Logic 101

Sigh.

Two weeks in a row Matt Slick, Andrew Rappaport and the rest on BTWN have tried to save face after I explained my critique of their argument. Seeing as they are still just as confused as before I went on (and possibly more so), I have decided to spell out a few more issues here. They say it is an issue of wording. In reality, it is an issue of logic. As demonstrated already, they don’t get this because they don’t understand logic.

So, the first version of the argument has the first premise as this:

1) ‘Either god or not-god accounts for logic’.

This is how Slick actually said it, word-for-word, at various times on BTWN, in debates with people, on his radio show, etc. It is also a horrible train-wreck of a sentence. So what is wrong with this sentence? The problem is the placement of the ‘not’. Negation is a ‘truth-functional monadic operator’. What this means in more plain terms is just that it prefixes individual formulas (which is what makes it monadic), and the new formula it makes when it has been applied has a truth-value which is a product of the truth-value of the original proposition (which is what makes it truth-functional). So, an example will help. Here is a proposition:

2) Washington was the first president of America.

If we want to negate this proposition, we stick a ‘not’ in front of it as follows:

3) Not-(Washington was the first president of America).

The way negation works is by making the new formula have the opposite truth-value to the original one. Say 2) is true, then 3) (the negation of 2) is false. Also, say 2) is false, then 3) is true. Negation toggles between truth-values.

We can say 3) a little more perspicuously as

4) It is not the case that (Washington was the first president of America).

This means the same as 3).

In English, the grammar is messy and not logically regimented, meaning that we often express the same thing by having the negation in the middle of the sentence rather than at the start, as follows:

5) Washington was not the first president of America.

However, this is just a difference of wording, and 3), 4) and 5) all express exactly the same proposition. In propositional logic, if we set p = ‘Washington was the first president of America’, then we would write all three of these formally as follows:

6) ~(p)

In first-order logic, where we have terms for names and simple properties, we would express it differently. We would have a term for the name ‘Washington’, say ‘w’, and a term for the property ‘…was the first president of America’, say ‘F’. So we would write 2) as follows:

7) Fw

With the negation being:

8) ~(Fw)

Now, to return to Slick’s first premise, the negation does not prefix a proposition, but rather just a term in a proposition. It says that ‘not-god’ accounts for logic. But, as we have just seen, negation prefixes propositions not names. It is as if Slick’s premise would be written in first-order logic as

9) Ag or A~(g)

(where ‘g’ is ‘God’ and ‘A’ is ‘…accounts for logic’).

But because the negation is prefixing not the proposition ‘Ag’ but the name ‘g’ inside the proposition, it makes no sense. It is not a well-formed formula, and so cannot be given a truth-value. It is like the way ‘President first the was America Washington’ is just nonsense, and so neither true nor false. So if we take Slick literally, and phrase the argument exactly as he does, then the first premise isn’t really a premise at all, but a meaningless string of words.

If I said ‘either Bob broke into my house, or not-Bob broke into my house’, you would think I had difficulty talking properly. ‘Not-Bob’ isn’t a person, and obviously he didn’t break into my house. Phrasing it as not-Bob is literally meaningless.

To make it a well-formed formula, the closest thing would be:

10) Ag or ~(Ag)

But now we have a dichotomy as the first premise, and if we use disjunctive syllogism we are going to be inevitably back to triviality (as I literally proved in my original post). Let’s quickly give the argument both ways just in case anyone is still unsure how it goes:

Pr1. Ag or ~(Ag)

Pr2. ~(Ag)                  (i.e. negating the first option)

Con. ~(Ag)                  (i.e. concluding the second option)

 

Pr1. Ag or ~(Ag)

Pr2. ~~(Ag)                  (i.e. negating the second option)

Con. Ag                        (i.e. concluding the first option)

So Slick doesn’t want to repair his train wreck of a sentence, 1), into 10), because it is check-mate for the argument if he does that. No debate. Game over.

So it looks like the choice is between a meaningless first premise (i.e. 9) and a trivial argument (i.e. if we use 10). Well, we can read 1) a little differently, a little more charitably. There is another reading of 1) which is not meaningless. So go back to the example of me saying the following:

11) Either Bob broke into my house, or not-Bob broke into my house.

Instead of reading this as ‘Either Bob broke into my house, or it is not the case that he broke into my house (which would make the subsequent argument trivial again), we could read it as follows:

12) Either Bob broke into my house, or someone else broke into my house.

Now, we can express this perfectly well in first order logic, using quantifiers. These are devices which use variables (rather than names). So one quantifier is called the ‘existential’ quantifier, ‘∃’. To say ‘something is red’, we would use the variable ‘x’ and the predicate ‘R’ for ‘…is red’ and the existential quantifier as follows:

13) ∃x(Rx)

This says ‘There is a thing x such that x is red’, or more colloquially ‘something is red’. So when someone says 12, the implicit assumption is that someone broke into the house, and either it was Bob, or it wasn’t Bob. We can express this as follows:

14) ∃x(Bx) and ((x = b) or ~(x = b))

It says ‘there is a thing x such that x broke into my house, and that thing x is either identical to Bob, or it is not identical to Bob’. More colloquially, ‘either Bob broke into my house or someone else did’. Stating it this way excludes the idea that nobody broke into the house, and presumably you would only say 12) if you knew that someone had broken in.

So we could read Slick’s first premise more charitably along those lines, and build in explicitly the claim that something accounts for logic to the premise, and than say that either that thing is identical to god or it is not identical to god, as follows:

15) ∃x(Ax) and ((x = g) or ~(x = g))

This says ‘there is something that accounts for logic, and that thing is either identical to god, or it is not identical to god’. More colloquially, ‘either  god accounts for logic, or something else does’.

So, it looks like we have made some progress towards finding a more charitable way to cash out the logical form of the first premise. 15) is well-formed, so not meaningless, and it doesn’t lead to triviality the same way as 10) did. So, is this the desired destination for Slick’s argument form? I say no. Here’s why.

There is good reason for thinking that nothing accounts for logic, which would make 15), though elegantly formed, false. Here is Aristotle, in the Metaphysics (book IV, section 4) discussing whether the law of non-contradiction can be demonstrated:

“But we have now posited that it is impossible for anything at the same time to be and not to be, and by this means have shown that this is the most indisputable of all principles.-Some indeed demand that even this shall be demonstrated, but this they do through want of education, for not to know of what things one should demand demonstration, and of what one should not, argues want of education. For it is impossible that there should be demonstration of absolutely everything (there would be an infinite regress, so that there would still be no demonstration); but if there are things of which one should not demand demonstration, these persons could not say what principle they maintain to be more self-evident than the present one.”

This much debated passage seems to be suggesting that non-contradiction cannot be demonstrated from some other foundation, because it is the foundation for demonstration itself. Some things, he suggests, must be the end of demonstration and explanation, lest there be an infinite regress of explanation. If so, then it seems that we may have some reason to suppose that no ‘account’ of this principle of logic can be given. Here is another philosopher, David Lewis, making a similar point:

“Maybe some truths just do have true negations [i.e. maybe non-contradiction doesn’t hold].  … The reason we should reject this proposal is simple. No truth does have, and no truth could have, a true negation. Nothing is, and nothing could be, literally both true and false. This we know for certain, and a priori, and without any exception for especially perplexing subject matters … That may seem dogmatic. And it is: I am affirming the very thesis that Routley and Priest [i.e. philosophers who deny non-contradiction] have called into question and-contrary to the rules of debate-I decline to defend it. Further, I concede that it is indefensible against their challenge. They have called so much into question that I have no foothold on undisputed ground. So much the worse for the demand that philosophers always must be ready to defend their theses under the rules of debate.” (Lewis, Logic for Equivocators, (1998), p 434 – 435).

Lewis, probably the most influential analytic philosopher of the late 20th Century, and no stranger to defending controversial theses adeptly, simply offers no argument in support of non-contradiction. He seems to be implying that the very call to account for it is impossible to answer.

Now, obviously, Aristotle and Lewis can be wrong. I disagree with both about different things (future contingents and realism about possible worlds, respectively), so just citing them as authorities is not a way of establishing the thesis they argue for. However, what this does is highlight the difficulties associated with establishing 15), as it requires explicitly what Aristotle and David Lewis are very insistent cannot be granted; a reason for thinking that non-contradiction holds, or an ‘account’ of non-contradiction.

So this does not say that 15) is false. But it does show that it would be almost impossible to establish it. Matt Slick, an admittedly learned theologian, who has had no training in philosophy or logic, would have to solve a puzzle that has literally been too difficult for the greatest philosophers and logicians in history to solve: how to justify non-contradiction.

With these considerations in mind, we can see how Herculean the task would be to justify the premise. Possibly something accounts for logic, but how do you show that? How do you show that it is not just a brute given foundation?

One thing is clear: Slick’s original way of pumping up the intuition that 1) is true is to cite the fact that either god exists or it is not the case that he exists. But this dichotomy is not the same premise, and could be true even when 15) is false. So it is no help. The fallacy of begging the question, that I accused him of before, was not just that he gave a premise that was a potentially dubitable disjunction instead of a dichotomy; it was that he offered the dichotomy as justification for the premise. That is the essence of the false dichotomy, and now it is clear what the task is for justifying 15), it is obvious that it will not work again.

There is nowhere for this argument to go. It is over, even if they claim that it isn’t. Even if they claim that I was making a point about ‘wording’, or that I was drunk (which I wasn’t), or any other ad hominem. The task is too great to be overcome by Slick, and if it is too difficult for Aristotle or David Lewis, I am not holding my breath that anyone will be able to justify 15) either.

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