Editing Propositional calculus (section)

== Formalization ==
Propositional logic is typically studied through a [[formal system]] in which [[well-formed formula|formulas]] of a [[formal language]] are [[interpretation (logic)|interpreted]] to represent [[propositions]]. This formal language is the basis for [[Proof calculus|proof systems]], which allow a conclusion to be derived from premises if, and only if, it is a [[logical consequence]] of them. This section will show how this works by formalizing the {{section link||Example argument}}. The formal language for a propositional calculus will be fully specified in {{section link||Language}}, and an overview of proof systems will be given in {{section link||Proof systems}}.

=== Propositional variables ===
{{Main article|Propositional variable}}
Since propositional logic is not concerned with the structure of propositions beyond the point where they cannot be decomposed any more by logical connectives,<ref name=":13" /><ref name=":1" /> it is typically studied by replacing such ''atomic'' (indivisible) statements with letters of the alphabet, which are interpreted as variables representing statements ([[Propositional variable|''propositional variables'']]).<ref name=":1" /> With propositional variables, the {{section link||Example argument}} would then be symbolized as follows:

:'''Premise 1:''' <math>P \to Q</math>
:'''Premise 2:''' <math>P</math>
:'''Conclusion:''' <math>Q</math>

When {{mvar|P}} is interpreted as "It's raining" and {{mvar|Q}} as "it's cloudy" these symbolic expressions correspond exactly with the original expression in natural language. Not only that, but they will also correspond with any other inference with the same [[logical form]].

When a formal system is used to represent formal logic, only statement letters (usually capital roman letters such as <math>P</math>, <math>Q</math> and <math>R</math>) are represented directly. The natural language propositions that arise when they're interpreted are outside the scope of the system, and the relation between the formal system and its interpretation is likewise outside the formal system itself.

=== Gentzen notation ===
If we assume that the validity of [[modus ponens]] has been accepted as an [[axiom]], then the same {{section link||Example argument}} can also be depicted like this:

:<math>\frac{P \to Q, P}{Q}</math>

This method of displaying it is [[Gerhard Gentzen|Gentzen]]'s notation for [[natural deduction]] and [[sequent calculus]].<ref name=":40"/> The premises are shown above a line, called the '''inference line''',<ref name=":3" /> separated by a '''comma''', which indicates ''combination'' of premises.<ref name=":34"/> The conclusion is written below the inference line.<ref name=":3" /> The inference line represents ''syntactic consequence'',<ref name=":3" /> sometimes called ''deductive consequence'',<ref name=":7"/>> which is also symbolized with ⊢.<ref name=":6"/><ref name=":7" /> So the above can also be written in one line as <math>P \to Q, P \vdash Q</math>.{{refn|group=lower-alpha|The turnstile, for syntactic consequence, is of lower [[Order of operations|precedence]] than the comma, which represents premise combination, which in turn is of lower precedence than the arrow, used for material implication; so no parentheses are needed to interpret this formula.<ref name=":34" />}}

Syntactic consequence is contrasted with ''semantic consequence'',<ref name="ms16"/> which is symbolized with ⊧.<ref name=":6" /><ref name=":7" /> In this case, the conclusion follows ''syntactically'' because the [[natural deduction]] [[Rule of inference|inference rule]] of [[modus ponens]] has been assumed. For more on inference rules, see the sections on proof systems below.