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!
