Editing Propositional calculus (section)

=== Semantic truth, validity, consequence ===
Given <math>\varphi</math> and <math>\psi</math> as [[formula (mathematical logic)|formulas]] (or sentences) of a language <math>\mathcal{L}</math>, and <math>\mathcal{I}</math> as an interpretation (or case){{refn|group=lower-alpha|Some of these definitions use the word "interpretation", and speak of sentences/formulas being true or false "under" it, and some will use the word "case", and speak of sentences/formulas being true or false "in" it. Published ''reliable sources'' ([[WP:RS]]) have used both kinds of terminological convention, although usually a given author will use only one of them. Since this article is collaboratively edited and there is no consensus about which convention to use, these variations in terminology have been left standing.}} of <math>\mathcal{L}</math>, then the following definitions apply:<ref name="metalogic" /><ref name=":19" />

* '''Truth-in-a-case:'''<ref name=":21" /> A sentence <math>\varphi</math> of <math>\mathcal{L}</math> is ''true under an interpretation'' <math>\mathcal{I}</math> if <math>\mathcal{I}</math> assigns the truth value '''T''' to <math>\varphi</math>.<ref name=":19" /><ref name="metalogic" /> If <math>\varphi</math> is [[logical truth|true]] under <math>\mathcal{I}</math>, then <math>\mathcal{I}</math> is called a ''model'' of <math>\varphi</math>.<ref name="metalogic" />
* '''Falsity-in-a-case:<ref name=":21" />''' <math>\varphi</math> is ''false under an interpretation'' <math>\mathcal{I}</math> if, and only if, <math>\neg\varphi</math> is true under <math>\mathcal{I}</math>.<ref name="metalogic" /><ref name=":20" /><ref name=":21" /> This is the "truth of negation" definition of falsity-in-a-case.<ref name=":21" /> Falsity-in-a-case may also be defined by the "complement" definition: <math>\varphi</math> is ''false under an interpretation'' <math>\mathcal{I}</math> if, and only if, <math>\varphi</math> is not true under <math>\mathcal{I}</math>.<ref name=":19" /><ref name="metalogic" /> In [[classical logic]], these definitions are equivalent, but in [[Non-classical logic|nonclassical logics]], they are not.<ref name=":21" />
* '''Semantic consequence:''' A sentence <math>\psi</math> of <math>\mathcal{L}</math> is a ''[[Logical consequence|semantic consequence]]'' (<math>\varphi \models \psi</math>) of a sentence <math>\varphi</math> if there is no interpretation under which <math>\varphi</math> is true and <math>\psi</math> is not true.<ref name=":19" /><ref name="metalogic" /><ref name=":21" />
* '''Valid formula (tautology):''' A sentence <math>\varphi</math> of <math>\mathcal{L}</math> is ''logically valid'' (<math>\models\varphi</math>),{{refn|group=lower-alpha|Conventionally <math>\models\varphi</math>, with nothing to the left of the turnstile, is used to symbolize a tautology. It may be interpreted as saying that <math>\varphi</math> is a semantic consequence of the empty set of formulae, i.e., <math>\{\}\models\varphi</math>, but with the empty brackets omitted for simplicity;<ref name="BostockIntermediate" /> which is just the same as to say that it is a tautology, i.e., that there is no interpretation under which it is false.<ref name="BostockIntermediate" />}} or a ''tautology'',<ref name="ms33"/><ref name="ms34"/>ref name="ms32<ref name=":29" /> if it is true under every interpretation,<ref name=":19" /><ref name="metalogic" /> or ''true in every case.''<ref name=":21" />
* '''Consistent sentence:''' A sentence of <math>\mathcal{L}</math> is ''[[Consistency|consistent]]'' if it is true under at least one interpretation. It is ''inconsistent'' if it is not consistent.<ref name=":19" /><ref name="metalogic" /> An inconsistent formula is also called ''self-contradictory'',<ref name=":1" /> and said to be a ''self-contradiction'',<ref name=":1" /> or simply a ''contradiction'',<ref name=":30" /><ref name=":31" /><ref name=":32" /> although this latter name is sometimes reserved specifically for statements of the form <math>(p \land \neg p)</math>.<ref name=":1" />

For interpretations (cases) <math>\mathcal{I}</math> of <math>\mathcal{L}</math>, these definitions are sometimes given:

* '''Complete case:''' A case <math>\mathcal{I}</math> is ''complete'' if, and only if, either <math>\varphi</math> is true-in-<math>\mathcal{I}</math> or <math>\neg\varphi</math> is true-in-<math>\mathcal{I}</math>, for any <math>\varphi</math> in <math>\mathcal{L}</math>.<ref name=":21" /><ref name="ms35"/>
* '''Consistent case:''' A case <math>\mathcal{I}</math> is ''consistent'' if, and only if, there is no <math>\varphi</math> in <math>\mathcal{L}</math> such that both <math>\varphi</math> and <math>\neg\varphi</math> are true-in-<math>\mathcal{I}</math>.<ref name=":21" /><ref name="ms36"/>

For [[classical logic]], which assumes that all cases are complete and consistent,<ref name=":21" /> the following theorems apply:

* For any given interpretation, a given formula is either true or false under it.<ref name="metalogic" /><ref name=":20"/>
* No formula is both true and false under the same interpretation.<ref name="metalogic" /><ref name=":20" />
* <math>\varphi</math> is true under <math>\mathcal{I}</math> if, and only if, <math>\neg\varphi</math> is false under <math>\mathcal{I}</math>;<ref name="metalogic" /><ref name=":20" /> <math>\neg\varphi</math> is true under <math>\mathcal{I}</math> if, and only if, <math>\varphi</math> is not true under <math>\mathcal{I}</math>.<ref name="metalogic" />
* If <math>\varphi</math> and <math>(\varphi \to \psi)</math> are both true under <math>\mathcal{I}</math>, then <math>\psi</math> is true under <math>\mathcal{I}</math>.<ref name="metalogic" /><ref name=":20" />
* If <math>\models\varphi</math> and <math>\models(\varphi \to \psi)</math>, then <math>\models\psi</math>.<ref name="metalogic" />
* <math>(\varphi \to \psi)</math> is true under <math>\mathcal{I}</math> if, and only if, either <math>\varphi</math> is not true under <math>\mathcal{I}</math>, or <math>\psi</math> is true under <math>\mathcal{I}</math>.<ref name="metalogic" />
* <math>\varphi \models \psi</math> if, and only if, <math>(\varphi \to \psi)</math> is [[logically valid]], that is, <math>\varphi \models \psi</math> if, and only if, <math> \models(\varphi \to \psi)</math>.<ref name="metalogic" /><ref name=":20" />