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What is the difference between Modus Ponens and an implication?

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

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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:

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


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)

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Wait, if I know $[(P \rightarrow Q) \land P] \rightarrow Q$ and I know $(P \rightarrow Q) \land P$, how do I conclude $Q$ without your second form of modus ponens? I think the second form is the primary form, for proofs at least. – Thomas Andrews Mar 27 '13 at 17:30
Thanks, @Henning for the improved formatting! – amWhy Mar 27 '13 at 18:07

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).

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The short of it is that an implication, $P\implies Q$, read "$P$ implies $Q$", is simply a statement, while modus ponens is a method to draw a conclusion from an implication.

Other people have gone into some details about what these are, I just want to explore some "gotchas" about implication.

Implication, in mathematical logic, is a funny thing, because we intuitively interpret the word "implies" as meaning "follows logically from." But implication in mathematical logic is different.

For example, the statement, "$\text{(I am a walrus)}\implies 0=0$," is true. There is no sense in which me being a walrus leads us to the conclusion that $0=0$, but in mathematical logic, the implication is true.

In elementary logic, $P\implies Q$ is true if and only if either $\lnot P$ is true or $Q$ is true. In particular, if $Q$ is true, $P\implies Q$ is true, whatever $P$ is.

Also, it is a natural language tendency to add an implicit "for all" when we talk implications. For example, if someone told you: $$(z>0)\implies (x>y\implies xz>yz)$$ the natural reading is to interpret this as meaning "for all $x,y,z$." That reading of modus ponens is risky, because we can sometimes conclude the wrong thing is we try to "lazily" use modus ponens. For example, if you happen to know that $z=2$, then you can conclude that the statement $x>y\implies xz>yz$. But this is no longer true "for all $x,y,z$," but only for this specific value of $z$. So you have to be careful how you apply it.

In particular, then, in formal logic, when reading $P\implies Q$, you should neither read it intuitively to mean that $Q$ is some direct logical consequence of $P$, nor read it with an implicit generality.

So, to use an example from an answer above,

If I am smiling, then I am happy.

is only a statement about right now. If you want to make it a universal, you have to explicitly say,

At any time, if I was smiling at that time, then I was happy at that time.

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