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terms built with logical connectives and the only inference rule

In the case of propositional systems the axioms are terms built with logical connectives and the only inference rule is modus ponens.

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The only rule of inference needed for this argument

The only rule of inference needed for this argument is modus ponens.

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the inference rule used

Note that the goals always match the affirmed versions of the consequents of implications (and not the negated versions as in modus tollens) and even then, their antecedents are then considered as the new goals (and not the conclusions as in affirming the consequent) which ultimately must match known facts (usually defined as consequents whose antecedents are always true); thus, the inference rule which is used is modus ponens.

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the 'cut rule' (for single conclusion sequent calculi)

This means modus ponens is the 'cut rule' (for single conclusion sequent calculi).

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a rule of inference and deductive argument form

Modus ponens, otherwise known as affirming the antecedent or implication elimination, is a rule of inference and deductive argument form.

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the best known deductive inference rule

Modus ponens is the best known deductive inference rule.

for “affirming mode”

Modus ponens is Latin for “affirming mode”.

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mode of affirming'

Modus ponens ('mode of affirming') is a logical rule of inference based on conditional propositions.

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an inference rule often used in mathematical reasoning

One is that although modus ponens is undoubtedly an inference rule often used in mathematical reasoning, mathematical reasoning is by no means the only one.

a valid form of deductive inference -

I agree that modus ponens is a valid form of deductive inference - if 1 and 2 are true for some value of p and q, then 3 is always true.