Editing Propositional calculus (section)

=== Interpretation (case) and argument ===
{{Main|Interpretation (logic)}}
For a given language <math>\mathcal{L}</math>, an '''interpretation''',<ref name=":24"/> '''valuation''',<ref name=":29" /> '''Boolean valuation''',<ref name="ms24"/> or '''case''',<ref name=":21" />{{refn|group=lower-alpha|The name "interpretation" is used by some authors and the name "case" by other authors. This article will be indifferent and use either, since it is collaboratively edited and there is no consensus about which terminology to adopt.}} is an [[assignment (mathematical logic)|assignment]] of ''semantic values'' to each formula of <math>\mathcal{L}</math>.<ref name=":21" /> For a formal language of classical logic, a case is defined as an ''assignment'', to each formula of <math>\mathcal{L}</math>, of one or the other, but not both, of the [[truth value]]s, namely [[truth]] ('''T''', or 1) and [[false (logic)|falsity]] ('''F''', or 0).<ref name="ms25"/><ref name=":19"/> An interpretation that follows the rules of classical logic is sometimes called a '''Boolean valuation'''.<ref name=":29" /><ref name="ms26"/> An interpretation of a formal language for classical logic is often expressed in terms of [[truth tables]].<ref name="metalogic"/><ref name=":1" /> Since each formula is only assigned a single truth-value, an interpretation may be viewed as a [[Function (mathematics)|function]], whose [[Domain of a function|domain]] is <math>\mathcal{L}</math>, and whose [[Range of a function|range]] is its set of semantic values <math>\mathcal{V} = \{\mathsf{T}, \mathsf{F}\}</math>,<ref name=":2" /> or <math>\mathcal{V} = \{1, 0\}</math>.<ref name=":21" />

For <math>n</math> distinct propositional symbols there are <math>2^n</math> distinct possible interpretations. For any particular symbol <math>a</math>, for example, there are <math>2^1=2</math> possible interpretations: either <math>a</math> is assigned '''T''', or <math>a</math> is assigned '''F'''. And for the pair <math>a</math>, <math>b</math> there are <math>2^2=4</math> possible interpretations: either both are assigned '''T''', or both are assigned '''F''', or <math>a</math> is assigned '''T''' and <math>b</math> is assigned '''F''', or <math>a</math> is assigned '''F''' and <math>b</math> is assigned '''T'''.<ref name="metalogic" /> Since <math>\mathcal{L}</math> has <math>\aleph_0</math>, that is, [[Denumerably infinite|denumerably]] many propositional symbols, there are <math>2^{\aleph_0}=\mathfrak c</math>, and therefore [[Cardinality of the continuum|uncountably many]] distinct possible interpretations of <math>\mathcal{L}</math> as a whole.<ref name="metalogic" />

Where <math>\mathcal{I}</math> is an interpretation and <math>\varphi</math> and <math>\psi</math> represent formulas, the definition of an ''argument'', given in {{section link||Arguments}}, may then be stated as a pair <math>\langle \{\varphi_1, \varphi_2, \varphi_3, ..., \varphi_n\} , \psi \rangle</math>, where <math>\{\varphi_1, \varphi_2, \varphi_3, ..., \varphi_n\}</math> is the set of premises and <math>\psi</math> is the conclusion. The definition of an argument's ''validity'', i.e. its property that <math>\{\varphi_1, \varphi_2, \varphi_3, ..., \varphi_n\} \models \psi</math>, can then be stated as its ''absence of a counterexample'', where a '''counterexample''' is defined as a case <math>\mathcal{I}</math> in which the argument's premises <math>\{\varphi_1, \varphi_2, \varphi_3, ..., \varphi_n\}</math> are all true but the conclusion <math>\psi</math> is not true.<ref name=":21" /><ref name=":13" /> As will be seen in {{section link||Semantic truth, validity, consequence}}, this is the same as to say that the conclusion is a ''semantic consequence'' of the premises.