This is a preview of subscription content, log in via an institution to check access.
Notes
At one point Vann says that the view I took in my earlier papers would rule out weakening classical logic to deal with the paradoxes (if it were not also weakened elsewhere), since that would be adopting different standards for truth than elsewhere. I don’t see this: the view could be that the weakened logic is the only really general logic, but that typical predicates in physics, mathematics, sociology and so forth obey excluded middle as a non-logical axiom, which is enough to make their logic effectively classical.
I’ve shifted from quantificational arguments to quantificational theorems for notational simplicity, but the points extend to arguments.
The significance of this does not depend on the issue raised in my precis, about whether these are genuine rules of inference or merely rules of theorem closure. (On the other hand, my remarks in the paragraph after next do depend on that.).
In the model theory I employ (which uses a partially ordered value space with a top value 1), an argument is weakly valid if in every model, if the premises have value 1 then so does the conclusion; and strongly valid if in every model, the meet of the values of the premises is less than or equal to the value of the conclusion. (When the set of premises is empty, both reduce to the claim that the conclusion has value 1; so when talking about the validity of sentences as opposed to arguments, there is no difference between strong and weak.).
We need only get from A to ∀x(x instantiates [v | v=v ∧ A]), from that and [v | v=v ∧ A] = [v | v=v ∧ B] to ∀x(x instantiates [v | v=v ∧ B]), and from that to B.
I say ‘fully accept’ and ‘fully reject’ rather than simply ‘accept’ and ‘reject’, to put aside issues related to the lottery and preface “paradoxes”. If for instance A 1 through A 1,000,000,000 represent the claims that given tickets in a one-billion-ticket lottery will win, one might fully accept their disjunction and not quite fully reject each one individually: one assigns it minuscule but non-zero degree of belief. Something like this can happen even in the two-disjunct case, if one is willing to lower the threshold T of belief required for acceptance sufficiently far, and raise the corresponding co-threshold 1−T that one has to go below for rejection. In what follows, ‘accept’ and ‘reject’ will always mean ‘fully accept’ and ‘fully reject’, where these involve degree of belief 1 and 0 respectively.
Greg might worry that invoking degrees of belief in describing the normative demands of logic requires excessive computational demands. (See Restall 2005.) I try to disarm such worries in Field (2009)—though the discussion there is in terms of single-conclusion logic, since I hadn’t thought of the generalization to multiple conclusion logic until preparing this response to Greg.
There are serious limitations on the adequacy of this model, especially as regards sentences with unbounded quantifiers: this comes from the fact, discussed at length in the book, that there is no model in which we can identify the truth of such sentences with their having value 1 in the model. But a model with limitations can still be useful, and the present point is that to the extent that it can be applied to acceptance it can be applied to rejection.
References
Evans, G. (1978). Can there be vague objects? Analysis, 38, 208.
Field, H. (2001). Truth and the absence of fact. Oxford: Oxford University Press.
Field, H. (2009). What is the normative role of logic? Proceedings of the Aristotelian Society, Suppl 83, 251–268.
Restall, G. (2005). Multiple conclusions. In P. Hajek, L. Valdes-Villanueva & D. Westerstahl (Eds.), Proceedings of the 12th international congress of logic, methodology and philosophy of science (pp. 189–205). King’s College Publications.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Field, H. Replies to commentators on Saving Truth From Paradox . Philos Stud 147, 457–470 (2010). https://doi.org/10.1007/s11098-009-9470-y
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11098-009-9470-y