Modus Ponens, Modus Tollens:

From Rosen 6th Edition, p. 63:

Consider the following argument...

This has the form:

pq
p

q

This form of argument is calls Modus Ponens (latin for "mode that affirms")


Note that an argument can be valid, even if one of the premises is false. For example, the argument above doesn't say whether you do or don't have a current password.
Maybe you do, and maybe you don't . But either way, the argument is still valid.


Consider this argument:


Now, in real life, we might say: that's not a valid argument. For example, maybe you can't log into the network because your Ethernet cable is bad, or because the network is down for maintenance, or any one of a zillion other reasons.

But in fact, this is a valid argument in logic. If we accept the two premises, then the conclusion follows. One of the premises is "If you have a current password, you can log into the network". There are no ifs, ands, or buts.

So, this illustrates an important point: when working with logic problems it is important to take the statements literally and at face value. Don't read things into the problems that aren't there.

We sometimes use problems that seem to be about the real world—we do this to make the problems more interesting and relevant, and to give us some insight into what the symbols mean. But the problems, ultimately are not in the real world—they are about a mathematical model of the real world, where we make a lot of simplifying assumptions. For example, the hard and fast absolute statement:
According to this statement, a current password is sufficient for you to be able to log in.

So, if we assume this to be true, as we do in the argument below, and we assume that you can't log into the network, then we can definitely conclude: you don't have a current password. So here it is again:

This is an argument of the form:

¬q
p
q

¬p

This form of argument is called modus tollens (the mode that denies).

Both modus ponens and modus tollens have "universal forms":

Universal modus ponens:

x((P(x)→Q(x))
P(a), where a ∈ {domain of the predicate P}

Q(a)

E.g. All fish have scales. This salmon is a fish. Therefore, this salmon has scales.

Universal modus tollens:

x((P(x)→Q(x))
¬Q(a), where a ∈ {domain of the predicate P}

¬P(a)

E.g. All surfers are hot. Conrad is not hot. Therefore Conrad is not a surfer.

Try to come up with your own examples of modus ponus, modus tollens, universal modus ponens, and universal modus tollens.