Editing Modus ponens (section)

===Algebraic semantics===

In mathematical logic, [[algebraic semantics (mathematical logic) | algebraic semantics]] treats every sentence as a name for an element in an ordered set. Typically, the set can be visualized as a [[lattice (order) | lattice]]-like structure with a single element (the "always-true") at the top and another single element (the "always-false") at the bottom. Logical equivalence becomes identity, so that when <math>\neg{(P \wedge Q)}</math> and <math>\neg{P} \vee \neg{Q}</math>, for instance, are equivalent (as is standard), then <math>\neg{(P \wedge Q)} = \neg{P} \vee \neg{Q}</math>. Logical implication becomes a matter of relative position: <math>P</math> logically implies <math>Q</math> just in case <math>P \leq Q</math>, i.e., when either <math>P = Q</math> or else <math>P</math> lies below <math>Q</math> and is connected to it by an upward path.

In this context, to say that <math display="inline">P</math> and <math>P \rightarrow  Q</math> together imply <math>Q</math>—that is, to affirm ''modus ponens'' as valid—is to say that the highest point which lies below both <math>P</math> and <math>P \rightarrow  Q</math> lies below <math>Q</math>, i.e., that <math>P \wedge (P \rightarrow  Q) \leq Q</math>.{{efn|The highest point that lies below both <math>X</math> and <math>Y</math> is the "[[Join and meet|meet]]" of <math>X</math> and <math>Y</math>, denoted by <math>X \wedge Y</math>.}} In the semantics for basic propositional logic, the algebra is [[Boolean algebra (structure) | Boolean]], with <math>\rightarrow</math> construed as the [[material conditional]]: <math>P \rightarrow  Q = \neg{P} \vee Q</math>. Confirming that <math>P \wedge (P \rightarrow  Q) \leq Q</math> is then straightforward, because <math>P \wedge (P \rightarrow  Q) = P \wedge Q</math> and <math>P \wedge Q \leq Q</math>. With other treatments of <math>\rightarrow</math>, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.