Lexikon&op=content&tid=7579">mall ] [ ref="module.php?name=Lexikon&op=content&tid=7648">Mathematica ] [ ref="module.php?name=Lexikon&op=content&tid=8032">mo ] [ ref="module.php?name=Lexikon&op=content&tid=8040">mod ] [ ref="module.php?name=Lexikon&op=content&tid=8079">module ] [ ref="module.php?name=Lexikon&op=content&tid=8167">mp ] [ ref="module.php?name=Lexikon&op=content&tid=8228">ms ] [ ref="module.php?name=Lexikon&op=content&tid=8258">mu ] [ ref="module.php?name=Lexikon&op=content&tid=8384">N ] [ ref="module.php?name=Lexikon&op=content&tid=8386">na ] [ ref="module.php?name=Lexikon&op=content&tid=8460">nc ] [ ref="module.php?name=Lexikon&op=content&tid=8472">ne ] [ ref="module.php?name=Lexikon&op=content&tid=8627">ng ] [ ref="module.php?name=Lexikon&op=content&tid=8630">ni ] [ ref="module.php?name=Lexikon&op=content&tid=8660">nl ] [ ref="module.php?name=Lexikon&op=content&tid=8672">nn ] [ ref="module.php?name=Lexikon&op=content&tid=8675">no ] [ ref="module.php?name=Lexikon&op=content&tid=8718">norm ] [ ref="module.php?name=Lexikon&op=content&tid=8760">ns ] [ ref="module.php?name=Lexikon&op=content&tid=8787">nu ] [ ref="module.php?name=Lexikon&op=content&tid=8799">numbers ] [ ref="module.php?name=Lexikon&op=content&tid=8820">O ] [ ref="module.php?name=Lexikon&op=content&tid=8964">om ] [ ref="module.php?name=Lexikon&op=content&tid=9014">op ] [ ref="module.php?name=Lexikon&op=content&tid=9071">operator ] [ ref="module.php?name=Lexikon&op=content&tid=9457">pe ] [ ref="module.php?name=Lexikon&op=content&tid=9550">ph ] [ ref="module.php?name=Lexikon&op=content&tid=9651">pl ] [ ref="module.php?name=Lexikon&op=content&tid=9870">pound ] [ ref="module.php?name=Lexikon&op=content&tid=9908">pr ] [ ref="module.php?name=Lexikon&op=content&tid=10234">quantifier ] [ ref="module.php?name=Lexikon&op=content&tid=10253">query ] [ ref="module.php?name=Lexikon&op=content&tid=10319">range ] [ ref="module.php?name=Lexikon&op=content&tid=10364">rc ] [ ref="module.php?name=Lexikon&op=content&tid=10385">re ] [ ref="module.php?name=Lexikon&op=content&tid=10767">ro ] [ ref="module.php?name=Lexikon&op=content&tid=10820">row ] [ ref="module.php?name=Lexikon&op=content&tid=10887">ru ] [ ref="module.php?name=Lexikon&op=content&tid=10914">rw ] [ ref="module.php?name=Lexikon&op=content&tid=10918">S ] [ ref="module.php?name=Lexikon&op=content&tid=10922">sa ] [ ref="module.php?name=Lexikon&op=content&tid=11010">sc ] [ ref="module.php?name=Lexikon&op=content&tid=11070">scope ] [ ref="module.php?name=Lexikon&op=content&tid=11150">se ] [ ref="module.php?name=Lexikon&op=content&tid=11281">set ] [ ref="module.php?name=Lexikon&op=content&tid=11314">sh ] [ ref="module.php?name=Lexikon&op=content&tid=11376">si ] [ ref="module.php?name=Lexikon&op=content&tid=11506">sit ] [ ref="module.php?name=Lexikon&op=content&tid=11509">sj ] [ ref="module.php?name=Lexikon&op=content&tid=11651">so ] [ ref="module.php?name=Lexikon&op=content&tid=11790">spec ] [ ref="module.php?name=Lexikon&op=content&tid=11934">st ] [ ref="module.php?name=Lexikon&op=content&tid=12133">su ] [ ref="module.php?name=Lexikon&op=content&tid=12246">sy ] [ ref="module.php?name=Lexikon&op=content&tid=12359">T ] [ ref="module.php?name=Lexikon&op=content&tid=12360"> ] [ ref="module.php?name=Lexikon&op=content&tid=12588">th ] [ ref="module.php?name=Lexikon&op=content&tid=12715">tn ] [ ref="module.php?name=Lexikon&op=content&tid=12721">to ] [ ref="module.php?name=Lexikon&op=content&tid=12787">tr ] [ ref="module.php?name=Lexikon&op=content&tid=12896">tt ] [ ref="module.php?name=Lexikon&op=content&tid=12970">type ] [ ref="module.php?name=Lexikon&op=content&tid=12986">ua ] [ ref="module.php?name=Lexikon&op=content&tid=13030">um ] [ ref="module.php?name=Lexikon&op=content&tid=13175">us ] [ ref="module.php?name=Lexikon&op=content&tid=13229">V ] [ ref="module.php?name=Lexikon&op=content&tid=13252">va ] [ ref="module.php?name=Lexikon&op=content&tid=13260">value ] [ ref="module.php?name=Lexikon&op=content&tid=13274">var ] [ ref="module.php?name=Lexikon&op=content&tid=13275">variable ] [ ref="module.php?name=Lexikon&op=content&tid=13310">ve ] [ ref="module.php?name=Lexikon&op=content&tid=13366">vi ] [ ref="module.php?name=Lexikon&op=content&tid=13725">win ] [ ref="module.php?name=Lexikon&op=content&tid=13877">ws ] [ ref="module.php?name=Lexikon&op=content&tid=14024">Y ] [ ref="module.php?name=Lexikon&op=content&tid=14160">~ ]
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