Logic and its Limits
[Many of the arguments in this discussion were drawn directly from the Retrieving Reason Podcast by Kelly Fitzsimmons-Burton. Links to her series on reason will be included with this content]
Logic, reason and rationality play a fundamental role in allowing us to know what is true and understand the world. I will also argue that it holds a primary position in allowing us to make the right decisions and actions as well. However, from the outset it can be difficult to know what logic is, let alone what its limits might be. That is what I aim to do here. First, I aim to unpack what logic is, how it relates to reason and what role it plays in our ability to know the truth. After this, I will consider how logic is limited. Namely, I will explain how the adoption of logic begs the question as well as how logic cannot demonstrate its own value.
The Nature of Logic
We commonly site logic and its importance whenever we are arguing whether this be in philosophy or in common disagreement; we intuit logic and its value. Yet, if we were ever stopped and asked, we might find it strange and difficult to explain what logic actually is because this is not commonly known.
The easiest way to understand logic is as a rule: it is the rule that contradictions cannot be true. Therefore, when we spot a contradiction in an argument, we can say that this rule has been broken and the argument is illogical. For example, we might accept the claim that there are pandas in China, but if one of then makes the claim that there are no pandas in China at all, that person’s argument is now illogical because both of those claims cannot be true at the same time; there are either pandas in China, or there aren’t. Therefore, one of those claims has to go in order for the argument to retain its logic or, at least has to be qualified in some way.
Once logic is understood as this rule of consistency, some confusion or, at least, some misunderstanding about its relevance and application can arise. I will claim that adherence to logic as a rule for understanding the truth is essential, but a retort to this claim would be that logic is in some sense optional if we want to make any meaningful sense of things and a firm adherence to logic is either too cold or too closed off to be a rule that ought not ever be broken. I will attempt to gradually iron-out these criticisms.
To begin with, let’s expand the definition of logic by considering the laws of thought. Each one of these laws is a self-evident truth taken to be assumed when we employ logic as a rule. They are so called “Laws of Thought” because it is argued that one cannot even think coherently if one of these laws are broken. And so, what are these laws? There are 3 of them and they are:
1. The Law of Identity,
2. The Law of Non-Contradiction, and
3. The Law of the Excluded Middle.
These might sound technical, but in fact if there is anything that makes logic difficult to understand it might be just how obvious its principles are, much like explaining water to a fish. Keeping this in mind, let’s unpack these laws.
1. The Law of Identity:
A=A
A thing is what it is,
Truth is truth, a tree is a tree and a cup of tea is a cup of tea.
This might be the most obvious of the three laws and perhaps the most difficult to clarify, for, what more is there to say? The general idea is that if things are not what they are; if, say, truth is not truth, things begin to fall apart.
2. The Law of Non-Contradiction:
A ≠ Non-A
A thing cannot be itself and not itself at the same time and in the same respect.
If an object is blue all-over, it cannot be red all-over unless this is at a different time or we mean something different by our terms.
This law probably requires the most qualification of the three, but it remains relatively simple. Two opposing claims cannot be true at the same time. It is the “same-time-and-same-respect” clause that requires a bit more work. For example, we might posit that a piece of art is beautiful in one sense and not beautiful in another… doesn’t this break the law? No. It would only break the law if we were to posit that the artwork is at once beautiful and not beautiful in the same sense; only then do we have a contradiction. Or, we could say that we perceived the artwork to be beautiful on the Monday and not on the Tuesday without contradiction. It is only when we claim that the artwork was beautiful and not beautiful at the same time that our claim becomes contradictory and begins to stop making sense.
3. The Law of the Excluded Middle
A is either A or Non-A
A thing is either itself or not itself. It cannot be both.
Either there are pandas in China or there aren’t.
Again, this is fairly straight forward. Simply put, there is no mid-point between a truth and its opposite; things are either true, or they aren’t. Either the artwork is beautiful, or it isn’t. Now, of course, you might protest that the artwork may be only slightly beautiful… doesn’t this break the law? No, because the artwork is either slightly beautiful, or it isn’t. Therefore, it could be that any number of things can be claimed to be true. It just so happens that whatever that claim happens to be, the claim either is or isn’t true, and is not both at once.
In fact, this brings me to something that I would like to touch upon which is that the laws of thought don’t reduce our thinking to binary thinking: it doesn’t simply dictate that things must be black and white. Truth comes in shades of grey, and logic allows for this. We saw this briefly with the example above, but to go further, let’s consider why the laws state that A ≠ Non-A, instead of stating that A ≠ B. I am conscious of how using these terms can make the discussion seem too abstract and so, allow me to re-frame them. Saying that A ≠ Non-A runs parallel to claiming that a tree is not a non-tree. On the other hand, saying that A ≠ B is parallel to claiming that a tree is not, for example, a plant. Now, of course, the distinction is important because a tree can be a tree and a plant at the same time. Therefore, of A is a tree and B is a plant, A can equal B because a tree can be a plant without contradiction. However, notice what the law states: that A ≠ Non-A… this is very different because if A is a tree, it can also be a plant but it cannot be not a tree (if it is, in fact, a tree). This is why the laws do not reduce things to simple binary truths: they simply hold that things are what they are. Being what they are, they may be many other things at the same time – reality tends to be complex like that. Nevertheless, in so far as things are themselves, they cannot be not themselves also. Or, another way it was put by Kelly Fitzsimmons Burton is that we might say that things be black, white or grey and when we state with logic that things are either black, or they’re not, we’re claiming that they’re white. But that’s not what logic holds. If things are not black, they could still be every gradation of grey up to white, but it is still the case that things are either black, or they’re not.
Hopefully, this has clarified why an adherence to logic does not force binary thinking and hopefully we now have an outline of the fundamentals of logic. However, before we move on to its problems, there are still a number of things to unpack about it.
First, a side-note about logic’s relation to reason… Now, reason is another term that can be used in a number of ways but as it relates to logic here, we are considering reason as a faculty: as in, we have the faculty of reason, and this allows us to do reasoning. The faculty of reason is simply the faculty that allows us to think logically. Therefore, when we say that we have the faculty of reason, we say that we have the ability to think logically, and when we say that we are reasoning, we say that we are putting this faculty to use and are actively using logic.
The second thing to note, and something that carries over from unpacking the laws of thought is why the laws cannot be broken if we want to think in any kind of meaningful or coherent way. In fact, this might be the most important thing about logic. The best way to demonstrate why adherence to the laws is so important might be by considering what breaking the laws entails.
Let’s take the first law: A=A; a thing is itself…
To break this law would be to hold to the contrary and to claim that a thing is not itself. Could we make sense of this? For example, could we understand the claim that “this tree is not a tree.”? not the claim that “this tree is a tree in one sense yet not in another.”, but the claim that “in the sense that this tree is a tree, it is also not a tree.”? Let’s take this one step further. Let’s say that you ask someone for some tea and the ingredients are hot water, milk and sugar. Recognise that in common use this assumes that all of the ingredients are what they are: that water is water, that milk is milk, and so on… but now you are functioning as if the law of identity doesn’t apply and the person comes back to you with a drink made of soot, mud an gravy. Imagine trying to argue that you asked for tea and not soot without invoking the first law: your friend could argue that tea is not tea but is, in fact anything other than tea. Therefore, your friend brought you soot. What’s worse is that if things aren’t what they are, even soot isn’t soot because then it would still be what it is, and so, you can’t ask for soot expecting tea because nothing stops your friend from coming back with a mug full of petrol. If what I’m describing to you sounds insane, that’s because it is (perhaps even technically so)… can you imagine a sate in which up is not up, but left or right or backwards, or where words are not words or sound is not sound but each thing is always something completely different and even different from the thing that it has apparently changed to? Now, could you even imagine it, let alone try to make a coherent argument or go about your day…
Hopefully this has demonstrated why the first law cannot be broken, but there is a more concise way to do this: if things are not what they are, we could say that “truth is not truth.” and then we could ask “is this true?” if we say yes, truth is not truth and therefore the claim itself cannot be true. Therefore, the alternative that we are left with is to say that the claim is wrong, and that truth is truth and, therefore, that the law must be followed.
That’s just for the first law, and I don’t want to labour the point for too long but if you want to understand why the other two laws cannot be broken, try to consider similar questions when you imagine an attempt to break laws two and three.
In any case, we must follow the laws, or else things stop making sense. Therefore, reason; logic and our adherence to it are not relegated to technical study but are, in fact, relevant to all of our thinking if anything is to make any sense. This is of course not to posit that our thinking will be logical simply because we possess reason, nor that we will always be conscious of it, but in so far as our thoughts and arguments can make sense, they need to be logical and, in so far as they are illogical, they don’t.
Now then, that should suffice for a fair introduction to logic: I have outlined what it is, the laws it is based upon, its relation to reason, and why the laws cannot be broken. Now we may turn to consider the limits of Logic.
The Limits of Logic
As it happens, despite how fundamental and even necessary logic is, it cannot be the beginning or the end of our truth-seeking efforts. Despite the fact that the laws of logic cannot be broken, I will argue that there are at least two important criticisms that can be levelled against it: the first is that logic, by its own nature, begs the questions; and the second is that logic cannot demonstrate its own value.
Problem 1: Begging the Question
The outline of this problem is that the laws of thought are assumed justifications for the laws of thought, ultimately, rendering them circular. Therefore, the laws of thought and – by extension – logic beg the question: why is logic initially justified?
To clarify this problem, let’s return to the example of why the first law cannot be broken. The argument goes that, if the first law were to be broken, things are not themselves and therefore, truth is not truth. Yet, this argument breaks down because if truth is not truth this very claim cannot be true… However, notice what principles we have to invoke in order to make this argument…
To clarify the problem further still, let’s return to the first law: A thing is what it is; another iteration of which is A=A and yet another iteration of which is that “truth is truth”, which helps with our current purposes. Notice what is at work here: when we judge that the claim “truth is truth” is, in fact, true, we are already working under the assumption that this claim has the status of being either true or false (which is the third law). Moreover, we are functioning under the assumption that the claim cannot be true and false at once (which is the second law) and, of course, the first law is the claim itself. Therefore, to justify the first law, we are already assuming the applicability of all three laws at once. Indeed, the same would be true for the other two laws. For example, to assert that “truth cannot be non-truth at the same time in the same respect” (the second law) also assumes that this claim will either be true or that it won’t be (the third law) and assumes that truth is truth (the first law). Finally, to claim that “a thing is either true or it is not” (the third law) also assumes that it cannot be both at once (the second law) and that truth is truth (the first law).
This demonstrates two things. The first is that it shows how – in a way that jams the mental gears – all three laws essentially say the same thing and are three sides of the same coin (although, this isn’t necessary to unpack here).
The second thing that it demonstrates is that the laws of thought assume the laws of thought and, therefore, logic assumes logic. In this regard, logic is internally self-re-enforcing once it gets going, but we may still ask what justifies it in the first instance prior to it getting off of the ground. Nevertheless, to posit that logic may not ultimately justify logic does not take from how fundamental it is: it is still the case that things don’t make any sense without it and, in fact, this is a point I will hone in on when developing the rest of my overall argument.
Problem 2: Logic does not demonstrate its own value
This problem is pretty straight-forward: could you use logic alone to prove that logic is important? Imagine trying to appeal to a friend, or a stranger on the street, attempting to explain to them why logic is so important… could you use a purely logical argument to do so? For example, you might insist that A=A, or that A ≠ Non-A (but may equal B), and they might ask you “so what?” Of course, I have argued that we cannot make sense of things without logic, but could we use logic or reason alone to demonstrate that making sense of things is valuable or worth doing? Indeed, if that man on the street has been living – apparently – perfectly well so far, without knowing about logic, why would it be any concern to him now based upon the dictates of logic alone? That is, of course, unless that man simply happens to have a raw interest in the matter.
This problem reveals something else about logic, which is that, unless you count the laws of though themselves, logic has no content. Even an apparently pure logical statement such as 2+2=4 uses symbols and concepts of quantity in order to function. Again, logic is simply the mechanism by which we track that “2 is 2”, which allows the claim to make sense.
Having no content, logic also lacks value. Now, it’s important to be careful here because the use of logic does have value content. That’s because the moment someone uses logic, they are at least implying that logic has value enough to be used. Yet, this would only demonstrate that logic has value once it’s in use, but this is not that same as logic proving its own value prior to being used. In light of this, the only other alternative appears to be to posit that logic must always have value because we can never truly think illogically or ever be illogical. However, at this point I would first argue whether this allows logic to prove its own value, or puts that onus on something else, and following on from this, I would proceed to point out that it is exactly upon this point that we may see the collision between the centrality of logic in our truth-seeking efforts and the argument that use Is fundamental to knowing what is true.