Free-variable Tableaux

Wolfgang Schwartz asks here if there is a “canonical” way to build free-variable tableaux which are guaranteed to close if the original formula is valid. It seems to me that this must be the case, since free-variable tableaux are a complete proof method. But maybe I don’t understand the question.

The point of free-variable tableaux is to postpone substitutions of strong quantifiers until such a substitution results in a closed branch. So instead of expanding a branch containing ∀x A(x) by A(t) for all possible t, you keep the free x around until you get a formula ¬ B on the branch where A(x) and B unify. Closure, i.e., the actual substitution of x by a term only happens when this is the case. So at each stage, you should check if you can apply Closure (i.e., if you can close the branch) but you don’t actually apply it until you can.

The Status of Logic in Philosophy

It is a commonly accepted view (among logicians working in philosophy [departments]) that while logic was considered central to philosophy in the mid-20th century, it has since moved closer and closer to the margins. It is said, e.g., that while in the 1950s and 60s it was common to find “pure” logicians working in philosophy departments (and consequently, that as a pure logician you could find a job in a philosophy department), this is no longer the case to a similar extent. It is also believed that while a few decades ago logicians enjoyed a reputation (among philosophers) as people who were doing important (and hard) work, now much of logic is just not considered philosophy anymore. I wonder how accurate this cluster of suspicions (henceforth “the belief”) actually is .

Anecdotal evidence certainly suggests that it’s difficult to find a job as a logician. But is it more difficult than finding a job, period? And is it more difficult now than it was 30 or 40 years ago?

Anecdotal evidence also suggests that logic is not held in particularly high regard among some (many?) philosophers. But is this more true now than it was 30 or 40 years ago?

These are difficult questions to answer; difficult to answer, that is, other than by adducing more anecdotal evidence. One would think that if the belief is correct, then the number of logic jobs advertised should have declined over the last 30 years (as a percentage of all philosophy jobs). One would also think that the philosophy curriculum would have changed reflecting the changing status of logic in philosophy as a whole. Thirdly, it would likely be the case that as fewer logicians were trained and hired by philosophy departments, there should be fewer logic papers published by logicians in philosophy departments.

I have no idea of how to come up with hard data on the first two cases, but I’d be interested to hear anecdotal evidence, especially on the second point. Does your department offer fewer advanced logic courses now than it did 20 or 30 years ago? Did your department use to require a graduate logic course of its grad students but no longer does?

There is some hard data on the third issue, thanks to the Web of Science. I’ve done some searches on logic papers in the past three decades. The data is a little skewed, I’m sure, since they don’t have extensive coverage of the period 1975-1984, but the results are nevertheless interesting. I tried to compare output by people working in philosophy departments (in the US, Canada, the UK, Ireland, Australia, and New Zealand) in logic journals (The Journal and Bulletin of Symbolic Logic, and the Journal of Philosophical Logic). The results are interesting:

1975-1984 1985-1994 1995-2004
Journal of Symbolic Logic +
Bulletin of Symbolc Logic
Journal of Philosophical Logic 32/252
Total logic papers 52/3,406

In the first two lines, I gave the ratio of papers by authors in US/ Canadian/ British/ Irish/ Australian/ New Zealand philosophy departments to the total number of papers in the respective journals. In the last line, you have the ratio of logic papers to all papers in the index (by authors in US/Canadian etc. departments). It looks like the output of logicians in English-speaking philosophy departments has increased since the 1970s.

I conjecture that what happened in philosophy vis-à-vis logic is not that logic has become (seen as) less central, but that as philosophical logic has matured over the last 50 years, it has been integrated into the appropriate areas of philosophy. So perhaps it isn’t logic per se that’s seen as less central to philosophy, but the kind of logic you could still commonly find being done in philosophy departments in the 50s and 60s, which wasn’t that much different from the logic done in math departments.

UPDATE: Updated the table to include New Zealand, as well as Ireland, Scotland, and Wales.

UPDATE: Followup posted here.

UPDATE: More followups:
Thoughts on logic in philosophy grad programs
Survey of logic in US grad programs
Brian Weatherson’s JfP analysis and logic jobs
Session on logic in graduate philosophy programs at the ASL Spring Meeting/APA Pacific
Logic in America vs Logic in Europe