Friday, July 16, 2010


There is no position so obviously wrong that someone on the internet won't passive-aggressively defend it.

Making Your Own Fun


I thought I'd take some time to work through this one.

The question asks, essentially, whether "If anything is possible, then something is possibly impossible" is satisfiable. The antecedent establishes the existence of a world - at this world, which we'll call "world-0", everything is possible. This means that this world accesses a world where A and a world where not-A, for all A. I'll interpret "something is impossible" to mean that some p is not possible, i.e., that for some p, it is not possible that p. Converting modal operators, this means that necessarily not-p.

If, at world-0, everything is possible, then world-0 accesses a world where necessarily not-p. That world accesses either no other worlds, or only worlds where not-p.

Is "If anything is possible, then something is possibly impossible" satisfiable? Sure. Say world-0 accesses, inter alia, world-1. It also accesses any worlds sufficient to make "anything is possible" true at world-0. World-1 accesses no worlds. Then world-0 accesses a world at which not-p is necessary. In other words, from world-0, it is possibly impossible for p to be true.


Monday, July 12, 2010

Kicking Away the Classical Ladder

This part of the postscript to An Introduction to Non-Classical Logic struck me as important:
It may fairly be asked what logic I have been using to specify and reason about the semantics of the various logics we have been dealing with. The procedures employed have not been formal ones, of course. Like most mathematics, matters have been left at an informal level. They could be formalised in a standard set theory, such as Zermelo Fraenkel set theory, based on classical logic. But to someone, such as an intuitionist or paraconsistent logician, who takes such reasoning not to be correct, at least in part, things cannot be left like this. The classical ladder must, so to speak, be kicked away.
If non-classical logics are correct formalizations of some kinds of thinking, it's worth wondering why metalogical reasoning about those logics should be done in classical terms. If non-classical logics were just, so to speak, games of symbol-manipulation, then abandoning their principles when doing "correct" reasoning would make sense. But non-classical logics have philosophically respectable semantics, thanks to recent scholarship. Not only do the proof-theories of these logics make internal sense, but those proof-theoretic methods mirror semantic interpretations that make intuitive sense and link the symbol-manipulation of the logics to real ways of thinking about real things.

I think there is an analogy with non-Euclidean geometries. At first, they were perhaps viewed as logically-consistent but meaningless consequences of the nature of geometrical reasoning. Recent advances in the sciences have shown that, in truth, non-Euclidean geometries are fair representations of actual things. Non-classical logics are fair representations of things: modal logic of possibility and necessity, fuzzy logic of vagueness, and so on. Though these logics will always be developing alongside metaphysical accounts of the nature of the reality each logic is describing, they have already come into their own. There is no barrier to using non-classical logic to reason about logic itself. This is to my mind (and probably to others) a very difficult thing to attempt. The classical ladder is comfortable and natural. Thinking about logic paraconsistently, for example, is strange.

But it might just be the way forward.

Saturday, July 10, 2010


We still get hits for "Alexis Fitts".

Who is this person?

Well...when we last met her, she was pooh-poohing racial discrimination.

Now she has a degree in English (lol) and is working for Mother Jones.


Friday, July 09, 2010

The Two Philosophies

This is philosophy:
Kripke's Soundness and Completeness Theorems establish that a sentence of L is provable in intuitionistic predicate logic if and only if it is forced by every node of every Kripke structure. Thus to show that (¬∀x¬P(x) → ∃xP(x)) is intuitionistically unprovable, it is enough to consider a Kripke structure with K = {k, k′}, k < k′, D(k) = D(k′) = {0}, T(P, k) empty but T(P, k′) = {0}. And to show the converse is intuitionistically provable (without actually exhibiting a proof), one only needs the consistency and monotonicity properties of arbitrary Kripke models, with the definition of forcing.
Moschovakis, J., "Intuitionistic Logic", The Stanford Encyclopedia of Philosophy.

This is also philosophy:
From this same perspective we will have to consider symptoms and incidents outside the norm as indices of a potential labour of subjectification. It seems to me essential to organize new micropolitical and microsocial practices, new solidarities, a new gentleness, together with new aesthetic and new analytic practices regarding the formation of the unconscious. It appears to me that this is the only possible way to get social and political practices back on their feet, working for humanity and not simply for a permanent reequilibration of the capitalist semiotic Universe.
Guattari's The Three Ecologies. Tr. Ian Pindar and Paul Sutton.

Some similarities might pop out to a person not accustomed to reading such things. Indeed, they both probably sound utterly incomprehensible. The difference is that the first selection is talking about something; logic has a specialized, difficult vocabulary because it has become an object of precise analysis in recent decades. The second selection is impenetrable because the author is trying to clothe his empty, worthless speculation with gaudy verbiage.

But both passages are exercises in philosophy. What happened? Well, I don't want to make this an analytic vs. Continental pissing contest. Analytical philosophers make blunders too, and sometimes hide their limited understanding behind pompous vocabulary. Continentals do hit upon something true and meaningful, on occasion. What I want to do instead is draw a distinction between good and bad philosophy. Good philosophy increasingly means rigorous philosophy. Although insights might be arrived at without mathematical rigor, it is out of fashion in certain branches of philosophy to proceed except with clear definitions and transparent inferences. This is a good thing, for whatever is lost in creativity is gained in exactness. It's not a trend philosophy had to take, but it's good it did. Of course, as I just demonstrated, not all schools of thought or branches of philosophy even attempt to apply analytical rigor to their investigations. I get the sense, though, that those who abandon such scientific tools are slowly being marginalized. At the very least, the precise method is gaining more adherents, so even if a large part (perhaps even the majority) of the profession clings to Hegelian obscurantism, a critical mass of philosophers is now thinking clearly and expressing its arguments coherently.

Still, philosophy has a problem. It has a reputation of being...well, nothing but Guattari and his ilk - incoherent, pretentious, and ultimately useless. It doesn't help that the heavy analytic stuff is damned hard, so undergrads and even grad students tend to avoid it and focus on the softer stuff. Intellectual trends in the humanities make that easier, too - it's more fashionable in the liberal arts as a whole to talk about gender and subjectivity and social construction. People actually doing philosophy sometimes do it from other departments, or in interdisciplinary programs like Berkeley's Group in Logic and the Methdology of Science. Those who are studying in other fields and come at philosophy at amateurs will also get exposed overwhemingly to the "soft" stuff. The study of philosophy in philosophy departments, then, has a reputation for being, well, bullshit.

This can be awfully frustrating to someone like me - someone inside philosophy who agrees that far too much philosophy is wasted effort. When I point out that logic is, after all, philosophy, it's easy to dismiss logic as a branch of mathematics. Indeed, because, as I mentioned, the philosophy in philosophy departments is so continentalized, logic courses are often conducted in mathematics or computer science departments. This administrative classification is meaningless, though - philosophy, whatever it is (and its range is broad and vague), retains its nature no matter the circumstances of its academic study.

In short, yes, philosophy is ridiculous, but that's not due to the objects of its study but to (some of) the subjects doing that study. My tastes are increasingly turning to logic, so criticism of philosophy can seem bizarre - reading a completeness proof (and finishing it as an exercise!) is not an instance of sloppiness and subjectivism, whatever it is.

I'll have more to say soon. Priest has me thinking about metalogic and I'm hoping to get my hands on Dummett's book on intuitionism. It apparently uses sequent calculus (;_____________________________;) instead of tableaux, but I'll give a strictly inferior proof-theory a try anyway.

I continue to just have lots of free time to do that!

Tuesday, July 06, 2010


What's brown and sounds like a bell?

Eh, Google it. I was just thinking how much jokes like that play on the fluid shift between use and mention. There's a Hegel quote to be made, but it's disconcertingly hard to find the right passage - proof I need to do more reading and less interneting.

I know there are several SCOTUS cases I could blog - I think it probable I will. And there's always random stuff to talk about. For instance, did you know that I now own a Fender Stratocaster?

I'll have more to say when I am in the proper mood. Triple-digit heat works its way through even the air conditioning.