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.

Prover9 and Mace4

Just found two fantastic programs and a GUI for exploring first-order classical models and also automated proof, Prover9 and Mace4.  There are many other theorem provers and model checkers out there.  This one is special as it comes as a self-contained and easy to use package for Windows and Macs.

There are many impressive examples built in which you can play with.  To start easy, I gave it a little syllogism:

all B are A
no B are C

with existential presupposition, which is expressed simply:

exists x a(x).
exists x b(x).
exists x c(x).
all x (b(x) -> a(x)).
all x (b(x) -> -c(x)).

and asked it to find a model. Out popped a model with two individuals, named 0 and 1:

a(0).
- a(1).

b(0).
- b(1).

- c(0).
c(1).

So individual 0 is an A, a B, but not a C. Individual 1 is not an A, nor a B, but is a C.

Then I requested a counterexample to the conclusion no C are A:

a(0).
a(1).

b(0).
- b(1).

- c(0).
c(1).

The premises are true in this model, but the conclusion is false.

Finally, does the conclusion some A are not C follow from the premises?

2 (exists x b(x)) [assumption].
4 (all x (b(x) -> a(x))) [assumption].
5 (all x (b(x) -> -c(x))) [assumption].
6 (exists x (a(x) & -c(x))) [goal].
7 -a(x) | c(x). [deny(6)].
9 -b(x) | a(x). [clausify(4)].
10 -b(x) | -c(x). [clausify(5)].
11 b(c2). [clausify(2)].
12 c(x) | -b(x). [resolve(7,a,9,b)].
13 -c(c2). [resolve(10,a,11,a)].
16 c(c2). [resolve(12,b,11,a)].
17 $F. [resolve(16,a,13,a)].

Indeed it does. Unfortunately the proofs aren’t very pretty as everything is rewritten in normal forms.  One thing I want to play with is how non-classical logics may be embedded in this system.

A non-judgmental reconstruction of drunken logic

Simmons (2007) makes a helpful contribution to the logical modelling of real arguments by an addition of the shot glass modality to intuitionist logic.  A snippet:

Per Per Martin-Löf [7], something is true when witnessed by an object of knowledge, which lends itself to an obvious question of whether the truth of a proposition can be obviated by the presence of alcohol, seeing as alcohol has an clearly negative impact on one’s knowledge [1]. The possibility of the analytical truth of a proposition becoming questionable under the influence is also evidenced by discussion as to whether conference submissions that can be understood while drunk are novel enough to be worth accepting.

I think the following inference rule which I discovered while living in the homeland of Martin-Löf still requires further investigation:

Reference

Robert J. Simmons.  A non-judgmental reconstruction of drunken logic.  Presented at SIGBOVIK 2007, April 1, 2007. Winner of the Best Paper raffle. [PDF]

Visualising the lambda-calculus

Remember the -calculus? (I used to have as a logo on my phone—maybe one for a t-shirt…)

These are amazing [found via reckless intuitions]:

• Alligator eggs
• Visual Lambda (see also this thesis) [found via unprincipled explosion]

Dov Gabbay’s papers

Just noticed that Dov Gabbay’s webpage now (well, could have been for a while) has a load of his papers in PDF.  Also check out the wonderful interview with Dov Gabbay in Ta! and a more recent one in the Reasoner.  Here’s some of his advice on encouraging communication between different communities from the latter:

The different communities I mentioned before will communicate more to each other. But you can accelerate the process. For example, it can take ten years to a PhD student to find the connections between voting theory and belief revision, or you can go ahead and organize a conference on it! Sooner or later the communities will talk to each other. It is like a boy and a girl on a trip. They are very compatible and they like each other. Sooner or later something will happen, but you can accelerate it by putting them together in the same room the first night of the holiday. One way or the other, it will happen.

Fun with integer sequences (A016103)

I needed a closed form of twice the sequence defined by A016103 (see if you can guess the connection with psychology :-)). When looking at its description, “Expansion of “, I wasn’t quite sure how to calculate a closed form. But the nearby sequences helped narrow down the search to something of the form … and this seems to do the trick:

I found it by brute-force search, looking only at the first four terms of the sequence. This matches the sequence in the encyclopaedia entry at least for these values:

0, 1, 15, 151, 1275, 9751, 70035, 481951, 3216795

I can’t go further—I’m using R and haven’t yet found to time to get it to use long integers :-(

But aren’t sequences fun? :-)

It’s funny how the same names keep popping up…

I first heard of Per Martin-Löf through his work in intuitionist logic, which turned out to be important in computer science (see Nordström, Petersson, and Smith, 1990).  His name has popped up again (Martin-Löf, 1973), this time in the context of his conditional likelihood ratio test, apparently used by Item Response Theory folk to assess whether two groups of items test the same ability (see Wainer et al, 1980).  Small world.

References

Martin-Löf, P. (1973). Statistiska modeller. Anteckningar fran seminarier lasaret 1969–1970 utarbetade av rolf sundberg. Obetydligt ändrat nytryck, october 1973 (photocopied manuscript). Institutet för Säkringsmatematik och Matematisk Statistik vid Stockholms Universitet.

Bengt Nordström, Kent Petersson, and Jan M. Smith. (1990). Programming in Martin-Löf’s Type Theory. Oxford University Press.

Howard Wainer, Anne Morgan and Jan-Eric Gustafsson (1980).  A Review of Estimation Procedures for the Rasch Model with an Eye toward Longish Tests.  Journal of Educational Statistics, 5, 35-64

“Semantics”

“… there can hardly be any question that what ’semantics’ conveyed and conveys to the mind of the general reader is a theory of meaning, which Tarski’s theory most emphatically was not. By calling his theory ’semantics,’ Tarski opened the door to endless misunderstandings on this point. There has been significant damage to logic arising from such misunderstandings, from confusion of model theory or ’semantics’ improperly so-called with meaning theory or ’semantics’ properly so-called.”
—From Tarski’s Tort by John P. Burgess

More on interpretation

From a piece by Jonathan Wolff on academic humour (hat tip: the marvellous Leiter Reports):

The logician in question, the late George Boolos, used to give a lecture in which he went through a number of popular phrases that, when analysed in terms of standard logic, mean something quite different from how we normally understand them.

The example everyone remembers is the popular song lyric “everybody loves my baby, but my baby don’t love nobody but me”. From this, it logically follows that “I am my baby”.

I guess the idea is you formalise this as:

∀x. loves(x, My Baby)
∀x. loves(My Baby, x) → x = Me

In this formalisation, loves(My Baby, My Baby) follows from the first premise. Then from the second premise, we get My Baby = Me.

Hopefully Most Reasonable People restrict the domain over which the first x quantifies…

ETA: Actually I should have known there’d be individual differences in interpretation. See the comments.