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I am just beginning to study proofs. What is the difference between modus ponens and implies/if-then when relating to proofs?

I figure that we are using modus ponens to advance the proof. If we were using $P \implies Q$, couldn't I start with a false P in order to falsely "prove" Q? In the limited number of proofs I have seen it seems like we just use $P \implies Q$. I am thinking it is either shorthand for modus ponens if we assume what we start out with is true?

How should I understand $\implies$ in proofs?

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  • $\begingroup$ You can't do what concerns you because to use modus ponens, you need to establish both $P$ and also $P \Rightarrow Q$. That means you won't be able to use a false premise for a bogus proof. When using $P \Rightarrow Q$ in mathematical writing, in context it's often obvious that $P$ has already been established. $\endgroup$ Commented Feb 18, 2021 at 23:42
  • $\begingroup$ To clarify, since P is already established. When we say $P \to Q$ we are using modus ponens just not saying it every time? $\endgroup$
    – JCK
    Commented Feb 19, 2021 at 3:32
  • $\begingroup$ That's correct, yes. $\endgroup$ Commented Feb 19, 2021 at 4:36

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Modus ponens, specifically is an inference rule. An inference rule is says that a conclusion is true if its premises are true. In other words, the inference rules of a system tell you what manipulations you are allowed to perform that don't introduce erroneous conclusions.

Here's modus ponens symbolically.

$$ \frac{A \to B \;\; \text{and} \;\; A}{B} $$

$A \to B$ is a premise. $A$ is a premise as well. $\text{and}$ exists outside the formal language and just tells us that we have both premises. $B$ is the conclusion. The line also exists outside the formal language and tells us how the premises and the conclusion are related.

Modus ponens allows you to take a conditional that you already know and its premise and infer its conclusion. This rule is fundamental to how conditionals are used.

Implies, which I've written $\to$, is how you represent the meaning of if A, then B. It is equivalent to the following:

$$ A \to B \;\; \text{is equivalent to} \;\; (\lnot A) \lor B $$

Another way of looking at it is that $A \to B$ is false if and only if $A$ is true and $B$ is false.

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