Editing Propositional calculus (section)

==== Semantics via. truth tables ====
{{Logical connectives sidebar}}
Since logical connectives are defined semantically only in terms of the [[truth value]]s that they take when the [[propositional variable]]s that they're applied to take either of the [[Principle of bivalence|two possible]] truth values,<ref name=":1" /><ref name=":21" /> the semantic definition of the connectives is usually represented as a [[truth table]] for each of the connectives,<ref name=":1" /><ref name=":21" /><ref name=":37" /> as seen below:

{| class="wikitable" style="margin:1em auto; text-align:center;"
|-
! <math>p</math>
! <math>q</math>
! <math>p \land q</math>
! <math>p \lor q</math>
! <math>p \rightarrow q</math>
! <math>p \Leftrightarrow q</math>
! <math>\neg p</math>
! <math>\neg q</math>
|-
| {{Success|}}T || {{Success|}}T || {{Success|}}T || {{Success|}}T || {{Success|}}T || {{Success|}}T || {{Failure|}}F || {{Failure|}}F
|-
| {{Success|}}T || {{Failure|}}F || {{Failure|}}F || {{Success|}}T || {{Failure|}}F || {{Failure|}}F || {{Failure|}}F || {{Success|}}T
|-
| {{Failure|}}F || {{Success|}}T || {{Failure|}}F || {{Success|}}T || {{Success|}}T || {{Failure|}}F || {{Success|}}T || {{Failure|}}F
|-
| {{Failure|}}F || {{Failure|}}F || {{Failure|}}F || {{Failure|}}F || {{Success|}}T || {{Success|}}T || {{Success|}}T || {{Success|}}T
|}

This table covers each of the main five [[logical connective]]s:<ref name=":5" /><ref name=":0" /><ref name=":3" /><ref name=":12" /> [[Logical conjunction|conjunction]] (here notated <math>p \land q</math>), [[Logical disjunction|disjunction]] ({{math|''p'' ∨ ''q''}}), [[Material conditional|implication]] ({{math|''p'' → ''q''}}), [[Logical biconditional|biconditional]] ({{math|''p'' ↔ ''q''}}) and [[negation]], (¬''p'', or ¬''q'', as the case may be). It is sufficient for determining the semantics of each of these operators.<ref name=":1" /><ref name="ms27"/><ref name=":21" /> For more truth tables for more different kinds of connectives, see the article "[[Truth table]]".