Modus Ponens vs implication?

Is there a difference between Modus Ponens and an implication?

If so, could you please give a simple example to help understanding?

Solution 1:

An implication is simply a proposition that asserts "if $P$, then $Q$": $$P \rightarrow Q$$

Modus ponens is a rule of inference that tells us when we have the conjunction of an implication and the hypothesis (i.e. antecedent) of the implication: $P\rightarrow Q$ together with $P$, we are justified to conclude that $Q$. This is best expressed as follows:

$$ \begin{array}{rl} & P\rightarrow Q \\ & P \\ \hline \therefore & Q\end{array} $$

Example:

If there is a smile on my face, then I am happy. (Implication)

If there is a smile on my face, then I am happy. There is a smile on my face. Therefore, I am happy. (Using modus ponens)

Solution 2:

Modus ponens is a rule of inference that tells us under what circumstances we can infer a sentence from other given sentences. In particular, it says that given sentences $P \rightarrow Q$ and $P$ we may infer $Q$.

Implication is used in many ways. One simple way is that an implication is simply any sentence of the form $P \rightarrow Q$.

A slightly deeper concept is (logical) implication (or entailment), which is a relationship between statements: when we say $P$ implies $Q$ we mean that $P$ cannot be true without $Q$ also being true.

While it is true that there is a great similarity between the first and last concepts it is perhaps best to think of modus ponens as living in the syntactic side of logic (where we deal only with strings of "meaningless" symbols), while logical implication lives on the semantic side (where we speak of truth in models).