Being tired of seeing that a significant number of philosophers and other people tend to think that the problems raised by philosophical scepticism can be solved by just saying: “Well, if the arguments/claims of sceptics are valid/true, then they would refute themselves, therefore they can't be valid/true”, (and that's it! They think that's all it takes to solve the problems of scepticism that have troubled many of the greatest philosophers for so many years!) which seems to me to be a lazy and simplistic response, I have written this thread:

One of the arguments used against skeptics since the times of ancient Greece and down to the present day, starts from the following implication: "If there is no proof, there is proof" (¬p → p)

They say that it must necessarily be true, because if the skeptics affirm without proof that there is no proof, they would be as untrustworthy as those who affirm without proof that proof exists. And if they affirm it with a demonstration, they are contradicting themselves, because then there would be at least one demonstration: the one proving that there is no proof.

The ancient skeptics themselves have already responded to this argument in Adversus Mathematicos and in Outlines of Pyrrhonism.

I for my part will give a somewhat different answer to that argument:

Let p be: "Proof exists" (Meaning: “There is at least one proof”).

1: ¬p (Sceptical hypothesis)

2:¬p is equivalent to ¬ (p v p) (1, Idempotent Law)

3: ¬(p v p) is equivalent to ¬(¬¬p v p) (2, Double negation)

4: ¬(¬¬p v p) is equivalent to ¬ (¬p → p) (3, Implication)

5: ¬p is equivalent to ¬(¬p → p) (1,2,3,4 Transitivity)

6: p v ¬p (Law of the Excluded Middle)

7: ¬p → ¬p (6, Implication)

8: ¬p → ¬(¬p → p) (5,7 Substitution)

9: ¬p ^ [¬p → ¬(¬p → p)] (1,8 Conjunction)



This shows that there is no logical contradiction in denying the implication ¬p → p if proof does not exist, and therefore those who disagree with the sceptics cannot hold that the implication ¬p → p is true to show that p is true without begging the question, since this implication is only true if p, that is: "Proof exists", is true.

It will be answered: "But what you have just done here is certainly a demonstration, therefore there is at least one demonstration." Then the skeptic will answer: But if what I just did is a proof, then its conclusion: ¬p^ [¬p →¬(¬p → p)] must be true, therefore ¬p is true, therefore ¬(¬p→p) is true, therefore (¬p→p) is false, therefore ¬p is true and p is false.

The problem is that logic seems to self-destruct, and the skeptic then asks: Where is the error in [Insert here a seemingly valid argument that has “therefore there is no proof” as its conclusion, that the opponent of the sceptic has criticised by saying that ¬p→p] then? Since if the conclusion follows from the premises, and the premises are admited to be true, then the conclusion, “proof does not exist”, must also be admited.

He would continue: “I myself admit that I do not know if there is an error or not, nor if what I have just stated is a proof , but my argument is that by not being able to determine even if proof is more likely to exist than for it not to exist, we should suspend judgment”

David Hume, in the conclusion of section I of his Treatise of Human Nature, understood the problem quite well:

If the sceptical reasonings be strong, say they, ’tis a proof, that reason may have some force and authority: if weak, they can never be sufficient to invalidate all the conclusions of our understanding. This argument is not just; because the sceptical reasonings, were it possible for them to exist, and were they not destroy’d by their subtility, wou’d be successively both strong and weak, according to the successive dispositions of the mind. Reason first appears in possession of the throne, prescribing laws, and imposing maxims, with an absolute sway and authority. Her enemy, therefore, is oblig’d to take shelter under her protection, and by making use of rational arguments to prove the fallaciousness and imbecility of reason, produces, in a manner, a patent under her hand and seal. This patent has at first an authority, proportion’d to the present and immediate authority of reason, from which it is deriv’d. But as it is suppos’d to be contradictory to reason, it gradually diminishes the force of that governing power, and its own at the same time; till at last they both vanish away into nothing, by a regular and just diminution. The sceptical and dogmatical reasons are of the same kind, tho’ contrary in their operation and tendency; so that where the latter is strong, it has an enemy of equal force in the former to encounter; and as their forces were at first equal, they still continue so, as long as either of them subsists; nor does one of them lose any force in the contest, without taking as much from its antagonist. ’Tis happy, therefore, that nature breaks the force of all sceptical arguments in time, and keeps them from having any considerable influence on the understanding. Were we to trust entirely to their self-destruction, that can never take place, ’till they have first subverted all conviction, and have totally destroy’d human reason.

Yet even Hume appears not to have been thoroughgoing enough: Just because in the past our nature has been such that sceptical reasonings haven't had a considerable influence on our understanding, that doesn't mean that in the future they won't have a considerable influence in our minds. As Bertrand Russell pointed out:

(...)the law of habit is itself a causal law.
Therefore if we take Hume seriously we must say: Although in the past the sight of an apple has been conjoined with expectation of a certain kind of taste, there is no reason why it should continue to be so conjoined: perhaps the next time I see an apple I shall expect it to taste like roast beef. You may, at the moment, think this unlikely; but that is no reason for expecting that you will think it unlikely five minutes hence. If Hume's objective doctrine is right, we have no better reason for expectations in psychology than in the physical world.