Tag: logic

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

Weary?

“Discussions of minor problems in traditional logic are in these days generally regarded as a bore. They are commonly prefaced by an apology, and the people who chance to see such things in print, wearily remark, ‘Good Lord, yet another one!’ Such is the fashion. We are apt to forget that boredom is often the shadow cast by an enthusiasm; and that the wailing from him who is bored, is the complaint of one who can neither escape from, nor enter into, the interest which, for these very reasons, is to him so exasperatingly stale and unprofitable. This does not mean that all men are under a moral obligation to delight in what interests any one of them. But neither should the lone individual tremble at the possibility that what interests him may weary others. Rather let him state his interest plainly, and to those who fail to share it, wish a kindly ‘God speed’ as they depart.”

– Roelofs, H. D. (1927). The Distribution of Terms. Mind, 36, 281-291.

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.

Success

“This series of lectures on proof-theory is a priori dedicated to mathematicians and computer-scientists, physicists, philosophers and linguists ; and, since we are no longer in the XVI—not to speak of the XVIII—century, it is doomed to failure. […] This being said, plain success is not the only possible goal ; mine might simply be the exposition of a disorder in this apparently well-organised universe, in which logic eventually took its place between two beer mugs and the Reader’s Digest, and does not disturb, no longer disturbs—a sort of fat cat purring on the carpet.”

–Jean-Yves Girard, The Blind Spot

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 and 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.

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…

A Connectionist Computational Model for Epistemic and Temporal Reasoning

Many researchers argue that logics and connectionist systems complement each other nicely. Logics are an expressive formalism for describing knowledge, they expose the common form across a class of content, they often come with pleasant meta-properties (e.g. soundness and completeness), and logic-based learning makes excellent use of knowledge. Connectionist systems are good for data driven learning and they’re fault tolerant, also some would argue that they’re a good candidate for tip-toe-towards-the-brain cognitive models. I thought I’d give d’Avila Garcez and Lamb (2006) a go [A Connectionist Computational Model for Epistemic and Temporal Reasoning, Neural Computation 18:7, 1711-1738].

I’m assuming you know a bit of propositional logic and set theory.

The modal logic bit

There are many modal logics which have properties in common, for instance provability logics, logics of tense, deontic logics. I’ll follow the exposition in the paper. The gist is: take all the usual propositional logic connectives and add the operators □ and ◊. As a first approximation, □P (“box P”) means “it’s necessary that P” and ◊P (“diamond P”) means “it’s possible that P”. Kripke models are used to characterise when a model logic sentence is true. A model, M, is a triple (Ω, R, v), where:

  • Ω is a set of possible worlds.
  • R is a binary relation on Ω, which can be thought of as describing connectivity between possible worlds, so if R(ω,ω’) then world ω’ is reachable from ω. Viewed temporally, the interpretation could be that ω’ comes after ω.
  • v is a lookup table, so v(p), for an atom p, returns the set of worlds where p is true.

Let’s start with an easy rule:

(M, ω) ⊨ p iff ω ∈ v(p), for a propositional atom p

This says that to check whether p is true in ω, you just look it up. Now a recursive rule:

(M, ω) ⊨ A & B iff (M, ω) ⊨ A and (M, ω) ⊨ B

This lifts “&” up to our natural language (classical logic interpretation thereof) notion of “and”, and recurses on A and B. There are similar rules for disjunction and implication. The more interesting rules:

(M, ω) ⊨ □A iff for all ω’ ∈ Ω such that R(ω,ω’), (M, ω’) ⊨ A

(M, ω) ⊨ ◊A iff there is an ω’ ∈ Ω such that R(ω,ω’) and (M, ω’) ⊨ A

The first says that A is necessarily true in world ω if it’s true for all connected worlds. The second says that A is possibly true if there is at least one connected world for which it is true.

A sketch of logic programs and a connectionist implementation

Logic programs are sets of Horn clauses, A1 & A2 & … & An → B, where Ai is a propositional atom or the negation of an atom. Below is a picture of the network that represents the program {B & C & ~D → A, E & F → A, B}.

A network representing a program

The thresholds are configured so that the units in the hidden layer, Ni, are only active when the antecedents are all true, e.g. N1 is only active when B, C, and ~D have the truth value true. The thresholds of the output layer’s units are only active when at least one of the hidden layer connections to them is active. Additionally, the output feeds back to the inputs. The networks do valuation calculations through the magic of backpropagation, but can’t infer new sentences as such, as far as I can tell. To do so would involve growing new nets and some mechanism outside the net interpreting what the new bits mean.

Aside on biological plausibility

Biological plausibility raises its head here. Do the units in this network model – in any way at all – individual neurons in the brain? My gut instinct says, “Absolutely no way”, but perhaps it would be better not even to think this as (a) the units in the model aren’t intended to characterise biological neurons and (b) we can’t test this particular hypothesis. Mike Page has written in favour of localists nets, of which this is an instance [Behavioral and Brain Sciences (2000), 23: 443-467]. Maybe more on that in another post.

Moving to modal logic programs and nets

Modal logic programs are like the vanilla kind, but the literals may have one of the modal operators. There is also a set of connections between the possible worlds, i.e. a specification of the relation, R. The central idea of the translation is to use one network to represent each possible world and then apply an algorithm to wire up the different networks correctly, giving one unified network. Take the following program: {ω1 : r → □q, ω1 : ◊s → r, ω2 : s, ω3 : q → ◊p, R(ω1,ω2), R(ω1,ω3)}. This wires up to:

A network representing a modal logic program

Each input and output neuron can now represent □A, ◊A, A, □~A, ◊~A, or ~A. The individual networks are connected to maintain the properties of the modality operators, for instance □q in ω1 connects to q in ω2 and ω3 since R(ω1, ω2), R(ω1, ω3), so q must be true in these worlds.

The Connectionist Temporal Logic of Knowledge

Much the same as before, except we now have a set of agents, A = {1, …, n}, and a timeline, T, which is the set of naturals, each of which is a possible world but with a temporal interpretation. Take a model M = (T, R1, …, Rn, π). Ri specifies what bits of the timeline agent i has access to, and π(t) gives a set of propositions that are true at time t.

Recall the following definition from before

(M, ω) ⊨ p iff ω ∈ v(p), for a propositional letter p

Its analogue in the temporal logic is

(M, t) ⊨ p iff t ∈ π(p), for a propositional letter p

There are two extra model operators: O, which intuitively means “at the next time step” and K which is the same as □, except for agents. More formally:

(M, t) ⊨ OA iff (M, t+1) ⊨ A

(M, t) ⊨ KA iff for all u ∈ T such that Ri(t,u), (M, u) ⊨ A

Now in the translation we have network for each agent, and a collection of agent networks for each time step, all wired up appropriately.

Pages 1724-1727 give the algorithms for net construction. The proof of soundness of translation relies on d’Aliva Garcez, Broda, and Gabbay (2002), Neural-symbolic learning systems: Foundations and applications.

Some questions I haven’t got around to working out the answers to

  • How can these nets be embedded in a static population coded network. Is there any advantage to doing so?
  • Where is the learning? In a sense it’s the bit that does the computation, but it doesn’t correspond to the usual notion of “learning”.
  • How can the construction of a network be related to what’s going on in the brain? Really I want a more concrete answer to how this could model the psychology. The authors don’t appear to care, in this paper anyway.
  • How can networks shrink again?
  • How can we infer new sentences from the networks?

Comments

I received the following helpful comments from one of the authors, Artur d’Avila Garcez (9 Aug 2006):

I am interested in the localist v distributed discussion and in the issue of biological plausibility; it’s not that we don’t care, but I guess you’re right to say that we don’t “in this paper anyway”. In this paper – and in our previous work – what we do is to say: take standard ANNs (typically the ones you can apply Backpropagation to). What logics can you represent in such ANNs? In this way, learning is a bonus as representation should precede learning.

The above answers you question re. learning. Learning is not the computation, that’s the reasoning part! Learning is the process of changing the connections (initially set by the logic) progressively, according to some set of examples (cases). For this you can apply Backprop to each network in the ensemble. The result is a different set of weights and therefore a different set of rules – after learning if you go back to the computation you should get different results.

Do we reason when we think we reason?

Quick comment on David Miller, Do We Reason When We Think We Reason, or Do We Think?, Learning for Democracy 1, 3, 2005, 57-71.

Miller’s central conjecture is that it is not logical thinking or reasoning which drives intelligent thinking forward, but rather blind guessing, intuitive thinking. Conjectures don’t come from reasoning, and conjectures are what allow us to make progress. This is contrary to the doctrine of followers of critical thinking who ignore conjecture formation and argue that reasoning is all about justification and trying to persuade, “an attitude,” Miller suggests, “that reeks of authority, of the attitude of a person who wants to teach rather than to learn” (p. 62); they also hold that critical thinking is about finding flaws in arguments – Miller argues that it should be about finding flawed guesses.

I agree, with some caveats.

Miller makes the assumption that since a conclusion of a deductive inference is “implicitly or explicitly” included within its premises, that nothing new is discovered by drawing the conclusion. Every deductive argument, says Miller, is “question begging”. This can be defeated with a mathematical example. Given some set of axioms, e.g. Dedekind-Peano arithmetic, it is very difficult to prove anything that’s not trivially true. In fact many trivially true statements are difficult to prove! Drawing “question begging” inferences can be tricky and informative. However even in purely deductive mathematical reasoning, conjecture forming is crucial, so requires some sort of guessing of the flavour suggested by Miller. Proving statements in theories which include mathematical induction, for instance, often requires the proof of lemmas which need to be speculated somehow.

It is clear the premises of a deductive argument have to come from somewhere. This is the easiest way to attack deduction and show that it is not identical to “thinking”. A valid argument from a set of premises which are not true is useless. The moon is provably made from brie if we slip a contradiction into our premises (and use a logic in which B follows from A and ~A). But drawing inferences from a set of premises allows us to understand more about what they mean, how the different bits of knowledge we have relate to each other.

Also logic consists of more than rules of inference, premises, and conclusion to prove. Somehow the bits have to be glued together, often with a search mechanism of some kind, to draw the conclusions.

I don’t think it’s accurate to say that we don’t reason when we generate new conjectures. It may not feel like reasoning as a book on logic or probability describes it but the brain could very well still be doing something which can be accurately modelled using logic or probability. The missing ingredient is perception (a big chunk of which is top-down, dare I suggest deductive?), how we modify according to the environment we’re in. This, I reckon, allows us to grow new deductive machinery.

Now could it be that the search mechanism is what does the guessing for us, generates the conjectures?

The reasoning Miller discusses seems to be of the very conscious flavour, i.e. our culturally evolved reasoning technology. In a deductive calculus perhaps? We’re “reasoning” if and only if we’re consciously aware of doing something which resembles reasoning. So given this viewpoint on reasoning, a valid question to ask could be, would learning logic/probability help us to be more creative, say? Help us in our conversations? But I think reasoning systems developed by mathematicians and others can also be useful to analyse what we’re doing when it doesn’t feel like we’re reasoning.