# Particular case of an Implication

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 $p \to q$ sounds a bit more weird if $q$ is $\neg p$ compared to other situations where $q$ is arbitrary ?

One possible conjecture (which is probably wrong) is that people find that $p \wedge \neg p$ 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

• What exactly is your question? – nathan.j.mcdougall Apr 12 '15 at 6:30
• To quote myself : "... why between 1) and 2) people usually find 1) a bit more wierd than 2) ?" and "...why when p is false, the truth of the proposition p→q sounds a bit more wierd if q is ¬p compared to other sittuations where q is arbitrary ? " – nerdy Apr 12 '15 at 6:32
• Perhaps because propositions of the form p implies not p are inherently false, whereas propositions that are part of a more complex system (like evaluating the relationship between the moon's composition and Bill Gate's wealth,) are more difficult to evaluate initially and are subject to change in the system (e.g., Bill Gates could promise, in theory to turn the moon into cheese should he become poor.) – nathan.j.mcdougall Apr 12 '15 at 6:36
• See Relevance Logic : "consider the classically valid inference : The moon is made of green cheese. Therefore, either it is raining in Ecuador now or it is not. The conclusion seems to have nothing to do with the premise. Relevance logicians have attempted to construct logics that reject theses and arguments that commit “fallacies of relevance”." – Mauro ALLEGRANZA Apr 12 '15 at 6:37
• The sentence "If Bill Gates is poor then Bill Gates is rich" is different from the sentence "If Bill Gates were poor then Bill Gates would be rich". I hope that makes sense. – Najib Idrissi Apr 12 '15 at 9:16

I think that the distinction between the two lies in the possibility of the consequent.

In 1., the statement "Bill Gates is rich" is not absurd per se, it only is in the context of the premise "Bill Gates is poor". This is strengthened by the fact that Bill Gates is actually rich.

However, in 2., we have the claim "The moon is made of cheese", which is commonly accepted to be utterly false.

So, what I think happens subconsciously is that in case 2., we are automatically triggered to reject the conclusion because it's always false. This in turn leads to a hunt for the cause, i.e. vacuous truth.

On the other hand, with 1., it really requires some thought and interplay between antecedent and consequent to reach the conclusion that the statement as a whole must be true. This is because the statements are obviously intimately related, so we are triggered to do a more thorough investigation of the situation before reaching a conclusion.

• I think it's a good point that in 1 the conclusion is true in real life, but in 2 the conclusion isn't. – nathan.j.mcdougall Apr 12 '15 at 7:10

The language of implication is often used in common speech, but usually to mean something rather different from implication in mathematical logic.

In one sense, people may attach a causal relationship or at least a positive correlation between the clauses linked by "if". For example, if you have high blood pressure, your life expectancy is less than the average person's. There is a positive correlation between high blood pressure and low life expectancy.

That is, there is a tendency to interpret "if A then B" as "A tends to cause B" or "observing A makes B more likely."

Neither of the implications fits either of these interpretations. For implication (2), however, we have to use some knowledge of the real world to come to this conclusion; that is, it seems obvious (based on what we know about how the moon came to be, and how Bill Gates gained his wealth) that the composition of the moon is not predicted in any way by how much wealth Bill Gates has. For implication (1), we do not need to know anything about cosmology or cheese or even to know who Bill Gates is to reject causality or correlation; we merely need to know that "rich" is the opposite of "poor". Being poor cannot cause you to be rich, and it is not correlated with being rich.

Alternatively, there is one common way in which the language of implication is used when the antecedent is believed to be false, namely, to emphasize how sure one is that the antecedent is false. For example, a common English idiom to express disbelief in the proposition that Bill Gates is poor is, "If Bill Gates is poor then I'm a monkey's uncle." Since I am not a monkey's uncle (which the listener hopefully will accept as obviously true without question), the statement "Bill Gates is poor", combined with my implication, contradicts a known true fact and therefore is rejected.

A variant of this is, "If Bill Gates is poor then I'll eat my hat." The usual connotation of this is that it would be very unpleasant for me to have to eat my hat, so making this statement shows that I am quite confident Bill Gates is not poor (since otherwise I would be taking some risk of having to eat my hat later, in case it is shown that he is poor).

Implication (2) can be taken as yet another variant of this same idiom, namely, we use an absurdly false consequent (or in the eat-my-hat variant, one the speaker presumably hopes is false) in order to emphasize how unlikely the antecedent is to be true. Everyone knows the moon is not actually made of cheese (Wallace and Gromit notwithstanding), so clearly this implication says Bill Gates is not poor. And indeed Bill Gates is not poor. In this sense, implication (2) is absurd but true in common, non-mathematical language.

On the other hand, since Bill Gates is rich, there is not even a hope that the consequent of implication (1) is false. Implication (1) thus does not fit the pattern of this common idiom at all, and is not recognized as a non-mathematical way of saying that Bill Gates is not poor. To ascribe that meaning to it, one has to work through details of mathematical logic that are not part of our common intuition about the English language.