A psychoanalyst walks into a bar(red subject)

A psychoanalyst walks into a bar with a book on logic and set theory. He orders a whisky. And another. Twelve hours and a lock-in later, all he has to show for the evening is a throbbing headache and some indecipherable nonsense scribbled on a napkin.

That’s the only conceivable explanation for these diagrams from The Subversion of the Subject and the Dialectic of Desire in the Freudian Unconscious, by Jacques Lacan (published in the Écrits collection):

But, I hear you ask, surely this notation means something? After all, Lacan is famous and studied across the world, and f(x) is well-recognised as a function, f, applied to argument x. So the I(A) and s(A) must mean something?

Here is a brief interlude on functions. The Fibonacci sequence, which pops up in all kinds of interesting places in nature, can be defined as following:

f(0) = 0,
f(1) = 1,
f(n) = f(n-1) + f(n-2), for n > 1.

In English, this says that the first two numbers in the sequence are 0 and 1. The numbers following are obtained by summing the previous two: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …

If you tell it a number (e.g., 0, 1, 2, …) then it replies with the respective number in the sequence (first, second, third, …). It might look a bit scary if you haven’t seen the notation before, but check out these examples demonstrating how the arithmetic is carried out:

  • f(0)  =  0
  • f(1)  =  1
  • f(2)  =  f(1) + f(0)  =  1 + 0 = 1
  • f(3)  =  f(2) + f(1)  =  1 + 1 = 2
  • f(4)  =  f(3) + f(2)  =  2 + 1 = 3
  • f(5)  =  f(4) + f(3)  =  3 + 5 = 5
  • f(6)  =  f(5) + f(4)  =  5 + 3 = 8

The point here is that the function notation “does something”. It provides a way of defining and referring to (here, mathematical) objects.

Less well-known, but appearing in university philosophy courses, is the lozenge symbol, ◊, which means “possible” in a particular kind of logic called modal logic. It seems plausible that there is something meaningful here in Lacan’s use of the symbol too.

Here is Lacan, “explaining” his notation for non-mathematicians:

Huh?

Lacan doesn’t try to explain what the notion means; he doesn’t seem to want readers to understand. Maybe he is just too clever and if only we persevered we would get what he means. However, elsewhere in the same text Lacan uses arithmetic to argue that “the erectile organ can be equated with √(-1)”. Personally, I am unconvinced.

Alan Sokal and Jean Bricmont have written a book-length critique of Lacan’s maths and others’ similar use of natural science concepts. Having read lots of mathematical texts and seen how authors make an effort to introduce their notation, I think it’s entirely possible Lacan is a fraud. That might sound harsh, but just look at how he writes. I reckon anyone can see for themselves that Lacan is writing nonsense if they take a look and forget for a moment how famous he is.

 

Lightly edited 18 Sept 2018, hopefully making clearer!

Some troubling and interesting things about investigating reasoning

Competence models are typically created and explored by a small number of experts.  Boole, Gentzen, Kolmogorov, Ramsey, De Finetti, …  The authority can often be shifted to the mathematics.   However, although non-experts can usually understand a statement of the theorem to proved, often they can’t understand the details of the proof.

There are problems with being an expert.  If you stare too long at the formalism, then you lose your intuition, and can’t see why someone would interpret a task “the wrong” way.  Often there are a priori non-obvious interpretations.

And who decides what constitutes a permissible interpretation?  Some obvious ideas for this are open to debate.  For instance, is it always reasonable for people to keep their interpretation constant across tasks?  Or is it rational to change your mind as you learn more about a problem?  Is it rational to be aware of when you change your mind?

To complicate things further, various measures loading on g predict interpretations.  Does that mean that those who have better cognitive ability can be thought of as having reasoned to the correct interpretation?

Recognizing textual entailment with natural logic

How do you work out whether a segment of natural language prose entails a sentence?

There are two extreme positions on how to model what’s going on.  One is to translate the natural language into a logic of some kind, then apply a theorem prover to draw conclusions.  The other is to use algorithms which work directly on the original text, using no knowledge of logic, for instance applying lexical or syntactic matching between premises and putative conclusion.

The main problem with the translation approach is that it’s very hard, as anyone who has tried manually to formalise some prose will agree.  The main problem with approaches processing the text in a shallow fashion is that they can be easilly tricked,  e.g., by negation, or systematically replacing quantifiers.

Bill MacCartney and Christopher D. Manning (2009) report some work from the space in between using so-called natural logics, which work by annotating the lexical elements of the original text in a way that allows inference. One example of such a logic familar to those in the psychology of reasoning community is described by Geurts (2003).

The general idea is finding a sequence of edits, guided by the logic, which try to transform the premises into the conclusion.  The edits are driven solely by the lexical items and require no context.

Seems promising for many cases, easily beating both the naive lexical comparisons and attempts automatically to formlalise and prove properties in first-order logic.

References

Bill MacCartney and Christopher D. Manning (2009). An extended model of natural logic.  The Eighth International Conference on Computational Semantics (IWCS-8), Tilburg, Netherlands, January 2009.

Geurts, B. (2003). Reasoning with quantifiers. Cognition, 86, 223-251.

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

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:

$latex \frac{\Gamma \vdash A\mathit{, right?}}{\Gamma \vdash A}&s=2$

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]

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 $latex \frac{1}{(1-4x)(1-5x)(1-6x)}$”, 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 $latex a_1 \cdot b_1^n + a_2 \cdot b_2^n + a_3 \cdot b_3^n + a_4 \cdot b_4^n$… and this seems to do the trick:

$latex (4^n + 6^n – 2 \cdot 5^n) /2 $

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? 🙂