Wednesday, 1 July 2015

Does Logic Have its Own Subject Matter?

What is the subject matter of logic? This, of course, has been answered in many ways by many different philosophers. For example:

i) Logic gives us the structure of thought itself.

ii) Logic provides the foundations of language.

iii) Logic gives us the structure of the world in that, for example, it deals with its possibilities and impossibilities.

iv) Logic's essential subject matter is propositions, entailment, implication, inference, validity and consistency.

v) Logic doesn't have a subject matter (as such).

It's not the case, of course, that any of these definitions necessarily contradict the others. Indeed perhaps logic is accounted for by the sum of these seemingly distinct definitions.

  1. The Structure of Thought

It should be said 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 don't 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. In terms of thought, we can say that we can either think that p is the case or p isn't the case. Take Wittgenstein’s well-known expression of this 'tautology':

It will either rain this afternoon or it won't rain this afternoon.”

Perhaps this is a bad example because Wittgenstein himself said that the above “tells us nothing”. However, it's 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 can't 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:

We cannot think, or believe, both that it will and will not rain this afternoon.

Though we can say of the world:

It can't 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 can't also be painted red all over. That is a de re statement about the world (to use 20th-century jargon). We can also offer de dicto version. We can't 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.

For example, if one thought weren't equal to itself (if not numerically so), then thought itself would cease. Perhaps more relevantly. If we didn't 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; though at one point people didn't know that the evening star is the same as the morning star. 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.

  1. The Structure of Language

Despite what we said earlier about the mutual relatedness of thought and language (or “thought and talk” in Donald Davidson’s words), it can be said that many philosophers have believed that language has it own “logical grammar”. More relevantly, they've also said that such grammar is often hidden by the “ordinary grammar” of natural languages.

Both Ludwig Wittgenstein and Bertrand 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 uncovered the logical grammar underneath (if that’s the correct word) statements such as “The king of France is bald”.

To repeat: we're asking what is the subject matter of logic. In this case it's the logical grammar of an everyday expression.

Russell argued that the English sentence (or statement) “The king of France is bald” 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”), there is an implicit commitment to the existence of the king of France. The king of France didn't in fact exist. Someone who doesn't exist can't be either bald or not bald. So, to cut Russell’s long and complex story short, this expression is effectively “meaningless” because of the description's “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've also predicated baldness of this person (or non-person). Perhaps all I needed was a supply of quotation marks to escape from what Quine called “Plato’s beard”.

In retrospect, it can be said that rather than the logical grammar of that expression being hidden beneath its everyday grammar, its grammar (both logical and everyday) is explicit and therefore on the surface; which is more or less what the 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 or ambiguities (or so Russell thought at that time).

Similarly, the logical positivists’ “verification principle” told us that certain “metaphysical” expressions are meaningless either because they can't 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 is the logical and philosophical mix-ups of metaphysical and ordinary-language expressions. Only the expressions of science, maths and logic are free, so the logical positivists believed, from such sins.

  1. Formal Logic

More formally and technically, it can be said that inference, consequence, validity, entailment, etc. 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's still nevertheless the case that all language and thought mainly abides by the principles laid down by formal logic. And even if this isn't always the case, when thought and language don't do so, this effectively renders them 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 when they do so.

Of course we now come across the well-known problem of which way does the logical arrow point. Does it point from language-users to logic or from logic to language-users? 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 naturally reason and use logic?

I suspect that the arrow points in both directions. After all, many people can improve their logical skills by studying logic in some form or even acquire brand new logical techniques. Similarly, it's hard to believe that the majority of people reason correctly by sitting down and thinking about the laws 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 (or at least scientists) in their everyday lives. C. S. Peirce believed that it is scientists and the methods of science that should be the true subject matter of logic. And Dewey called such bricks-and-mortar logic “everyday inference”.

So even if formal logic needn't (or doesn't) study everyday reasoning, we can safely say that everyday reasoning rarely breaks the rules of formal logic. This fact isn't unlike Donald 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, then the “radical interpretation” of other communities (or even aliens from outer space) would be rendered impossible. Either that or we'd need to assume that such communities were collectively insane.

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