# Soundness of a deductive argument

I am a student who was just introduced to philosophy.

On a test, we were given the following multiple-choice question on soundness of a deductive argument:

“Anything that travels in time necessarily changes the past. But necessarily, nothing changes the past. Thus, necessarily, nothing travels in time.”

The argument is:

1. invalid and sound

2. invalid and unsound

3. valid and sound

4. valid and unsound

I don't have the answer with me, and my peers and I are torn between 3 and 4. Given that there is only one possible answer, which should it be?

The answer to this question depends in part of the symbolization resources you have available to you.

If we translate as follows:

Anything that travels in time necessarily changes the past

to

T -> C

and

But necessarily, nothing changes the past.

to not C

Thus, necessarily, nothing travels in time.

and the conclusion to not T.

Then we wind up with:

1. T -> C
2. not C
3. Therefore not T

which is a valid deductive argument -- in fact, it's an example of modus tollens.

here T = "to travel in time" and C = "to change the past".

But the soundness remains unclear or at least outside of the scope of what seems fair in a problem, because for it to be sound, the two premises must be true in addition to the form being valid.

A careful reader will note that the above symbolization drops the following:

1. necessarily.
2. anything vs. nothing.

If we want to add them, then we need a more advanced form of symbolization, which gives us:

1. ∀x(Tx -> ◻Cx)
2. ◻~∀x(Cx)
3. Therefore, ◻~∀x(Tx)

Here, ∀x = for any x, T = travels in time, ◻ = necessity, Cx = changes the past.

which seems to also be valid (at least on the inference rules I can think of). But again, the same problem arises with respect to soundness.

if your instructor is into goofy tricks, then linguistically, it seems to not be sound, because if something traveled in time from the present to the future, that doesn't seem to alter the past -- so the first premise seems demonstrably false.