User:Knorlin/sandbox

In mathematical logic, algebraic semantics treats every sentence as the name of an object in an ordered set. Typically, the set can be visualized as a 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 $\neg{(P \wedge Q)} $ and $\neg{P} \vee \neg{Q}$, for instance, are different names for the same lattice element: $\neg{(P \wedge Q)} = \neg{P} \vee \neg{Q}$. Logical implication becomes a matter of relative position: $P$ logically implies $Q$  just in case $P \leq Q$, i.e., when $P$  is connected to $Q$  by an upward path.

In this context, to say that $P$ and $P \rightarrow  Q$  together imply $Q$ —that is, to affirm modus ponens as valid—is to say that $P \wedge (P \rightarrow  Q) \leq Q$. In the simpliest algegraic semantics, for basic propositional logic, $\rightarrow$  is construed as the material conditional; that is $(P \rightarrow  Q) = (\neg{P} \vee Q) $. Then the algebra is Boolean, and confirming that $P \wedge (P \rightarrow  Q) \leq Q$  is straightforward. With other treatments of $\rightarrow$, the semantics becomes more complex, the algebra may be non-Boolean, and the validity of modus ponens cannot be taken for granted.