Editing Modus ponens (section)

==Correspondence to other mathematical frameworks==

===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.

===Probability calculus===
If <math>\Pr(P \rightarrow Q) = x</math> and  <math>\Pr(P) = y</math>, then <math>\Pr(Q)</math> must lie in the interval <math>[x + y - 1, x]</math>.{{efn|Since <math>\neg P</math> implies  <math>P \rightarrow Q</math>, <math>x</math> must always be greater than or equal to <math>1 - y</math>, and therefore <math>x+y-1</math> will be greater than or equal to <math>0</math>. And since <math>y</math> must always be less than or equal to <math>1</math>, <math>x+y-1</math> must always be less than or equal to <math>x</math>.}}<ref name="Hailperin, T. 1996">{{cite book |last1=Hailperin |first1=Theodore |title=Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications |year=1996 |page=203 |publisher=Associated University Presses|location=London|isbn=0934223459}}</ref> For the special case <math>x = y = 1</math>, <math>\Pr(Q)</math> must equal <math>1</math>.

===Subjective logic===
''Modus ponens'' represents an instance of the binomial deduction operator in [[subjective logic]] expressed as:

<math display="block">\omega^{A}_{Q\|P} = (\omega^{A}_{Q|P},\omega^{A}_{Q|\lnot P})\circledcirc \omega^{A}_{P}\,,</math>

where <math>\omega^{A}_{P}</math> denotes the subjective opinion about <math>P</math> as expressed by source <math>A</math>, and the conditional opinion <math>\omega^{A}_{Q|P}</math> generalizes the logical implication <math>P \to Q</math>. The deduced marginal opinion about <math>Q</math> is denoted by <math>\omega^{A}_{Q\|P}</math>. The case where <math>\omega^{A}_{P}</math> is an absolute TRUE opinion about <math>P</math> is equivalent to source <math>A</math> saying that <math>P</math> is TRUE, and the case where <math>\omega^{A}_{P}</math> is an absolute FALSE opinion about <math>P</math> is equivalent to source <math>A</math> saying that <math>P</math> is FALSE.  The deduction operator <math>\circledcirc</math> of [[subjective logic]] produces an absolute TRUE deduced opinion <math>\omega^{A}_{Q\|P}</math> when the conditional opinion <math>\omega^{A}_{Q|P}</math> is absolute TRUE and the antecedent opinion <math>\omega^{A}_{P}</math> is absolute TRUE. Hence, subjective logic deduction represents a generalization of both ''modus ponens'' and the [[Law of total probability]].<ref>Audun Jøsang 2016:92</ref>