Logical Pluralism

In this post I will raise an issue for the logical pluralism of Beall & Restall (hereafter 'B&R') - a much-discussed, topic-revivifying view in the philosophy of logic. My study of their view was prompted by Mark Colyvan, whose course on Philosophy of Logic at Sydney Uni I'm helping to teach this year. Thanks to Mark for encouragement and discussion. 

I'll start off in this post by looking at B&R's central thesis, and arguing that it fails to capture an interesting, controversial position in the philosophy of logic. The problem is not that the view is false, but that it's "too easily" true.

I take their 2000 paper ‘Logical Pluralism’ as the starting point, but where more detail is needed, I draw on their 2005 book Logical Pluralism.

Their claim concerns the following schema:

Generalised Tarski Thesis (GTT): An argument is valid_x if and only if, in every case_x in which the premises are true, so is the conclusion. (2005, p. 29.)

(This is more precise than the corresponding schema, (V) (for 'validity'), from their 2000 as it makes clear what is allowed to vary.)

They write:

Logical pluralism is the claim that at least two different instances of GTT provide admissible precisifications of logical consequence. (2005, p. 29.)

Being an existential, numerical claim (‘There are at least two…’), there are many ways the view could be true. Later in this series of posts I'll look at the ways they imagine it coming out true - B&R hold that 'cases' may be taken to be worlds, Tarski-style models, or situations (in the sense of Barwise & Perry's situation theory, at least in the first instance). But here I want to highlight presumably unintended ways in which it comes out true, or at least appears to me to do so. If I'm right about this, these unintended ways threaten to rob their view of its apparent bite.

According to B&R, ‘cases’ may be models ‘Tarskian style’ (2000, p. 480) or ‘along Tarskian lines’ (2005, p. 29).

But this permits differences over exactly what a model is (even for a given language L - to fix ideas, let's consider the language of first-order logic).

For instances, do models provide assignments to variables, or just to names? That depends on how you like to treat quantification when defining 'true on a model'. (Another option, taken by Tarski, eschews assignments to variables in favour of a trick involving sequences of objects.)

Some think which option you take here is philosophically significant - see for instance Smith's 'Truth via Satisfaction?'. But few I think would want to say that not all of these options lead to Tarskian style models (in a broad sense).

But this doesn't actually matter, since there other differences, which seem definitely trivial, over what exactly a model is taken to be in various presentations of first-order logic.

Is a model a tuple of the form <D, I> (where D is the domain and I contains semantic information about all non-logical terms)? Or is it a tuple of the form <D, P, N>, where predicates’ extensions are given separately from names’ referents? Or do we bundle these ingredients informally, as is often done in introductory texts? (That is, do we think that a model is just: a domain, extensions for predicates etc., without thinking of the model as a mathematical object in its own right?)

The point is that these differences, while pretty unimportant, do lead to real differences
over which objects are the ‘cases’. And so to different ‘precisifications’ of the notion of logical consequence.

And so it seems that, if you believe that there is more than one slightly different way of doing broadly Tarksi-style model theory, then you should be a logical pluralist in Beall & Restall's sense. But that seems like the wrong outcome! And in some sense, not what they mean. B&R wanted to carve out and develop a distinctive philosophical view, one which would for instance conflict with the views of someone who thinks that nothing that is not some form of classical logic counts as 'logic'.

References

Beall, Jc & Restall, Greg (2000). Logical pluralism. Australasian Journal of Philosophy 78 (4):475 – 493.
Beall, Jc & Restall, Greg (2005). Logical Pluralism. Oxford University Press.
Smith, Nicholas J. J. (2017). Truth via Satisfaction? In Pavel Arazim & Tomas Lavicka (eds.), The Logica Yearbook 2016. London: College Publications. pp. 273-287.