View source for Semantics of logical connectives (FOL)

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==Definition==
Let {{M|\mathscr{L} }} be a given [[first order language]] and let {{M|C}} denote the collection of all {{link|logical connective symbols|FOL}}<ref group="Note">Usually this means:
* {{M|\neg}}, {{M|\vee}}, {{M|\wedge}}, {{M|\rightarrow}} and {{M|\leftrightarrow}}
However we can define almost all of these symbols in terms of the others, keep this in mind.</ref>, the ''semantics of a logical connective symbol''{{rMLFFISWL}} is a [[function]] from the [[Cartesian product]] of one or more [[Truth values (FOL)|set of truth values]] to the truth values, written {{M|\mathbf{B}_*}} where {{M|*}} is a logical connective symbol, {{M|*\in\{\neg,\vee,\wedge,\rightarrow,\leftrightarrow\} }}, this mapping is commonly called the [[truth table]]<ref name="MLFFISWL"/> of the connective.

Suppose {{M|X}} and {{M|Y}} are variables which may only take on truth values (eg, are the [[semantics of formulas (FOL)|semantics of a formula]]) then we define {{M|\mathbf{B}_*}} as follows:
===[[Uniary]] connectives===
[[Truth table]] for [[logical negation]]
{| class="wikitable" border="1" style="text-align:center;min-width:12em;"
|-
! style="min-width:3em;" | {{M|\mathbf{X} }}
! {{M|\mathbf{B}_\neg(X)}}
|-
! {{M|T}}
| {{M|F}}
|-
! {{M|F}}
| {{M|T}}
|}
===[[Binary]] connectives===
[[Truth table]] for {{M|\vee}}, {{M|\wedge}}, {{M|\rightarrow}}, {{M|\leftrightarrow}}, that is [[logical or]], [[logical and]], [[logical implication]] and [[logical equivalence]] respectively:
{| class="wikitable" border="1" style="text-align:center;"
|-
! style="min-width:4em;" | {{M|\mathbf{X} }}
! style="min-width:4em;" | {{M|\mathbf{Y} }}
! {{M|\mathbf{B}_\vee(X,Y)}}
! {{M|\mathbf{B}_\wedge(X,Y)}}
! {{M|\mathbf{B}_\rightarrow(X,Y)}}
! {{M|\mathbf{B}_\leftrightarrow(X,Y)}}
|-
! {{M|F}}
! {{M|F}}
| {{M|F}}
| {{M|F}}
| {{M|T}}
| {{M|T}}
|-
! {{M|F}}
! {{M|T}}
| {{M|T}}
| {{M|F}}
| {{M|T}}
| {{M|F}}
|-
! {{M|T}}
! {{M|F}}
| {{M|T}}
| {{M|F}}
| {{M|F}}
| {{M|F}}
|-
! {{M|T}}
! {{M|T}}
| {{M|T}}
| {{M|T}}
| {{M|T}}
| {{M|T}}
|}
==See next==
* {{Link|Semantics of formulas|FOL}}
==Notes==
<references group="Note"/>
==References==
<references/>
{{Formal logic navbox|plain}}
{{Definition|Formal Logic}}