# Vacuous Conditionals and How We Feel About Them

Let's take the following propositions :

1 - "If Bill Gates is poor then Bill Gates is rich".
2 - "If Bill Gates is poor then the moon is made of cheese".

Both propositions are inevitably true under the usual interpretation (our current world,) and I completely understand the reason.

The reason is that we don't have a situation where the antecedent is true and the consequent is false, so the proposition is true (even if it is vacuously true).

But there is something which is intriguing me.

Even though, some say that both 1) and 2) are a bit weird (the issue some people have with vacuous truth) , why between 1) and 2) people usually find 1) a bit more weird than 2) ?
In another words, why when p is false, the truth of the proposition "p -> q" sounds a bit more weird if q is ~p compared to other situations where q is arbitrary ?

One possible conjecture (which is probably wrong) is that people find that "p and ~p" ought to be false and perhaps in the natural language the implication resembles a bit conjunction. But I don't know if it does.

Anyways, does anyone have an idea about the issue?

P.S : I wanna emphasize here that my problem is not with relevance or vacuous truth, but rather the difference between 1) and 2) ( both of which suffer the problem of relevance ).
Thanks

Aristotle, the first person to write inclusively about logic, had a definite opinion on the subject. In the original system of syllogisms, these two 'weird' statements are not true, the first one is false and the second is undetermined. Later in the history of logic, mathematicians simplified deduction by injecting our new 'iffier' version of 'if', and scientists and some of the general public just came along, because it greatly improves the efficiency of the system.

Traditionally, the rule of induction known as modus ponens is "From A, when A implies B, Deduce B", it says nothing about when A is false, and it doubts the meaningfulness of "A implies B" when A is false or unknown. Similarly, Aristotle did not believe that there is only a single important quantifier "For All" and its dual "There Exists". If you do not use the simpler version of 'if', you need at least four: "All", "No", "Some", and "Not Every".

But the new mathematical logic was precise and efficient enough to diagnose that its own paradoxes really are unavoidable (it is not that they are absent from Aristotle, just harder to see.) In light of these problems, one response is Intuitionism, which wants to see mathematics as something more psychological than we currently do.

If math is really psychology, the scientific study of human abstraction, and not just advanced accounting, or some blessedly pure domain whose subject matter lies beyond the stars, then we need to pay more attention to your feeling of 'weirdness'. This approach revives the notion of doubting the meaningfulness of "A implies B" when A is unknown or false. But no one wants to lose the 'slickness' of modern logic that would result if we changed back the meaning of 'if' and accepting the extra complexities of Aristotelian quantification.

Instead, they accommodated the psychology by weakening our notion of negation, rejecting the principle known as tertium non datur, or the Law of the Excluded Middle, "What is not true is false. There is no third option". Psychologically, it is not really convincing to humans that everything that is not true must be false. It is not true that all unicorns are white, but it is not really false either, given that there are no unicorns. The implication about Gates and the moon is neither true nor false.

At the same time, we do not have the same problem with the idea that if unicorns were white then they would be black. That seems clearly false. "A implies not A" is still "not A or not A" which is still "not A". When A = "Gates is false", the psychological reaction, and Aristotle's, is the right one in intuitionistic logic.

So this system accepts the new meaning of 'if' and still backs up the feeling that your first statement is not true, and your second statement is not certain, but changing the meaning of 'not' in such a way that these statements are not true, but they are also not false.

It can be interpreted as a case of a more general concept of modality in logic. "True" can have multiple meanings in the same system, with different bases of reference. "Tertium non datur" may be true in (macroscopic) physics, and it can still be consistent, given strict rules about things like abstractly-defined collections not being real. But it is not true in fiction.

So that physics is using a specific mode of interpretation and fiction or intuitionistic mathematics is using a different one. They give up different things in order to feel safe from nonsense. You can't have it all, or you get a bunch of paradoxes that prevent you from working efficiently. But you can safely do different jobs differently.