Sunday, January 25, 2026

Weak Argument Against Vacuous Truths

Vacuous truths never felt intuitive to me, ever since I first learned about them.

Here's one reason why.  It's a weak reason and not a strong reason.  But it's strong enough in my potentially deranged mind for me to feel I should mention.  Forgive me if there are any typos, because they are pretty easy to make here.

Look at the statement NOT(if A then B) and the statement A and NOT B.  They both have the EXACT same truth table, if you draw the truth table out or ask a competent AI program to do it for you.

But I would argue A and NOT B is actually a stronger statement than NOT(if A then B), making the statements not equivalent.  A and NOT B is clearly not a conditional statement.  A happens no matter what.  Does A also happen no matter what in the statement NOT(if A then B)?

Suppose A doesn't happen and B does.  Is this an instance of NOT(if A then B)?  What if A doesn't happen and B also doesn't happen?  Also an instance of NOT(if A then B) perhaps?  I would say anything whatsoever that is anything other than if A then B fits the criteria of NOT(if A then B).  

Let's try an example to flesh out this line of thought.

Think of the actions of A and B being the following:

A = the action that you honked your car horn yesterday
B = the action that your car windshield broke.

There are four possibilities involving your horn and windshield yesterday:

1) your horn honked and your windshield broke
2) your horn honked and it didn't break
3) your horn did not honk and your windshield broke
4) your horn did not honk and your windshield didn't break.

Which of these 4 possibilities fit the 2 statements NOT(if A then B) and A and NOT B?  

A and NOT B would say that you definitely honked your horn yesterday and your windshield did not break.  Only possibility #2 fits that.

NOT(if A then B) would say it's not true that if you honked your horn yesterday, then your windshield broke.

If you did honk your horn, then only possibility #2 would fit that.  If you didn't honk your horn, then we don't know what would have happened for sure.  But mostly likely honking your horn wouldn't actively have caused your windshield to break.  This is NOT A and NOT B. Possibility #4.

Is possibility #4 a subset of the statement "it's not true that if you honked your horn yesterday, then your windshield broke"?  I would say yes, based on probabilistic assumptions alone. The only way this would have been true is if some mischievous individual installed some device in your car on purpose that would smash your windshield every time you honked your horn.  At least if you interpret the conditional if/then as definite cause and effect and not just random occurrence of each.  

We could alter our truth table for this specific example, but if we wanted to make a general table, we'd probably have to set up a U for unknown on vacuous truths.  Or maybe M for meaningless.

Still, there are pros and cons to using U for vacuous truths. They're more conservative, but they leave the logical system open instead of closed.  Plus they introduce a ternary or three-valued logic system (which some logicians already have invented) and they're a bit more complicated than logics with truth values only limited to T or F.

So even though I don't think vacuous truths always adhere to reality despite being pushed as such, they do allow closure in a fairly simple logical system, and there are definite benefits to that.  Not worth changing, but probably worth questioning more than they're currently questioned in introductory logic classes.



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