Editing Propositional calculus (section)

=== Propositional connective semantics ===
{{Main article|Logical connective|Truth function}}An interpretation assigns semantic values to [[atomic formula]]s directly.<ref name=":24" /><ref name=":21" /> Molecular formulas are assigned a ''function'' of the value of their constituent atoms, according to the connective used;<ref name=":24" /><ref name=":21" /> the connectives are defined in such a way that the [[Truth value|truth-value]] of a sentence formed from atoms with connectives depends on the truth-values of the atoms that they're applied to, and ''only'' on those.<ref name=":24" /><ref name=":21" /> This assumption is referred to by [[Colin Howson]] as the assumption of the ''[[Truth function|truth-functionality]] of the [[Logical connective|connectives]]''.<ref name=":13" />

==== 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]]".

==== Semantics via assignment expressions ====
Some authors (viz., all the authors cited in this subsection) write out the connective semantics using a list of statements instead of a table. In this format, where <math>\mathcal{I}(\varphi)</math> is the interpretation of <math>\varphi</math>, the five connectives are defined as:<ref name="BostockIntermediate" /><ref name=":29" />

* <math>\mathcal{I}(\neg P) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{F}</math>
* <math>\mathcal{I}(P \land Q) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{T}</math> and <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \lor Q) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(P) = \mathsf{T}</math> or <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \to Q) = \mathsf{T}</math> if, and only if, it is true that, if <math>\mathcal{I}(P) = \mathsf{T}</math>, then <math>\mathcal{I}(Q) = \mathsf{T}</math>
* <math>\mathcal{I}(P \leftrightarrow Q) = \mathsf{T}</math> if, and only if, it is true that <math>\mathcal{I}(P) = \mathsf{T}</math> if, and only if, <math>\mathcal{I}(Q) = \mathsf{T}</math>
Instead of <math>\mathcal{I}(\varphi)</math>, the interpretation of <math>\varphi</math> may be written out as <math>|\varphi|</math>,<ref name="BostockIntermediate" /><ref name="ms28"/> or, for definitions such as the above, <math>\mathcal{I}(\varphi) = \mathsf{T}</math> may be written simply as the English sentence "<math>\varphi</math> is given the value <math>\mathsf{T}</math>".<ref name=":29" /> Yet other authors<ref name="ms29"/><ref name=":43"/> may prefer to speak of a [[Model theory|Tarskian model]] <math>\mathfrak{M}</math> for the language, so that instead they'll use the notation <math>\mathfrak{M} \models \varphi</math>, which is equivalent to saying <math>\mathcal{I}(\varphi) = \mathsf{T}</math>, where <math>\mathcal{I}</math> is the interpretation function for <math>\mathfrak{M}</math>.<ref name=":43" />

==== Connective definition methods ====
Some of these connectives may be defined in terms of others: for instance, implication, <math>p \rightarrow q</math>, may be defined in terms of disjunction and negation, as <math>\neg p \lor q</math>;<ref name="ms30"/> and disjunction may be defined in terms of negation and conjunction, as <math>\neg(\neg p \land \neg q</math>.<ref name=":29" /> In fact, a ''[[Functional completeness|truth-functionally complete]]'' system,{{refn|group=lower-alpha|A truth-functionally complete set of connectives<ref name=":2" /> is also called simply ''[[Functional completeness|functionally complete]]'', or ''adequate for truth-functional logic'',<ref name=":13" /> or ''expressively adequate'',<ref name="Smith2003"/> or simply ''adequate''.<ref name=":13" /><ref name="Smith2003" />}} in the sense that all and only the classical propositional tautologies are theorems, may be derived using only disjunction and negation (as [[Bertrand Russell|Russell]], [[Alfred North Whitehead|Whitehead]], and [[David Hilbert|Hilbert]] did), or using only implication and negation (as [[Gottlob Frege|Frege]] did), or using only conjunction and negation, or even using only a single connective for "not and" (the [[Sheffer stroke]]),<ref name=":18" /> as [[Jean Nicod]] did.<ref name=":2" /> A ''joint denial'' connective ([[logical NOR]]) will also suffice, by itself, to define all other connectives. Besides NOR and NAND, no other connectives have this property.<ref name=":29" />{{efn|[[Truth_table#Overview_table|See a table]] of all 16 bivalent truth functions.}}

Some authors, namely [[Colin Howson|Howson]]<ref name=":13" /> and Cunningham,<ref name="ms31"/> distinguish equivalence from the biconditional. (As to equivalence, Howson calls it "truth-functional equivalence", while Cunningham calls it "logical equivalence".) Equivalence is symbolized with ⇔ and is a metalanguage symbol, while a biconditional is symbolized with ↔ and is a logical connective in the object language <math>\mathcal{L}</math>. Regardless, an equivalence or biconditional is true if, and only if, the formulas connected by it are assigned the same semantic value under every interpretation. Other authors often do not make this distinction, and may use the word "equivalence",<ref name=":3" /> and/or the symbol ⇔,<ref name="ms32"/> to denote their object language's biconditional connective.