Tuesday, March 6, 2012

What it means to say that an expression is a logical constant

To: Pearl K.
From: Geoffrey Klempner
Subject: What it means to say that an expression is a logical constant
Date: 4 April 2007 11:08

Dear Sachiko,

Thank you for your email of 28 March, with your University of London Logic essay in response to the question, 'What does it mean to say that an expression is a logical constant?'

I have to admit straight away that this is not a topic that I have ever found gripping, but some philosophical logicians absolutely love it. Looking at the Stanford Encyclopedia article (which coincidentally I read this morning before I looked at your essay and saw that it is your one reference) you can see why. If you are playing the analysis game this is a very rich topic indeed, with apparently no end in sight to all the variations and disagreements.

Why did you choose this? Is this a topic that grips you? Why? As a general rule, I would say that you have to go with the topics that grip you and avoid those that don't.

The first thing we need to establish is why anyone should care whether or not there is a definitive way to demarcate the logical constants. You are on the right track in talking about distinguishing the logical form of different statements. However, the point if this is that we are interested in distinguishing valid and invalid arguments. We are interested in this because we are interested in knowledge. We want to believe things that are true and not believe things that are false. This is the crux, and the starting point for any investigation into the logical constants.

According to the standard text-book account of formal or logical validity, an argument is valid purely in virtue of its form. Or, more fully, it is valid if and only if under every substitution of non-logical terms, the conclusion is true in every case where the premise or premises is or are true.

It is recognized that not all good arguments are logically valid, i.e. valid purely in terms of their form. But logic is only concerned with formal validity.

But what's so special about formal validity? who cares whether an argument is formally valid or not? A naive thought is that if an argument is formally valid there is just no way of responding to it. The logical machinery just grinds the result out and you can't argue. Whereas if the argument depends on the particular meaning of a term or terms - in other words, if it is claimed to be analytically valid but not logically valid - there is room for debate over whether the terms do in fact have the meaning that they are assumed to have.

Consider an argument involving the set-inclusion operator, or modal terms - two of the disputed cases of constants which may or may not be 'logical' depending on the criterion you are using. Naive set theory is crippled by Russell's Paradox, and there are various more or less arbitrary ways to avoid the paradox. There are competing modal logics and no universal agreement on which is the logic which truly captures our notions of possibility and necessity. So, if an argument depends on either set inclusion or modality, there is potential for disagreement over whether it is valid or not depending on your preferred set theory or system of modal logic.

But then again, intuitionist logicians object to certain inferences from classical logic - those involving the use of the law of excluded middle followed by double negation elimination - so 'whether or not an argument can be disputed' seems a less than firm criterion.

Given what I've said about analysis, you can guess that my inclination would be towards a more pragmatic demarcation of the logical constants. There is a point to demarcating the logical constants, but this varies with our interests in different areas of inquiry. Moreover, the ability to distinguish formally valid argument in a particular context is not undermined by the fact that we may not have a clear notion of formal validity outside that context.

This is consistent with what you say (or what the Stanford encyclopedia says) about semantic value determination and sense determination. Leaving aside absurd suggestions like 'tonk' which fails the basic minimal requirement for logic, there are choices to be made on pragmatic grounds. Here, as you show, Grice is relevant.

As a matter of detail, I didn't find your examples 1) - 3) very convincing. The equivalence in logical form which you claim for 1) and 3) clearly depends on our understanding of the sense that 3) is intended to convey.

Consider for example these statements:

1') All children play with some toys

2') Some children play with all toys

3') Some toys are played with by all children

1') says that there is no child who does not play with a toy, whereas 3') says that there is a toy (or are toys) which is/are played with by all children, which is a very different thing.

Like you, I do feel a bit 'iffy' about your answer. Go with this if you feel gripped and prepared to do more work (AND if you can get hold of Sainsbury!) but not otherwise.

All the best,

Geoffrey