At the end of my time on the BibleThumpingWingnut, after a few hours (and about 4 whiskeys, at about 3AM), Tim introduced a new person into the discussion to ‘engage’ with me for a bit. This was Jason Petersen, who advocates a version of Clarkian presuppositionalism. Jason began by laying out an axiomatic demonstration of how you can go from the principle that the bible is the word of god to the conclusion that you can account for the laws of logic. After he explained his ‘axiom of revelation’, which is that the bible is true, he moved to a passage which contains the phrase ‘no lie is of the truth’. We got a bit stuck on this, as I objected that lies can be inadvertently true, as for example when someone intends to deceive, says something they believe is false, but which happens to be correct. I think that this would still count as a lie, but Jason disagreed, urging that we should use the biblical definition instead. I was tired and a bit drunk, so I may have missed what was going on at the time. I thought I should get a more sober reflection down here instead.

As I understand what was going on, Jason was starting with his axiom, and then deriving things from that, part of which included the law of non-contradiction. His point was (I believe), that ‘no lies are of the truth’ is an instance of someone stating the law of non-contradiction, i.e. ~(p & ~p). I think this is an exegetical stretch, and even if interpreted as generously as possible it gives a different law, the semantic principle of bivalence. So I say that ‘no lies are of the truth’ means ‘all lies are false’, which I said was false, due to my understanding of what lying means. But let’s assume that the intentional aspect of lying is not important, and as such lying just means saying a falsehood. This makes the sentence ‘no lies are of the truth’ analytically true (i.e. true by definition). Fair enough. It just means ‘no falsehood is true’. In other words, it means that if something is false, it is not also true. The principle of bivalence says that every proposition takes exactly one truth value: true or false; i.e. that if a sentence is true, it is not false, and vice versa. For some reason, Jason thinks that the sentence actually should be read as meaning ‘it is not the case that both p and not-p’; i.e. it is not the case that p and it is not-p. Notice that this doesn’t use the word truth at all. The difference may seem minor, but it allows that there can be logics where some proposition is neither true nor false (so no bivalence), but where it and its negation are still incompatible (so keeping non-contradiction), etc. Anyway, we can forgive the fact that a) the sentence is false (because I am right about what lying means), b) the sentence at best means something similar to the principle of bivalence, and c) it doesn’t mean the same as the principle of non-contradiction. We can forgive all of those and just assume that he was right. So let’s just say he starts from his revelational axiom, and then ‘derives’ the principle of non-contradiction. That seemed to be what he wanted to do. I say that this is horribly flawed anyway, despite the above.

So he has an axiom: everything in the bible is true (he actually says ‘the bible alone is the word of God written’). This basically just means that every proposition in the bible is true. So think of the bible as a set of propositions, B = {a, b, c, …} and that every member of the set is true. Then he says that he can go to one of those propositions, which is the law of non-contradiction (although he repeatedly dropped the ‘non’ for some reason). Therefore, the law of non-contradiction is true. In this way he derives it from his basic axiom.

So, assuming a = the principle of non-contradiction, the argument so far is:

Premise 1) a & b & c & … (i.e. all the elements of B)

Therefore, a

However, the inference from B to a (from all the things in the bible, to the one particular thing in the bible), relies on the inference rule called ‘conjunction elimination’; from p & q one can infer p:

Premise 1) p & q

Therefore, p

Therefore, Jason’s ‘axiom’ needs to be supplemented with, at least, the inference rules of classical logic, if he is to move off his axiomatic starting point to derive anything (even if it is contained as a conjunct in his conjunction). He doesn’t mention inference rules, but he must be assuming them or else he would be stuck with his axiom. So let’s be nice and give them to him. But that means he is assuming classical logic. And that means he is assuming the law of non-contradiction. So he doesn’t need to ‘derive’ the law of non-contradiction, as he would in fact be assuming it at the outset.

But maybe he has in mind a sort of non-classical logic, one that retains the ability to use conjunction elimination, but does not postulate as an axiom that there are no contradictions. But then the problem would be that there would be nothing to stop the paradoxical looking inference rule: ‘negation introduction’, which I have just made up, but would look like this:

Premise 1) p

Therefore ~p

Presumably, Jason would want to object that this rule is not part of his implicit set of inference rules. But the question would then be, why not? It seems to me that the only thing Jason could appeal to would be the fact that there cannot be a contradiction, which just is the principle of non-contradiction. And if he said that he would be admitting that he does presuppose non-contradiction after all, and does not derive it from an axiom.

The results for his logic if he did have negation introduction would be devastating. For a start, from his axiom B, one could derive ~B; from the axiom that the bible is true, one could derive that it is not the case that the bible is true. Even if he derived a from B (the principle of non-contradiction from the bible), one could also derive ~a from B (by deriving a from B, and ~a from a). So the bible would say there could be no contradictions, and it would say that it is not the case that there could be no contradictions.

The point is that negation elimination is to be avoided at all costs. The best way to avoid it is to start with it as an axiom that there are no true contradictions.