Neurath's Boat

Logical truth and logical consequence

As GFA notes, there has come to be something of a sentiment that logical consequence is a more fundamental notion than logical truth. He cites Read and Etchemendy; Dummett also takes this view (I've been reading Dummett's Frege: Philosophy of Language). GFA questions how anybody can say this when the two are (usually) equivalent; usually you can translate a logical truth into the claim that some consequence relation holds, and vice versa.

GFA does note a couple of exceptions to this equivalence. It is not exactly an exception, but is perhaps also relevant that in providing a minimal basis for a logical system, it is possible to give only rules of inference and no axioms (in fact, this is often done; the logic I'm teaching in my intro class this semester is a "natural deduction" system which uses this approach). On the other hand, it is not possible to give only axioms; some rule of inference is always needed ("axiomatic" systems normally have modus ponens, as well as some sort of substitution rule; substitution rules may be a special case, but modus ponens is clearly a rule of logical consequence). At least, the only way to give a purely axiomatic system would be to make every logical truth an axiom.

Whether all of this suffices to make consequence the more "fundamental" notion, I'm not sure. I am by nature very suspicious of claims that anything is more fundamental than anything else. On the other hand, I sympathize with some of the motives for saying that consequence is the more fundamental notion. Dummett notes that the 20th century saw quite a bit of controversy over the status of logical truths; whether they could be understood to be "analytic" (whatever that means anyway; another issue that was much fought over) and what status they did have if they couldn't be classified as analytic. Dummett seems to consider this largely ink spilled in vain (certainly nothing much was ever settled by all these debates), and also thinks there wouldn't have been so much fuss over it if people had been thinking in terms of consequences rather than logical truths. Perhaps there is more of an intuition that a logical truth needs to be about something, that something needs to make it true, than there is any corresponding intuition regarding logical consequences.

If such an intuition has indeed been a source of frivolous worries, then the equivalence of logical consequence and logical truth ought to be enough to undermine the intuition; if logical truth and logical consequence are equivalent, then it's possible, even if not compulsory, to give a reductive account of the former in terms of the latter, so intuitions that special explanations of logical truths are needed should already be undermined. But they're not precisely equivalent; as GFA's examples show, and as mine may also show, logical consequence is an ever so slightly broader notion. This surely wouldn't justify any extravagant metaphysical thesis that logical consequences are built into the structure of reality in a way that logical truths are not, but of course I don't myself think any extravagant metaphysical theses are ever justified, and if Dummett is right the great benefit of focusing on logical consequence is that such a metaphysical thesis has no intuitive appeal anyway. If we set aside such metaphysical concerns, though, we do seem to be left with a meaningful sense in which consequence is more fundamental. Still, perhaps the terminology is less than ideal, since the word "fundamental" has so many associations with the metaphysical concerns.

Nietzsche on the A Priori

I've been thinking about part 1 of Beyond Good and Evil recently, as I've been teaching it.  I like to teach a bit of Nietzsche in intro, and I like to use a large block of text because I think context is important to understanding Nietzsche.  In this particular case, I have now come to think that "On the Prejudices of Philosophers" has as a central goal advancing a certain doctrine of the a priori, one influenced by but in conflict with that of Kant.

Nietzsche of course rejects the a priori claims of traditional metaphysics, on the basis that they purport to represent a true world which is in reality utterly fanciful.  Such is already firmly stated in the preface of BGE.  He also speaks harshly of the synthetic a priori judgments of Kant, insisting forcefully that they are among the "falsest judgments" (5), but he at the same time claims that they are "indispensable," something he never claims about the judgments of dogmatic metaphysics.

What did he mean in saying they were false?  Well, it's clear that they don't represent the true world; nothing does.  There is no true world.  Further, however, they do not represent experience.  Nietzsche doesn't dispute Kant's claim that such judgments constitute part of what makes experience possible for us, but he denies (on my reading) that this means they really tell us anything about the world of experience (a point on which Kant says equivocal things; Nietzsche is clearly reading Kant as either being confused or mistaken when he speaks of the synthetic a priori claims as empirically true and necessary).  Nietzsche is concerned to insist that the judgments are inventions of ours, interpretations imposed by us.  He explicitly opens up the possibility of alternatives to Kantian logic in various places, questioning the causal interpretation of the world (22) and suggesting that the subject/object pattern of interpreting the world is a mistake (12, 17) and an artifact of language (20), but even if we had no alternatives, he would insist that Kant (and more so the Kantians) were nonetheless failing to "distinguish between 'invention' and 'discovery'" (11).

So logic and mathematics are inventions, not in any way given to us, either by a true world or by any kind of fanciful a priori intuitions (the "faculties" he mocks in section 11).  An additional virtue of attributing such a view of the a priori to Nietzsche is that it may further help explain the surprisingly positive reference to Nietzsche in Carnap's "Overcoming Metaphysics through the Logical Analysis of Language," which even has a somewhat Nietzschean title.  The view of logic and mathematics as human inventions accords well with Carnap's conventionalism, and the possibility of alternative interpretations and the focus on the usefulness of the interpretations for us both match up with Carnap's principle of tolerance and his emphasis on pragmatic considerations in language choice.  If this was Carnap's own reading of Nietzsche, it becomes less surprising that Nietzsche earned his praise for "almost entirely avoiding" the confusion which Carnap saw as infecting metaphysics.

Wittgenstein

So I've been reading Kenny's book on Wittgenstein.  He said he wrote it because he was concerned that important lessons Wittgenstein had to teach were being forgotten (as of the early 70s).  Of course, Wittgenstein's star has, if anything, sunk further since then.  I was not impressed with the lessons Kenny derived from Wittgenstein; I still think his reputation has declined for good reasons.  But it gave me an occasion to ponder what those reasons are, and I have a theory.