Language and logic (updated)

Some careful philosophical discussion by Monti, Parsons, and Osherson (2009):

There may well be a “language of thought” (LOT) that underlies much of human cognition without LOT being structured like English or other natural languages. Even if tokens of LOT provide the semantic interpretations of English sentences, such tokens might also arise in the minds of aphasic individuals and even in other species and may not resemble the expressions found in natural language. Hence, qualifying logical deduction as an “extra-linguistic” mental capacity is not to deny that some sort of structured representation is engaged when humans perform such reasoning. On the other hand, it is possible that LOT (in humans) coincides with the ‘‘logical form’’ (LF) of natural language sentences, as studied by linguists. Indeed, LF (serving as the LOT) might be pervasive in the cortex, functioning well beyond the language circuit [...].

Levels of analysis again. Just because something “is” not linguistic doesn’t mean it “is” not linguistic.

This calls for a bit of elaboration! (Thanks Martin for the necessary poke.)  There could be languages—in a broad sense of the term—implemented all over the brain. Or, to put it another way, various neural processes, lifted up a level of abstraction or two, could be viewed linguistically. At the more formal end of cognitive science, I’m thinking here of the interesting work in the field of neuro-symbolic integration, where connectionist networks are related to various logics (which have a language).

I don’t think there is any language in the brain. It’s a bit too damp for that. There is evidence that bits of the brain support (at the personal-level of explanation) linguistic function: picking up people in bars and conferences, for instance. There must be linguistic-function-supporting bits in the brain somewhere; one question is how distributed they are. I would also argue that linguistic-like structures (the formal kind) can characterise (i.e., a theorist can use them to chacterise) many aspects of brain function, irrespective of whether that function is linguistic at the personal-level. If this is the case, and those cleverer than I think it is, then that suggests that the brain (at some level of abstraction) has properties related to those linguistic formalisms.

Reference

Monti, M. M.; Parsons, L. M. & Osherson, D. N. (2009). The boundaries of language and thought in deductive inference. Proceedings of the National Academy of Sciences of the United States of America.

Free books

From LogBlog:

Exciting developments! The Association of Symbolic Logic has made the now-out of print volumes in the Lecture Notes in Logic (vols. 1-12) and Perspectives in Mathematical Logic (vols. 1-12) open-access through Project Euclid. This includes classics like

• Shoenfield’s Recursion Theory,
• Lindström’s Aspects of Incompleteness in the LNL,
• Sacks’ Higher Recursion Theory,
• Hájek and Pudlák’s Metamathematics of First-order Arithmetic,
• Shelah’s Proper and Improper Forcing,
• Barwise’s Admissible Sets and Structures, and
• Barwise and Feferman’s Model-theoretic Logics in the PiML.

Science for the half-wits

A bit from Jean Yves-Girard’s latest rant, The phantom of transparency:

Still under the heading « science for the half-wits », let us mention non monotonic « logics ». They belong in our discussion because of the fantasy of completeness, i.e., of the answer to all questions. Here, the slogan is what is not provable is false : one thus seeks a completion by adding unprovable statements. Every person with a minimum of logical culture knows that this completion (that would yield transparency) is fundamentally impossible, because of the undecidability of the halting problem, in other terms, of incompleteness, which has been rightly named : it denotes, not a want with respect to a preexisiting totality, but the fundamentally incomplete nature of the cognitive process.

Completeness is boring.  Maybe Y-G would be less confused if he viewed these logics as modelling information update, important given the “fundamentally incomplete nature of the cognitive process”.

Logic and Reasoning: do the facts matter?

Had a read of Logic and reasoning: do the facts matter? by Johan van Benthem. Covers much ground in a short space but I found it thought provoking. Here’s a quick sketch of the bits I liked.

Van Benthem mentions the anti-psychologism stance, briefly the idea that human practice cannot tell us what correct reasoning is. He contrasts Frege’s view with that of Wundt; the latter, he argues, was too close to practice; Frege was too far. He argues that if logics were totally inconsistent with real practice then they’d be useless.

Much logic is about going beyond what classical logic has to offer and is driven by real language use. Van Bentham cites Prior’s work on temporal structure, Lewis and Stalnaker’s work on comparative orderings of worlds, work on generalised quantifiers which was driven by the mess of real language and for instance produced formalisations of quantifiers like most and few. Generally, van Bentham argues, “one needs to move closer to the goal of providing more direct and faithful mathematical renderings of what seem to be stable reasoning practices.” You want your logic to be more natural, closer to the phenomena. Conceptions of mathematical logic were driven by the terms that appeared in rigorous proofs, so the linguistic stuff is just widening the set of practices that are modelled.

Correctability in a logic is more important than correctness, he argues. This is consistent with the goals of the non-monotonic logic crowd I know and love. I find this most interesting when looking at individual differences in reasoning processes: perhaps a correctability dimension is out there somewhere, if only we could measure it and its correlates. I have some ideas—stay tuned.

Divergences from competence criteria, he argues, suggest new practices. I still see many papers in which people are scored against classical logic. Failure should cause an attempt to work out what practice is being followed by a person rather than the more common concern of what went wrong and how we could bring people back.

Much more in this little paper…

The Reasoner

The first issue (PDF) of The Reasoner is out now. From the website:

The Reasoner is a monthly digest highlighting exciting new research on reasoning and interesting new arguments. It is interdisciplinary, covering research in, e.g., philosophy, logic, AI, statistics, cognitive science, law, psychology, mathematics and the sciences.

That’s a wee bit exciting! The first issue has articles on co-referential expressions; categorisations of the different kinds of academic, to move beyond crude “humanities” and “science” divisions; the use of statistics in the courtroom; and counterpossibles. Nice mix of topics.

Logicism

This is a very quick reply to a colleague who queried the validity of “logical accounts of logical operations”.

At least two issues there: the validity of using logics to give a functional characterisation of (measurable) behaviour versus using them to provide a causal mechanism for the generation of the behaviour. Can’t see any problem with the former: it’s just a linguistic analysis and allows you to sort people into categories, e.g. to enable associations cross-task or with individual difference measures. Or to put it another way, it’s just a very structured likert scale! Whether or not the causal mechanism construction is a valid enterprise collapses to an argument of what can be measured experimentally and what particular logical mechanism is used. If all you can get at is outputs then it doesn’t matter what your model looks like, really. Add RTs and things become a bit more constrained. If your experiments give you sequences of outputs then the temporal nature of the model is more important.

There’s more to logic than, say, Gentzen natural deduction with trees that look like:

There is unlikely to be a one-to-one mapping between these beasties or the theorem provers that can build them and the brain, but that doesn’t imply the non-existence of another logical mechanism which is more closely related to the grey stuff. If the word “logic” causes you grief then replace with “computational”.

I am interested in equilvalencies between logics, for instance how nonmonotonic logic may be embedded in classical logic, or classical probability, or connectionist networks. I’m convinced these theoretical results can point us in the right direction for building cognitive models. For instance you can fix on a particular computational mechanism—whatever you like—and then argue using the maths how everything you’ve done maps across to the other formalisms. So one model is just a point in an equivalence class of models.