## 4/21/2009

### Logical pluralism and special relativity

I spent the end of last week at UCSD. I had a great time, and I got some extremely useful feedback on a couple of things I've been working on. I'm especially grateful to my host, Chris Wüthrich.

At dinner, we had a helpful discussion about the logical pluralism project I've been working on. The basic hope is to figure out if Beall and Restall are right in saying that the notion of consequence really is "unsettled" or indeterminate. I've had a couple of past posts discussing whether 'entails' is ambiguous (as Beall and Restall sometimes suggest). The upshot of those posts was that, if 'entails' is indeterminate, its indeterminacy arises from having a (hidden) parameter. This hidden parameter is what Beall and Restall call a 'case', as in: 'An argument is valid = In every CASE where all the premises are true, the conclusion is too.' Different logics fall out of different specifications of cases.

The next question to ask is: Is there any evidence that 'entails' really does have a hidden parameter? One way to answer this is to look at an argument that (most) people think is impeccable, which concludes that some concept does harbor a hidden parameter -- and then ask whether 'entails' is like that. The case that springs to my mind is Einstein's argument that simultaneity is relative in special relativity, i.e., whether 2 events are simultaneous is relative to a 3rd parameter, an inertial trajectory.

Let's spell this out. Let e1 and e2 be events. ‘e1 and e2 are simultaneous’ cannot be true simpliciter; rather, it can only be true relative to a frame of reference. This suggests a possible analogy with logical pluralism: ‘P1... Pn entails C’ is not (or at least rarely -- B&R make two little exceptions) true simpliciter, but rather relative to a specification of cases. So ‘entails’ is analogous to ‘simultaneous with’ (or any other predicate of temporal order), and ‘case’ is like ‘frame of reference.’

The key question is then: does the evidence that Einstein appealed to in order to show that there is not one correct notion of simultaneity have an analogue in the logical case?

So what is the physicists’ justification for claiming that there is no preferred frame of reference in special relativity? The fundamental reason is that the laws of nature can be couched in such a way that they are the same in all inertial frames. For one of the frames to be physically privileged, there would have to be some essential physical difference between it and the others. And because there is no privileged or distinguished frame, there is no basis for elevating one standard of simultaneity above the others—and it is in that sense that simultaneity is relative in special relativity.

Can we draw an analogy to the logical case? To be explicit, the relativistic argument is basically this:

P1. Different frames of reference yield different notions of simultaneity.
P2. The laws of nature are the same in every frame of reference.
Thus, there is no privileged frame of reference.
Thus, simultaneity is relative, i.e. there are multiple acceptable notions of simultaneity.

And, as said above, we are taking specifactions of cases as analogous to frames, and simultaneity as analogous to validity. This yields:

P1. Different specifications of cases yield different notions of validity.
P2. _______ is/are the same in every specification of cases.
Thus, there is no privileged specification of cases.
Thus, validity is relative, i.e. there are multiple acceptable notions of validity.

The question is: what should -— what could -- go in the blank in P2? What is the same in every specification of cases? I can see two suggestions B&R might make: 1. Identity and transitivity hold in every specification of cases. But B&R reject this as far too weak. 2. The desiderata for a logical consequence relation (formality, necessity, normativity) are the same in every acceptable (or something similar) specification of cases (‘acceptable’ will rule out possible worlds, right?). But are these desiderata really like ‘the laws of nature’ in the physics case?

Of course, the evidence for one type of pluralism (e.g. about simultaneity) need not have the same form as evidence for another type of pluralism (e.g. about consequence). That is, even if one cannot create a plausible argument for pluralism about validity analogous to the Einsteinian argument for pluralism about simultaneity, that obviously says nothing about whether another, entirely different sort of argument would do the job. Does anyone out there see what it could be? At UCSD, Jonathan Cohen pushed me on this point over dinner; in particular, he pointed out that Einstein's argument is an instance of an argument from symmetry, and that the broader genus of (good) arguments from symmetry might well encompass an argument-type that would do the trick for Beall and Restall's logical pluralism. Can anyone point me to an example of such?

At 11/5/09,  Anonymous said...

I would venture that "consequence" is a primitive notion---can't break it down-- as with color (how do you know you see color?--because
you do). How do you know that you
think x follows from y? Someone else can hold that x does not follow from y--and proffer argument, says that it does not meet the definition of "follows" or some such---but if both sides maintain their positions then what will it come down to? ---something like "it arises within me that x does follow from y--is correct" or simply "x does follow from y!" --------how do the two sides explain their respective ideas ultimately other than to say "this is what showed up" ?

At 12/5/09,  Greg Frost-Arnold said...

Hello --

I basically agree with you. I certainly agree that a theoretical definition of 'consequence' is not what JUSTIFIES modus ponens. Everyone and their dog agree that modus ponens is good, but then they want a simple theory that captures what (in the main) distinguishes the intuitively good inferences from the intuitively bad ones. "What showed up" (in your words) is the data for a theory of consequence.

I think color is a nice analogy. We see all these things are red, and we are curious what they all have in common. And there is something: they all reflect light of a certain wavelength. So in some sense the notion of color CAN be broken down -- or at least analyzed/ accounted for in some way.