Accounting for logic – again

0. Introduction

In this post I will be looking at a blog entry on the BibleThumpingWingnut website, entitled ‘Christianity and Logic’. The entry is written by Tim Shaughnessy, and takes a Clarkian angle. Shaughnessy’s argument is basically that Christianity can provide an ‘epistemological foundation’ for logic, using Scripture as a sort of axiomatic basis for logic, and that ‘the unbeliever’ cannot provide such a foundation, or ‘account’, for logic. If this is the first time you are encountering this Clarkian view, have a look at this article by Clark. I have written on this topic before, and I think that many of those points are directly relevant here.

For instance, here I argue that there is no binary choice between Christianity and non-Christianity; there are different versions of Christianity, different monotheistic religions, different versions of theism, and different versions of atheism. This version of Christianity is just one tiny dot on a huge intellectual landscape. To argue by elimination that this version Christianity is correct, means you have to eliminate a possibly infinite variety of systems. Pitting (this version of) Christianity against ‘the unbelieving worldview’ is already to commit the fallacy of false dichotomy. We might want to call this version of it the ‘Bahnsen fallacy’, in honour of its main witness.

More specifically with regards to the broadly Clarkian idea of deriving logical principles from the Scriptures, I have argued here that this is incoherent. Derivation requires a logical framework, which is constituted in part by logical principles (or axioms); derivation is a logical notion, and thus presupposes logical principles.

There are some new points which seem to be worth raising however, given the particular presentation by Shaughnessy, and so I will be exploring those ideas here.

1. ‘What is logic?’ 

Shaughnessy’s view of logic seems to be entirely gained from the study of Clark, in that he is the only author cited (rather than, say, Aristotle or Frege) on the topic of what logic is. This is unfortunate, because it seems that  Shaughnessy is unaware of the controversy surrounding the topic. So, we see him state that logic is “the correct process of reasoning which is based on universally fixed rules of thought”. This idea, that logic is about laws of thought, is a historically significant idea, coming to prominence in the 18th and 19th centuries, but it has never been a universal consensus among logicians and philosophers. These days it is not widely represented among practising logicians and philosophers at all (see this for a quick overview). The reason for this is that in the contemporary setting logic has a much broader extension, and can cover systems which deviate wildly from how we might realistically model thought (which is the preserve of logicians and computer scientists working in artificial intelligence). Logic, thought of broadly as concerning valid inference for various types of argument forms, is not considered to be tied in any special manner to how we think. There may be a logic to how we think, but logic is not just how we think. Never-the-less, Shaughnessy makes no mention of this, and simply asserts that logic has this 18th century relation to cognition.

His out-of-date description of logic becomes confounded with outright misunderstandings when he spells out what he considers to be the three laws of thought. It is utterly standard, when going down this non-modern view, to list the three laws of thought as: ‘the law of identity’, ‘the law of non-contradiction’ and ‘the law of excluded middle’. What is odd is the way these are cashed out by Shaughnessy. For instance, the law of non-contradiction is cashed out as “A is not non-A”, and the law of excluded middle is cashed out as “A is either B or non-B”. It seems to me that there is a failure of Shaughnessy to distinguish clearly between different aspects of vocabulary. There is a fundamental difference between logical vocabulary that refers to things directly (like ‘Alex’, ‘London’, ‘your favourite type of ice cream’, etc) and those which express facts (‘Alex is in London’, ‘vanilla is your favourite type of ice cream’, etc). The first are called ‘terms’, and the latter are called ‘propositions’. Propositions can be thought of as made up of terms standing in certain relations to one another. Crucially, propositions are given truth-values, true or false; terms are not. So, ‘Alex’ isn’t true or false; but ‘Alex is in London’ is either true or false. In Shaughnessy’s expression of the law of non-contradiction, we have a letter ‘A’, which seems to be a term, as it is something we are predicating something to, but then the predicate we are ascribing to it is that it is “not non-A”. The problem is that we have a negation fixing to a term, ‘non-A’. As I have pointed out before, negation is a propositional operator, and its function is to switch the truth-value of the proposition is prefixes from true to false (or vice versa). If we prefix it to a referring term, like ‘A’, then (because terms don’t have truth values), the resultant operation is undefined.

The conventional way to express the law of non-contradiction is with a propositional variable, ‘p’, which ranges over all propositions, as follows:

¬(p ∧ ¬p)    (‘it is not the case that both p and not-p’)

If you want to express this using propositions where the relation of terms is explicit (i.e. in a first-order manner), then it would be as follows, where ‘Px’ is a predicate and ‘a’ is a term:

¬(Pa ∧ ¬(Pa))   (‘it is not the case that a both is and is not P’)

The same problem infects “A is either B or non-B”. The correct way to express this is just that for every proposition, either it is true, or it’s negation is true:

p ∨ ¬p     (‘either p or not-p‘)

It is bizarre to say that either ‘A is B or non-B’. There is no predicate ‘non-B’; rather, either B applies or it doesn’t. Take the proposition that I am 6 feet tall. Either I am 6′ or I am not. In the second case I don’t have a property, called non-6′. What would this property be? Every height other than 6′? I am not 6′, but I am also not every height other than 6′. I just am 5’11”. So the way Shaughnessy expresses excluded middle is also confused.

And it’s not like stating non-contradiction and excluded middle is extremely complicated; all it involves is: ‘p or not-p’, and ‘not both p and not-p’. He hasn’t simplified them for a non-specialist audience – he has just misrepresented them.

So we have an out-of-date view of logic, coupled with a technically incorrect presentation of the principles under discussion. It’s not a great start to an article about the nature of logic.

1.1 Logic in the Bible?

Perhaps Shaughnessy’s misrepresentation of the basic laws of thought is more understandable when we see where he is going with all of this. The ultimate point he will be driving at is that these laws are found in the Bible. Various snippets of the Bible are then presented as evidence of this, but because they don’t really fit that well with the laws when expressed properly, he has written them in such a way that the claim that they are found in the Bible becomes (slightly) easier to swallow. Here is what he has to say about it:

“The law of non-contradiction (A is not non–A) is an expression of the eternal character and nature of God, “for he cannot deny [contradict] himself” (2 Tim. 2:13). The law of identity (A is A) is expressed in God’s name, “I AM WHO I AM” (Exodus 3:14), and the law of the excluded middle (A is either B or non-B) is expressed in Christ’s own words, “He who is not with Me is against Me” (Luke 11:23).“

Let’s take these one at a time. It is hard to take them seriously, but I will try.

1.1.1 Non-Contradiction

In the book of Timothy, it is said that God cannot contradict himself. I say that this is completely irrelevant to the principle of non-contradiction. There is a difference between saying things, and things being true (or false). The law of non-contradiction is about the latter, not the former. It isn’t a rule which says ‘thou shalt not contradict thy self’. It says that there is no proposition for which both it and its negation are true. It doesn’t proscribe what you can or cannot say at all.

For example, I can contradict myself, and sometimes do. Does this mean I broke the law of non-contradiction when I did so? No, of course not. Imagine I say ‘It is sunny now, at 14:07’, and then a few minutes later, ‘It was not sunny then, at 14:07’. The two sentences I uttered were expressing (from different times) that it was and was not sunny at 14:07. Obviously, it would be a contradiction if both of these were true, as p and not-p would both be true (exactly what the law of non-contradiction forbids). But were they both true? That would mean that it was both sunny and not sunny at the same time. Conventionally thinking, this is impossible. Therefore, while I contradicted myself, I didn’t break the law of non-contradiction. I expressed a true proposition, and then when I uttered the negation of that proposition what I said was false (or vice versa). Contradicting yourself isn’t a case of breaking the law of non-contradiction.

Back to the Biblical example, God cannot contradict himself. So what? The law of non-contradiction is true even though people can contradict themselves. An example of a being, even an infinite one, who cannot contradict themselves, is not an example of the law of non-contradiction. To think that it is, is to mix up the idea of saying two contradictory things with two contradictory propositions both being true.

1.1.2 Identity

Shaughnessy does manage to state the law of identity correctly, which is that (for all referring terms) A = A. Everything is identical to itself. According to the example given, the law of identity is expressed in “I am who I am”, which is the answer God gives to Moses in the book of Exodus. It has always baffled me as to why this has been seen as a profound thing for God to say here. God tells Moses to go to the Pharaoh and bring the Israelites out of Egypt. Moses basically says, ‘who am I to do that?’ God says that he will be with Moses, but Moses wants a bit more reassurance for some reason:

Moses said to God, “Suppose I go to the Israelites and say to them, ‘The God of your fathers has sent me to you,’ and they ask me, ‘What is his name?’ Then what shall I tell them?”

God said to Moses, “I am who I am. This is what you are to say to the Israelites: ‘I am has sent me to you.’” (Exodus, 3: 13-14)

One of my favourite comedy series ‘Knowing Me, Knowing You’, staring Steve Coogan, features a pathetic TV chat show host, called Alan Partridge. In episode 2, he is interviewing an agony aunt called Dannielle, played by Minnie Driver, who is listing the things she likes in men:

Dannielle: Power is attractive. Sensitivity. Sense of humour. I like a man who knows who he is.

Alan: I’m Alan Partridge.

If you think that the law of identity is expressed by Exodus 3:14, then you should also hold that it is expressed in this little bit of Alan Partridge script.

I’m just going to leave that there.

1.1.3 Excluded Middle

In the last example, Jesus saying “He who is not with Me is against Me” is an example of someone expressing something stronger than the law of excluded middle. The logical law of excluded middle says that for every proposition, p, either it or its negation is true. There are two propositions being considered in the saying above, put together in the form of a disjunction. The two propositions are:

‘x is with Jesus’

‘x is against Jesus’

The combined disjunction is universal, in that it applies to everyone:

For all x: either x is with Jesus or x is against Jesus.

We could write this in first order logic as follows:

∀x (Wx ∨ Ax)

However, this isn’t a logical truth. There is no logical reason to stop someone being neither with nor against Jesus. The following is not a logical contradiction:

∃x (¬Wx ∧ ¬Ax)      (‘there is an x such that it is not with Jesus and it is not against Jesus’)

If Jesus had said ‘Either you are with me or not with me’, then he would have said something which would have been logically true (because of the law of excluded middle). It would have the following form:

∀x (Wx ∨ ¬Wx)

Therefore, when Jesus says that everyone is either with him or against him, something which goes beyond the law of excluded middle, and it is not a logical truth. Why this has been picked to be an instance of this law can only be put down to either the author not understanding what the law actually states, or being so determined to find something that fits the pattern that they wilfully ignore the fact that it doesn’t.

1.2 The problem

If we are thinking of the examples of someone not contradicting themselves, or of everyone being split into the ‘with’ or ‘against’ categories, then we have (at best) particular instantiations of these rules, but not examples of the rules. Consider the difference between:

a) A sign which said ‘do not step on the grass’.

b) Someone walking along the path next to the grass.

With regards to a), we would say that it had the rule, ‘do not step on the grass’, written on it. On the other hand, b) would just be an instance of the someone following the rule.

Finding Jesus saying ‘Either you are with me or you aren’t’ would be like finding someone walking next to the grass. Sure, it instantiates what the law of excluded middle is about, but it isn’t the rule. The rule is general. It says ‘nobody walk on the grass’, not just this guy in particular; excluded middle says ‘for all propositions, either p or not-p‘. The Bible nowhere makes generalised statements about language, reasoning or validity.

So the examples fail in that they aren’t actually instances of the rules (as the laws themselves are muddled by Shaughnessy), but they also fail because (even if we pretend that they do instantiate the rules) they aren’t examples of the rules. The Bible doesn’t have the law of excluded middle stated in it. It instantiates it, in that every proposition expressed in the Bible is either true or false, but that is not important at all. Every proposition expressed in any book is either true or false! Exactly the same goes for non-contradiction. There is nothing special about the Bible such that you can find the three rules of thought in it. If you want to see what a book looks like which explicitly has the rule of non-contradiction in it, read Aristotle’s Metaphysics, book IV, section 3:

“...the most certain principle of all is that regarding which it is impossible to be mistaken; for such a principle must be both the best known (for all men may be mistaken about things which they do not know), and non-hypothetical. For a principle which every one must have who understands anything that is, is not a hypothesis; and that which every one must know who knows anything, he must already have when he comes to a special study. Evidently then such a principle is the most certain of all; which principle this is, let us proceed to say. It is, the same attribute cannot at the same time belong and not belong to the same subject and in the same respect.“

For Aristotle, the basic declarative sentence (the basic proposition) is the ascription of an attribute (or property) to a subject, and this is explored explicitly by him at great length. So ‘Alex is happy’ is this type of sentence. When he says “the same attribute cannot at the same time belong and not belong to the same subject and in the same respect”, this is simply to say that there cannot be any proposition, such as ‘Alex is happy’, for which it is true that ‘Alex is happy’ and it is also true that ‘Alex is not happy’, i.e. we cannot have both p and not-p. In contrast to the Bible then, Aristotle does not just give an instance of a sentence of the same form as the law of non-contradiction, like ‘it is not that Alex is both happy and not happy’ – he reflects on this and states the general proposition in its generalised form. It is explicit. With the case of the Bible, we have shoddy eisegesis going on, where Aristotelian principles are being read into a text that doesn’t have them.

So far, not great. Shaughnessy makes the following claim:

“It is precisely because the laws of logic are embedded in Scripture that the Christian is able to establish from an epistemological standpoint that they are fixed and universal laws. Without this epistemological foundation, we cannot account for the laws of logic“

Well, given what I’ve written above, it should be pretty obvious that I disagree with that. The laws of logic are not in the Bible. Given this, by his own standards, Shaughnessy doesn’t have an ‘epistemological foundation’ and ‘cannot account for’ these laws. Too bad.

2. An epistemological foundation for logic

Shaughnessy then presents the standard presuppositional line, the one we all knew was coming, where they brag about how great their ‘account’ of logic is, and how rubbish ‘the other account’ is.

“The unbeliever cannot account for logic in his own worldview and therefore cannot account for his ability to think rationally. The challenge has been made many times to unbelievers to account for logic in their own worldview and it has always fallen short or gone unanswered. Never has an adequate response been given. In formal debates, the challenge is often ignored by the unbeliever, yet the challenge demands an answer because debates presuppose logic. The unbeliever is required to use logic in order to make his argument against Christianity consistent and intelligible, but only the Christian worldview can account for logic. He is therefore required to rob the Christian worldview in order to make his argument against Christianity intelligible.”

Ok, well we’ve all seen this over and over again. So I am going to meet the challenge head on, and provide a few different ‘accounts’ of logic, which could be ‘epistemological foundations’ for it.

First of all, what do we mean by and ‘epistemological foundation’ for something? Well, I take it to mean something in virtue of which we can come to know something. So, an epistemological foundation for x could be thought of an an answer to the question, ‘how is it that we are able to know about x?’

Given that, our question is: ‘How is it that we are able to know about logic (and in particular those logical laws)?’. In order to play the game right, I shall not appeal to God in any way, I will just go along with the idea that logical laws are things that have some kind of ontology capable of allowing reference to them, and I will just pretend that the three principles cited by Shaughnessy (identity, non-contradiction and excluded middle) really are ‘logical laws’, even though it is a clumsy and out-dated way to talk about logic. I will play the game anyway, just to be a good sport.

2.1 They are self-evident.

Here is the first way of answering that question: we are able to know about logical laws because they are self-evident truths. This just means that to think about them is to know that they are true. They don’t need anything else to support my knowledge of them, because they are self-evident. This is a really simple answer, and there isn’t much more to be said about it.

The response might be something like: “that’s rationalism! You are saying that all knowledge is rationally determined based on self-evident truths, like Spinoza!” Before we get into the standard disputes about rationalism and empiricism, I want to point out that I don’t need to also say that this is how I get knowledge generally. The question is about logical laws only. Maybe these are the only self-evident truths, and I gain knowledge about other parts of the world through empirical access, or mystical intuition, or because a ghost illuminates the right answer for me. Who cares? The point is that this plainly is an answer to the question ‘how could we know about logical laws?’. It doesn’t require a God of any type, so is available to an atheist (or a theist, or really anyone apart from those people who for some reason are committed to the view that there are no such things as self evident truths). They are pretty good candidates for self-evident truths if you ask me, and I would dispute the claim that there are candidates that are more plausible (is ‘cogito ergo sum’ more plausible as a self-evident truth than non-contradiction? They seem even, if anything). If anything is self evident, its the law of non-contradiction. So this view is plausible, at least on first blush.

If there is a secret cheat-card answer to this that presuppositionalist apologists have, I’ve never heard it. Remember the challenge: “The challenge has been made many times to unbelievers to account for logic in their own worldview and it has always fallen short or gone unanswered.” Well, that’s one account. Here is another one.

2.2 They are synthetic a priori knowledge

Here is my second proposal: we are able to know about logical laws because they are synthetic a priori truths. In the Critique of Pure Reason, Immanuel Kant summarises his views on this type of knowledge as follows:

“…if we remove our own subject or even only the subjective constitution of the senses in general, then all constitution, all relations of objects in space and time, indeed space and time themselves would disappear, and as appearances they cannot exist in themselves, but only in us. What may be the case with objects in themselves and abstracted from all this receptivity of our sensibility remains entirely unknown to us. We are acquainted with nothing except our way of perceiving them, which is peculiar to us, and which therefore does not necessarily pertain to every being, though to be sure it pertains to every human being.”

Synthetic a priori knowledge has the property that it is integral to how we see the world. It is subjective, in the sense that Kant explains above (that is, if we were to remove the subject, then it would also disappear), but it is also universal, in the sense that it applies to “every human being”. So, space and time may be known a priori, yet the knowledge is not simply analytic (i.e. true in virtue of the meaning of the words used), but synthetic (true because of more than just the meaning of the words used). What we know is the form of our intuition, which is a non-trivial fact about the way things are, but is also directly available to us, as subjects, a priori. We are programmed to see the world in a spatio-temporal way.

Kant has his own ways of demonstrating that this is the case, using transcendental arguments which inspired Van Til and should be familiar to all presuppositionalist apologists. Essentially you show that the contrary leads to a contradiction. So we have to see the world in terms of space and time, because the contrary view (where we do not see the world in such a way) leads to complete incoherence. Space and time are necessary presuppositions of the intelligibility of experience (a phrase presuppositionalists love to use). As such, we have transcendental proofs for them. Presuppositionalists, like the gang at BibleThumpingWingnut.com, should welcome this methodology, as it is basically the sophisticated version of the Van Tillian method they endorse themselves, only directed squarely at epistemological issues.

I say that we just point the synthetic a priori machinery at the laws of logic, and there we go, an epistemological foundation for the laws of logic. We know excluded middle, non-contradiction and identity as forms of intuition. Everyone has them (which explains their apparent universal character). If we try to conceive the world without them, we get incoherence (which shows their necessity).

On this view, we are not suggesting that these principles have metaphysical necessity. As good Kantians, we simply say that we cannot know about the numenal realm. But this should be perfectly acceptable to those presuppositionalists who throw the gauntlet of providing an epistemological foundation for the laws of logic. They are the ones, after all, who think that these principles are the ‘laws of thought’. On this reading of what they are, the Kantian line seems perfectly suited.

It would be really hard to imagine a presuppositionalist mounting a successful attack against this view, which didn’t also backfire and undermine their own transcendental arguments. You can’t have it both ways. If you are going to use transcendental arguments for God, I’m going to use them for what I want as well.

2.3 They are indispensable

Here is one last attempt. How do we know about the laws of logic? Well, they are indispensable to our best theories of science, so it is reasonable to believe in them. This is a version of the Quine-Putnam indispensability argument for the existence of mathematical entities. Here is how I see the argument going:

1. We are justified to believe in all the entities that are indispensable to our best scientific theories.
2. Laws of logic are indispensable to our best scientific theories.
3. Therefore, we are justified to believe in the laws of logic.

I’m not personally that convinced by premise 2, but presumably Shaughnessy and all those who throw down the presup gauntlet are. Premise 1 says that we have justification to believe in those things which are indispensable to our best theories, and I think this is going to be accepted by most people. We believe in viruses because our best science tells us that they exist. It is reasonable to hold the belief in viruses on this basis.

This argument doesn’t say that we have conclusively established that the laws of logic exist, but it provides justification. Presuming a broadly fallibilist idea of justification (as most contemporary professional epistemologists do), then even though the indispensability argument doesn’t ensure the laws of logic exist, it provides sufficient support for the belief that they do to be justified. So it allows us to have justified belief in the laws of logic existing. If that belief is also true, then we know that they exist. Thus, this is an explanation of how we come to know (as in ‘justified true belief’) that the laws of logic exist. Thus, it is an answer to how we can have knowledge of them, and ultimately part of an epistemic foundation, and an ‘account’, of them.

3. Conclusion

So, above are three distinct views about the epistemological foundations of logic. None of them required God, or Jesus, or Reformed theology at all. No doubt, they will continue, over at BibleThumpingWingnut.com, to claim that “The challenge has been made many times to unbelievers to account for logic in their own worldview and it has always fallen short or gone unanswered. Never has an adequate response been given“. In reality though, for those of us who have spent a long time doing philosophy seriously, these claims are easily countered. I’m not saying I have all the answers; I’m saying that they don’t. I don’t know what the ‘right answer’ is about the nature of logic, or how epistemology and logic fit together. It is an incredibly complicated area. As with philosophy, it may be something we will ultimately never answer. It may be that for some reason the question itself doesn’t make sense, but that this realisation doesn’t come for many generations yet. Maybe the answer was given in some obscure scroll, now long forgotten by history. All these possibilities remain. But to claim that there is only one answer to this sort of question is silly. I have thought up the three examples here by referencing well-known ideas in philosophy. I could have easily plundered the great works of philosophy to find dozens more (such as platonism, structuralism, formalism, intuitionism, plenitudinous platonism, etc, etc). Don’t be fooled into thinking that in such a rich and complicated area of philosophy as this, that there are any easy answers.