consequently.org/writing/logical_pluralism

A question - do you mean that there are many self-cohering logics, but which are inconsistent with each other, or do you mean there are many logics that are mutually consistent?

Posted by: John Wilkins at July 20, 2004 04:44 PM

For the detailed view I’d point you to the paper that kicked all of this off. (Or you can wait until the book gets released, I suppose.)

But let me try to answer the question here. To answer your question, we need to know what you mean by “consistent” in the question. What is it for one logic to be consistent with another one? It could mean a number of different things, and our answers would probably go in different ways.

Anyway, our position is a pluralism and not a relativism. We think that there are different logics, and that they differ from each other. That means, that we think that certain arguments are valid with respect to some logics and invalid with respect to others, and there’s no further question as to whether or not the argument is really valid.

That’s the line. Does it help answer your question?

Posted by: Greg Restall at July 21, 2004 04:56 PM

Interesting… In my ignorance I thought it might be that the different logics are all ‘models’ (in the scientific sense) of the notion of logical consequence (taken to be ‘out there’ in some dubious sense) representing how it enters our reasoning in various contexts, if that makes sense. But reading what you’ve just said, it looks a bit odder than that - in a good way!

Posted by: Iorwerth Thomas at August 31, 2004 10:27 PM

I am looking forward to this book. I have just now run across the paper that kicked it off on the Web. CSLI published my book Logic, Convention, and Common Knowledge: A Conventionalist Account of Logic a few years ago. It comes at logical pluralism, as you call it, from a slightly different angle, but I believe the two are highly complementary. I hope you agree and look forward to hearing your opinion. (In fact, you would not know it to look at the book, but what got me started down that road was looking more than a decade ago at Etchemendy’s book on consequence and Dummett’s treatment of deduction.) You can find links to it on my homepage, http://www.syverson.org

Posted by: Paul Syverson at November 4, 2004 03:22 AM

2005? I note that the OUP site have the paperback down as 2005 as well, yet my paperback (which arrived in Belfast yesterday!) says 2006.

Posted by: Paul (Robinson) at April 5, 2006 07:37 PM

You are so right, Paul. It’s 2006 in the book. I thought, in late 2005, that it would come out in 2005, and in fact, it did. I had a copy in my hot little hands in December. However, the publication date in the volume is what stands, and in this case, it’s 2006.

Posted by: Greg Restall at April 5, 2006 07:48 PM

You seem to focus on the ambiguity in “cases” in (V). However, whenever we have a many-valued logic, other sorts of ambiguity arise in the definition of validity, since there needs to be some standard for choosing the “designated” value or values to be preserved in logical consequence. The definition of validity as preservation of truth is also not necessarily equivalent to validity as absense of possible counterexamples. I take it that a counter example is a case in which the premises all have a designated value while the conclusion has something else. The something else could be “false” or any non-designated value, or a value of lower rank than the premises, or a counter-designated value. Anyway, I’m satisfied by your arguments for logical pluralism as they’ve appeared in papers over the years. I still haven’t read the entire book. I’m not quite clear on the difference between logical pluralism and logical relativism. Certainly there are ways of defining the latter so that there is no difference: the consequence relation is understood relative to a specification of what counts as a “case”, for example. What is to be avoided is anything goes relativism, but who would want to defend that?

Posted by: Muhammad Legenhausen at June 24, 2006 05:00 PM

It’s true that you could think of logical consequence as being relative to a specification of the class of cases —- and so logical relativism could be an appropriate enough description. We chose pluralism because it seemed to us that relativism was a better word for a doctrine that says that a feature is relative to a person or a community or a time or a location or something helpfully independently specified. Any pluralism (that says that there isn’t one X but a number of Xs: X1, X2, … Xn) could become a relativism by saying that being-X is really being-Xi, which is a relative predicate, relative to the choice of index i.

As to many valued logics: we could stretch the story a little bit to allow them smoothly enough in the way you specify. For every many-valued evaluation v (in the logic of your choice), let there be a v-case, which makes true everything getting a designated value under that evaluation, and untrue, everything else. The consequence relation specified by truth-preservation in all v-cases will be the logic of designated-value preservation in the many-valued logic you chose.

The hard part is to show that the choice you made makes some kind of sense and that talking of v-worlds picks out a relation worth calling a consequence relation, in the way we tried to do for classical, relevant and constructive logics.

Posted by: Greg Restall at June 24, 2006 05:12 PM

Thanks for the quick response. Suppose a gappy logic like Kleene’s with values 1,2,3, for true, neither, false. If 1 is designated and there is a case where the premisses are true and the conclusion neither true nor false, the argument is invalid. If 1,2 are both designated, a case where the premisses are neither true nor false and the conclusion is false is enough to invalidate it. But, in addition to these two choices of designation that Kleene introduced, it seems reasonable enough to think that inferences are not so bad when the worst possible cases are those mentioned above. One might want to hold that arguments are only invalid when there are cases in which the premises are true and the conclusion false. This would yield a different notion of validity than could be handled by the v-cases you mentioned (or Kleene’s designated values). Yet it seems intuitive enough: invalidity requires a counterexample case in which the premises are true and the conclusion false and the neithers just don’t count in determining validity.

Posted by: Muhammad Legenhausen at June 25, 2006 03:14 AM

Good point! To cope with this phenomenon (different choices of designated values) in the way I described, you’d have different classes of cases, as we’ve taken “truth in a case” to be an absolute notion.

The other difficulty in the case of ignoring the middle value and taking invalidity as when the premises are true and the conclusion false, is to show that the consequence relation is transitive, as any consequence relation defined using truth-preservation in cases is a transitive one.

Whether the notion of consequence you want in Kleene’s three valued logic turns out to be transitive depends on the details of the language. In the conjunction only fragment it is. If you have formulas which are guaranteed to get a constant value, for each of the three values (call these formulas T, N and F respectively), then the argument from T to N is valid, and the argument from N to F is valid, but obviously the argument from T to F is not.

So, whether we allow this kind of case or not will depend on detail. I don’t know if anyone has worked this kind of detail at comprehensively. (My student Conrad Asmus has thought about this a fair bit, but I’m not sure how far he’s got with it.)

Posted by: Greg Restall at June 25, 2006 09:13 AM

a reply to Mr. Haj Muhammad Legenhausen’s question:

The non-counterexample-criterion (invalidity = (premises are T & conclusion is F)) is equivalent to

validity = ~(premises are T & conclusion is F) validity = (if premises are T then conclusion is not F) validity = (if premises are T then conclusion is T or N)

so we can define two designated values (and so two types of cases): premises-designated-values and conclusion-designated-values, which I abbreviate PD and CD. Then we can generalize V-schema as following:

(V) validity = (if premises are PD then conclusion is CD)

Then we can define LOGICAL PLURALISM by this general validity and if it is found a new recalcitrant validity then we can generalize our (V) further.

Posted by: Asadollah Fallahy at September 14, 2006 05:58 PM