Editing Modus ponens (section)

== Formal notation ==

{{Unreferenced section|date=May 2025}}

{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center" class="wikitable floatright"
|-
! ''p''
! ''q''
! ''p'' → ''q''
|- style="background:#D1FFBD"
| T || T || T
|-
| T || F || F
|-
| F || T || T
|-
| F || F || T
|}

The ''modus ponens'' rule may be written in [[sequent]] notation as
:<math>P \to Q,\; P\;\; \vdash\;\; Q</math>

where ''P'', ''Q'' and ''P'' → ''Q'' are statements (or propositions) in a formal language and [[⊢]] is a [[metalogic]]al symbol meaning that ''Q'' is a [[Logical consequence|syntactic consequence]] of ''P'' and ''P'' → ''Q'' in some [[formal system|logical system]].

In classical two-valued logic, ''modus ponens'' is encoded in the [[truth table]] of the [[material conditional]] (implication) operator. A truth table lists all possible combinations of the truth values of the arguments, in this case ''p'' and ''q'', one case per row. ''Modus ponens'' is the case where both ''p'' → ''q'' and ''p'' may be assumed (denoted as true). Encoding ''modus ponens'' faithfully, ''q'' may also be assumed and therefore is also denoted as true. The truth table of implication also expresses other common inference rules, such as [[modus tollens]] on the fourth row, assuming ''p'' → ''q'' and not ''q'' therefore not ''p'', and the [[monotonicity of entailment]] on the first and third rows, assuming ''q'' and ''p'' → ''q'', expressing how ''p'' may or may not be assumed.