[with JC Beall] “Logical Pluralism,” Australasian Journal of Philosophy, 78 (2000) 475–493.
A widespread assumption in contemporary philosophy of logic is that there is one true logic, that there is one and only one correct answer as to whether a given argument is deductively valid. In this paper we propose an alternative view, logical pluralism. According to logical pluralism there is not one true logic; there are many. There is not always a single answer to the question “Is this argument valid?”
Comments
Dear professor Greg Restall,
My name is Che-Ping Su. I am interested in the debate about logical consequence. I have read page1 to page7 of this paper. After reading the paper, I have the following two questions. If you are too busy to reply, you could just tell me where I can find the answer.
About your first dimension of pluralism:
Could you explain more about what you mean by ”Tarskian account of validity is formal but necessary truth preservation account of validity is not”?
(It seems that I can find the answer in John MacFarlane’s paper, What Does it Mean to Say that Logic is Formal?)
In my opinion, the differenc between these two accounts is that the Tarskian account of validity allows fewer concepts to be used in the metatheory. But necessary truth preservation account of validity allows much more concepts to be used in the metatheory. Tarskian account of validity allows only ‘true’, ‘model’, ‘and’, ‘or’, ‘not’, ‘iff’, ‘for all’, and ‘for some’ these concepts in the metatheory. Tarskian account of validity doesn’t allow ‘blue’ and ‘colored’ these concepts to be used in the metatheory. But necessary truth preservation account of validity does.
Consider the argument from a is blue to a is colored.
Although first order logic can’t make this argument valid, we can just set “if Ba, then Ca” as a non-logical axiom. It is just like what we do in the first order axiomatic arithmetic: first order logic can’t make math induction valid, but we can just set it as a (non-logical) axiom. Therefore, I think not being able to validate the argument from ‘a is blue’ to ‘a is colored’ is not a big deal for first order logic.
By the way, I am interested in the debate about logical consequence. Could you please give me some advises about which part of mathematical logic is important(/ highly relevant) to the debate about logical consequence. In other words, if I want to engage in the debate about logical consequence, I should pay attention to which part of mathematical logic?
I will go to ESSLLI05. It will be very cool if we can have discussion during the event.
Yours sincerely,
Che-Ping Su
Posted by: Che-Ping Su at July 30, 2005 09:44 PM
Let me answer the questions one by one.
I think that necessary truth preservation could be construed as being formal only if you’re going to make “formal” mean nothing at all. You can let “blue” and “coloured” be in the metatheory of your theory, but that doesn’t make it a formal theory, and the fact that your metatheory will now render the argument from “that’s blue” to “that’s coloured” valid, doesn’t mean that it’s formal.
Of course we have a theory in which the argument from “that’s blue” to “that’s coloured” is a valid one. But that doesn’t mean that the argument is logically valid. I’m not sure what you mean by the point that not being able to validate the argument “is not a big deal for first order logic.” I think that it’s a virtue of first order logic that it doesn’t validate the argument, because it enables us to express more things — it enables us to consider circumstances (models) in which blue things aren’t coloured, in which there aren’t infinitely many numbers, in which … etc. I think that’s a good thing, not a bad thing.
Anyway, we can talk more about this at ESSLLI in a couple of weeks.
See you soon!
Posted by: Greg Restall at August 1, 2005 05:40 AM
Dear professor Greg Restall,
My name is Che-Ping Su. I am interested in the debate about logical consequence. I have read page1 to page7 of this paper. After reading the paper, I have the following two questions. If you are too busy to reply, you could just tell me where I can find the answer.
About your first dimension of pluralism:
Could you explain more about what you mean by ”Tarskian account of validity is formal but necessary truth preservation account of validity is not”?
(It seems that I can find the answer in John MacFarlane’s paper, What Does it Mean to Say that Logic is Formal?) In my opinion, the differenc between these two accounts is that the Tarskian account of validity allows fewer concepts to be used in the metatheory. But necessary truth preservation account of validity allows much more concepts to be used in the metatheory. Tarskian account of validity allows only ‘true’, ‘model’, ‘and’, ‘or’, ‘not’, ‘iff’, ‘for all’, and ‘for some’ these concepts in the metatheory. Tarskian account of validity doesn’t allow ‘blue’ and ‘colored’ these concepts to be used in the metatheory. But necessary truth preservation account of validity does.
Consider the argument from a is blue to a is colored. Although first order logic can’t make this argument valid, we can just set “if Ba, then Ca” as a non-logical axiom. It is just like what we do in the first order axiomatic arithmetic: first order logic can’t make math induction valid, but we can just set it as a (non-logical) axiom. Therefore, I think not being able to validate the argument from ‘a is blue’ to ‘a is colored’ is not a big deal for first order logic.
By the way, I am interested in the debate about logical consequence. Could you please give me some advises about which part of mathematical logic is important(/ highly relevant) to the debate about logical consequence. In other words, if I want to engage in the debate about logical consequence, I should pay attention to which part of mathematical logic?
I will go to ESSLLI05. It will be very cool if we can have discussion during the event.
Yours sincerely, Che-Ping Su
Posted by: Che-Ping Su at July 30, 2005 09:44 PM