What is the subject matter of logic? This, of course, has been answered in many ways by many different philosophers. Some have said that it is the structure of thought itself. Others, that it provides the foundations for language. That it gives us the structure of the world in that it deals with its possibilities and impossibilities. Propositions have also been said to be at the heart of logic. But more formally and technically, we can say that entailment, implication, inference, validity, consistency and all the rest are the subject matter of logic. It is not the case, of course, that any of these definitions necessarily contradict any of the others. Indeed, perhaps logic is accounted for by the sum of these seemingly distinct definitions. However, if one discounts abstract propositions, as some philosophers do, then perhaps this particular emphasis may be at odds with other accounts. That is just one example of a possible conflict between the previous definitions.
i)The Structure of Thought
To begin, we should say that many philosophers would say that the structure of thought mirrors the structure of language. Many others reverse this duality by saying that the structure of language mirrors the structure of thought. Whatever the answer is, these statements do not in themselves tell us what is logical about thought and language. So let us take thought firstly.
Aristotle famously offered us what he called the ‘laws of thought’. These laws are captured in terms of basic Aristotelian logical terms. Take the ‘law of excluded middle’. That is, p or not-p or A or not-A. In terms of thought, we can say that we can either think that p is the case or p is not the case. Take Wittgenstein’s well-known expression of this:
It will either rain this afternoon or it will not rain this afternoon.
Perhaps this is a bad example because Wittgenstein himself said that the above ‘tells us nothing’. However, it is still the case that we must think that either p will be the case or not-p will be the case. Aristotle might have said that this shows us that we cannot think contradictories. Thus thought is structured around the basic and fundamental fact that we cannot think both p and not-p, at least not at the same time.
Another interesting point to make about Aristotelian laws of thought is that they also apply to the world. More than that, they apply to everything. So instead of
p or not-p
we can have:
A or not-A
Here the ‘A’ above can be about any situation in the world. Take the rain example again. We can at one time say that
We cannot think, or believe, both that it will and will not rain this afternoon.
But we can say of the world:
It cannot both rain and not rain at one and the same time.
We can say here, then, that either the structure of thought mirrors the structure of language or that the structure of language mirrors the structure of thought. We can also say that both thought and language abide by the structure of the world in that neither thought nor language can break, as it were, the world’s own logical rules. So if a ball is painted blue all over it cannot also be painted red all over. That is a de re statement about the world, to use 20th century jargon. We can also become all de dicto. We cannot think
That ball is red all over.
at the same time as thinking
That [same] ball is blue all over.
The question many philosophers have asked, vis-à-vis thought and language, is whether or not our thoughts about the ball are determined by the logical structure of our language. Similarly, do both language and thought somehow shape how we perceive and understand the world or does the world somehow impose its structures on both thought and language?
Let’s take another Aristotelian law of thought: the law of identity. This has something of the blindingly obvious about it. That is, A = A. However, this law lies at the very heart of all thought, language and, well, everything. Therefore it deserves to be notated logically, as Aristotle did, and commented upon. If one thought were not equal to itself, if not numerically so, thought itself would cease. Perhaps more relevantly. If we did not know that one thought were identical to itself or to another, this would certainly be the case. In ontological terms, we need to recognise the identity of a thought-about-an-object over time, which itself provides us with the very basis of thought itself. Similarly, A may well equal A, but at one point people did not know that the evening star is the same as the morning star (or that the British Prime minister is the same person as Gordon Brown?). More relevantly, the names ‘the morning star’ and ‘the evening star’ have the same reference, to use Frege’s term. That self-identical reference being the planet Venus. Without the stability of the law of identity, thought would be rendered incoherent, if not void.
ii)The Structure of Language
Despite what we said earlier about the mutual relatedness of though and language, or ‘thought and talk’ in Donald Davidson’s words, we can say that many philosophers have believed that language has it own ‘logical grammar’. More relevantly, they have also said that such grammar is often hidden by the ‘ordinary grammar’ of natural language. Both Wittgenstein and Russell, in their early years, believed that everyday expressions of a particular type hide their logical grammar. Take Russell’s ‘theory of descriptions’ in which he uncovers the logical grammar underneath, if that’s the correct word, phrases such as ‘The king of France is bald’. To repeat. We are asking what is the subject matter of logic. In this case it is the logical grammar of an everyday expression. Russell argued that this English sentence or statement hides it true logical grammar. By careful analysis he showed that because of the use of the definite article, ‘The’ in ‘The king of France is bald’, this is an implicit commitment to the existence of the king of France. The king of France does not in fact exist. Someone who does not exist cannot be either bald or not bald. So, to cut Russell’s long and complex story short, this expression is effectively ‘meaningless’ because of the descriptions ‘empty’ definite description. Other philosophers have said that the expression’s logical grammar renders it simply false (for more or less the same reasons). And yet I myself have used the description ‘The king of France’ in my own sentences. I also predicated baldness of this person (or non-person). Perhaps all I needed was a supply of quotation marks to escape from what Quine calls ‘Plato’s beard’. Anyway. In retrospect we can say that rather than the logical grammar of that expression being hidden beneath its everyday grammar, its grammar, both logical and everyday, is explicit on the surface, which is more or less what late Wittgenstein argued. So instead of offering an analysis, what Russell actually did was offer us a new version of the given expression. More strongly, it can be said that he offered us an alternative. That alternative, however, is free from logical and ontological complications and ambiguities.
Similarly, the logical positivists’ ‘verification principle’ tells us that certain ‘metaphysical’ expressions are meaningless either because they cannot be verified or because they have no observational or experiential consequences. The logic of such expressions, therefore, rendered them ‘meaningless’ – yes, that word again! In this case, then, the subject matter of logic was the logical and philosophical mix-ups of metaphysical and ordinary language expressions. Only the expressions of science, maths and logic do not commit such sins, as it is expressed in Wittgenstein’s Tractatus.
iii)Formal Logic
More formally and technically, as we have already said, we can say that inference, consequence, validity, entailment and the rest are the true subject matter of logic. Having said that, as the philosophers and logicians have told us, even if formal logic does indeed give of the pure and unadulterated subject matter of logic proper, it is still nevertheless the case that all language and thought abides by the principles laid down by formal logic. And even if this is not always the case, when thought and language do not do so, this effectively renders such things incoherent, meaningless or useless. In other words, everyone in their everyday talk infers things, they deduce logical consequences, and they believe in validity and consistency. They just don’t use these technical terms. Of course we now come across the well-known problem of which way does the logical arrow point, as it were. Does it point from language-users to logic or from logic to language-users? That is, do people learn to reason correctly by a self-conscious use of pre-existing logical laws and rules? Or does logic simply codify the way people reason and naturally use logic? I suspect that the arrow points in both directions. After all, many people can indeed improve their logical skills by studying logic in some form or even acquire brand new logical techniques. Similarly, it is hard to believe that the majority of people reason correctly by sitting down and thinking about the laws and truths of logic. However, pragmatists like Dewey and C. S. Peirce believed that logicians and philosophers should codify and notate logical principles and rules simply by studying the way people reason in their everyday lives. C. S. Peirce believed it is scientists and the methods of science that should be the true subject matter of logic. Dewey called such bricks-and-mortar logic ‘everyday inference’. So even if formal logic need not, or does not, study everyday reasoning, we can safely say that everyday reasoning rarely breaks the rules and truths of formal logic. This is not unlike Davidson’s argument that we must assume that the beliefs of the pygmies and our own beliefs have been, and are, largely true. Similarly, we must also assume mass rationality rather than mass irrationality. If the latter were true, the ‘radical interpretation’ of other communities, or even aliens from outer space, would be rendered impossible. Either that or we would need to assume that such communities were collectively insane.
