A SEMANTIC REINTERPRETATION OF BINDING THEORY


Non-Redundancy: A Semantic Reinterpretation of Binding Theory
Philippe Schlenker (UCLA & Institut Jean-Nicod)

Within generative grammar, Binding Theory has traditionally been considered a part of syntax. Recent attempts to bring it to semantics have tried to explore other sides of the phenomena. The syntactic treatment presupposes that some sentencial structures, which would otherwise be interpretable, are ruled out by purely formal principles. Thus

(S) He(i) likes him(i)

would, according to earlier standard generative theories, yield a perfectly acceptable interpretation, but it is ruled out by Chomsky's Condition B, which in this case prohibits co-arguments from bearing the same index. Philipe Schlenker explores a semantic alternative in which Condition B, Condition C, the Locality of Variable Binding of Kehler 1993 and Fox 2000, and Weak and Strong Crossover effects follow from a non-standard interpretive procedure (modified from Ben-Shalom 1996).

According to his proposals, constituents are evaluated top-down under a pair of two sequences, the sequence of evaluation s and the quantificational sequence q. The initial sequence of evaluation always contains the speaker and the addressee (thus if John is talking to Mary, the initial sequence of evaluation will be jEm, as Schlenker assumes throughout).

The bulk of the work is then done by a principle of Non-Redundancy, which prevents any object from appearing twice in any sequence of evaluation. One may think of the sequence of evaluation as a memory register, and of Non-Redundancy as a principle of cognitive economy that prohibits any element from being listed twice in the same register.

See also:
Strong Crossover Violations and Binding Principles

Postal, November, 1997

SCOPE AND SYNTACTIC ISLANDS
This is another paper by Schlenker on the most popular issue in Linguistics: scope readings that transcend syntactic islands.

Scopal Independence: On Branching & Island-Escaping Readings of Indefinites & Disjunctions
by Philippe Schlenker

Abstract

Hintikka claimed in the 1970's that indefinites and disjunctions give rise to `branching readings' that can only be handled by a `game-theoretic' semantics as expressive as a logic with quantification over Skolem functions.
Due to empirical and methodological difficulties, the issue was left unresolved in the linguistics literature.
Independently, however, it was discovered in the 1980's that, contrary to other quantifiers, indefinites may scope out of syntactic islands.

We claim that (i) branching readings and the island-escaping behavior of indefinites are two sides of the same coin: when the latter problem is considered in full generality, a mechanism of `functional quantification' (Winter 1998, 2003) must be postulated which is strictly more expressive than Hintikka's, and predicts that his branching readings are indeed real, although his own solution was insufficiently general.

Furthermore, (ii) we show that, as Hintikka had seen, disjunctions share the behavior of indefinites, both with respect to island-escaping behavior and (probably) branching readings. The functional analysis can thus naturally be extended to them.

Finally, (iii) we suggest that the functional analysis can and should be reinterpreted in terms of a mechanism of double quantification, according to which an indefinite may contribute (a) an existential quantifier which has narrow scope, but which (b) includes in its restrictor a definite description over identifying properties, i.e. properties which, given a certain number of individual arguments, hold true of exactly one object.

PROPOSITIONAL IDENTITY

PROPOSITIONAL IDENTITY AND LOGICAL NECESSITY

by David B. Martens
10 pages. AJL, March 12, 2004

In two early papers, Max Cresswell constructed two formal logics of propositional identity, PCR and FCR , which he observed to be respectively deductively equivalent to modal logics S4 and S5 . Cresswell argued informally that these equivalences respectively 'give ... evidence' for the correctness of S4 and S5 as logics of broadly logical necessity.
In this paper, I describe weaker propositional identity logics than PCR that accommodate core intuitions about identity and I argue that Cresswell's informal arguments do not firmly and without epistemic circularity justify accepting S4 or S5 . I also describe how to formulate standard modal logics ( K,S2 , and their extensions) with strict equivalence as the only modal primitive.

EXCURSIONS IN NATURAL LOGIC
Here is an interesting paper in the Linguistic-Logic Interface:

Excursions in Natural Logic
by Edward L. Keenan

Abstract

Empirically we present some novel entailment patterns in English, and begin to characterise them semantically.

Despite reliable judgments of entailment these patterns have gone largely unnoticed in work on philosophical logic and natural language semantics, possibly because many of the sentence pairs instantiating a pattern naturally invoke quantifier types not studied in standard logic, indeed not even definable in first order logic.

Also in one large class of cases the judgments of entailment rely on relations between pairs of quantifiers, so it is more natural to consider each pair as a binary quantifier rather than an instance of iterated unary quantifiers.

CONEIVABLE AND POSSIBLE WORLDS SEMANTICS
Here is a very interesting debate I've found:

(Source: Desert Landscapes)

Conceivable Worlds Semantics
by Uriah Kriegel

Here's an idea I've toyed with for years and never had the time and courage to pursue. What better forum to float it than this blog? (How liberating.)

Possible Worlds Semantics (PWS) is the thesis that the intension of a concept (or an expression) is given by a function from possible worlds to extensions. I suggest Conceivable Worlds Semantics (CWS): the intension of a concept (or expression) is a function from conceivable worlds to extension.

In PWS, "Hesperus" and "Phosphorus" have the same intension, because there are no possible worlds in which they have different extensions. But In CWS, they do have different intensions, because there *are* conceivable worlds in which they have different extensions. (This may need a qualification that would entail relativization to conceivers/speakers.) Ditto for "water" and "H2O."

In PWS, the concept of an Escher triangle has no intension, or else has the same intension as every other concept of an impossible figure, because in every possible world its extension is the empty set. In CWS, the concept of an Escher triangle has a proprietary intension.

Straight externalists will be horrified by these monstrous functions. Straight internalists (like myself) will be delighted. In any case, the notion of *narrow content* can be defined in terms of these functions. Two-dimensionalists can use these functions as their primary intensions. (I'm not sure if ultimately there is a difference between centered world and conceivable worlds, but anyway primary intension could be defined in terms of the function I have in mind.)

One problem with this idea is that it assumes the notion of a conceivable world. It is not clear how to define conceivable worlds, and the only definition that comes to mind appeals to acts of conceiving, which are themselves contentful and will therefore presuppose a notion of content. There are also ontological worries here: some may argue that there *are* no conceivable yet impossible worlds (alternatively: that our ontology should be committed to those).

I'm sure there are other problems as well. But the emerging semantics has (have?) a certain elegance and simplicity to it that makes me very fond of it.

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Reactions

It seems there must be a difference between centered worlds and conceivable worlds, since the latter is allowed to have Escher triangles and whatnot. Could the problem of presupposing content be avoided by just allowing the intension to be functions on impossible worlds? Then the problems would collapse into the problem of countenancing impossible worlds, but this might be worth the cost if the account could deliver substantial gains. I am having a hard time seeing what the substantial gains might be over 2d though, since 2d alleges to solve the Hes-Phos problem you mention. If CWS is divorced from 2d, one advantage is that CWS would not be subject to objections that arise to 2d from its allowing modal and epistemic operators to take different objects and from Kripkean objections to it's descriptivism-ish-ness. The CWS account would presumably still be subject to some problems that plague PWS, however.

Comment by Chris -- 6/11/2004 @ 1:07 pm

Sounds a lot like what two-dimensionalists are up to. See e.g. The Nature of Epistemic Space , which engages in more or less the project you describe. The main potential difference is that standard two-dimensionalism doesn't assign a nontrivial intension to expressions that are a priori incoherent, whereas maybe you're thinking that conceivable worlds are more fine-grained in a way that allows these to have nontrivial intensions. (The case of an Escher triangle sort of suggests this, though I'm not sure that this notion is a priori incoherent.) For a brief discussion of fine-grained modal spaces along those lines, see the final section ("Hyperintensionality") of the paper linked above.

Comment by djc -- 6/11/2004 @ 2:12 pm

I attended a nice paper by John Hawthorne at a modality conference last month where he argued that the notion of an epistemic counterpart was unclear, and that this theatened the connection between conceivability and metaphysical possibility that Dave and George Bealer have defended. This relates to your worry about how to define conceivable worlds, so you might want to keep an eye out for that paper when it appears.

Comment by LA Paul -- 6/11/2004 @ 4:59 pm

Thanks for the suggestion for getting around my first problem, Chris. In particular, I would like to take it in the following direction.

Suppose we deny (as I'm inclined to do anyway) the distinctive axiom of S4:

Lp => LLp; or MMp => Mp

(Where "L" and "M" are, respectively, the necessity and possibility operators.) Then some worlds which are actually impossible are nonetheless possibly possible.

We can then define a function from possibly possible worlds to extensions. This will be a different function from the one from *actually* possible worlds to extensions. (This is so because once we're denying the S4 axiom, the two funnctions will have different domains. I know next to nothing about function individuation, but I think different domains should be sufficient for different functions. Actually, since functions can be represented as sets of ordered pairs, and sets are different when they have different members, functions with different domains *must* be different.)

There is a problem, however. Not every possibly possible world is conceivable. At least I think not. Definitely some impossible worlds are inconceivable (e.g., worlds where 2+2=5). But perhaps one could argue that such worlds are not only *actually* impossible, but *necessarily* impossible (and hence not possibly possible). I have no idea how one would go about arguing for that, though. Welcome to the mess of S3!

Comment by Uriah -- 6/11/2004 @ 5:19 pm

I think S4 is false too; basically for the reasons Nathan Salmon gives in his `89 Phil Review paper, "The Logic of What Might Have Been." In a nutshell, the argument depends on the necessity of origins. One might think that table T could not have originated from a hunk of matter too different from the hunk that it actually originated from. Suppose that the hunk it actually originated from could have differed by 10 atoms, and it still could have been the ancestor of T. So we go to the world in which a table is made from the hunk that is 10 atoms different from the one that actually resulted in T. This table could have originated from a hunk of matter that was 10 atoms different from the hunk from which it was originated. So we go to a third world where the hunk is 10 atoms different than the hunk in the second world, and 20 atoms different from the hunk in the first world. Call this hunk `H'. It is possibly possible, but not possible, for T to originate from H. So the characteristic axiom of S4 is false. The cool part is that this is not an argument from vagueness with an induction premise, so it does not get automatically resolved with the problem of vagueness, as the Ship of Theseus objection to S4 would. The cool thing for CWS (IWS?) is that although a world where T originates from H is impossible, it is not `necessarily' impossible. (That is, it's in the space of possible worlds, though it is inaccessible from the actual world.) Perhaps this could be used to draw the distinction that you seek. Anyway, it is fun to think about!

Comment by Chris -- 6/11/2004 @ 7:40 pm

Continue

RESTALL'S ASSERTION-DENIAL PARADOX
THE ASSERTION-DENIAL PARADOX

Assertion, Denial, Accepting, Rejecting, Symmetry and all that.

Greg Restall himself describes this paper as a very polemical and summarises his arguments as follows:

Proponents of a dialethic or "truth-value glut" response to the paradoxes of self-reference argue that "truth-value gap" analyses of the paradoxes fall foul of the extended liar paradox: "this sentence is not true." If we pay attention to the role of assertion and denial and the behaviour of negation in both "gap" and "glut" analyses, we see that the situation with these approaches has a pleasing symmetry: gap approaches take some denials not to be expressible by negation, and glut approaches take some negations to not express denials. But in the light of this symmetry, considerations against a gap view point to parallel considerations against a glut view. Those who find some reason to prefer one view over another (and this is almost everyone) must find some reason to break this symmetry.

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Some reactions:

I'm wondering whether Priest could just bite the bullet, allowing both the assertion and denial of the Strengthened Liar. (He's already biting some hard bullets anyway.)

In your paper you present the following argument: "The aim of denial is untruth. Anything not true is fit for denial, and only untruths are fit for denial. But pick a sentence, like SL that the glut theorist takes to be both true and untrue. If the aim of denial is untruth, then this is to be denied. But wait a minute! According to the friend of gluts, SL is true, and to be accepted. But then its denial fails because only untruths are to be denied, not truths. So, SL falls into the overlap between what is to be denied and what is to be asserted."

You say that Priest's reaction to the argument should be that not all negations express denials. (Does he say this somewhere? Not in "In Contradiction", as far as I can tell, but I haven't read his entire oeuvre.) However, given that he thinks the truth of a negation is equivalent to the falsity of the negated sentence, and given that some sentences are both true and false, it seems to me that he ought to allow the denial, as well as the assertion, of paradoxical sentences.

It is not necessarily a failure of assertion to assert falsehoods or a failure of denial to deny truths, as long as you assert only {t} and {t,f} sentences and deny only {f} and {t, f} sentences.

Posted by: Jeff Johnson at June 10, 2004 11:37 AM

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Jeff, these are good points. He could bite the bullet and say that SL is to be asserted and denied, but in "What Not? A Defence of a Dialethic Theory of Negation" (in the Gabbay and Wansing Negation Volume ) he explicitly argues that denying is not asserting a negation, and in the liar (and strengthened liar) is a case in point.

I think that biting the bullet here would be a very bad thing for him. Suppose I have a valid argument from Ato Band I accept Aand the negation of B. Am I making a mistake, for Priest? The answer has to be no , that isn't sufficient for making a mistake, because in the case of the liar paradox, we accept the argument from Lto Las valid, yet we accept Land its negation.

So, could it be that I accept Aand I merely fail to accept B? Surely that's no good either, because then I'm making lots of mistakes all over the place just by being ignorant. (There are plenty of logical consequences of things that I accept that I fail to accept. Accepting isn't deductively closed.)

What else could he say? I think that the thing to say is that a mistake has been made if I accept Aand deny B. This works for the friend of gaps, the friend of gluts, and the foe of both, provided that we don't allow accepting and denial of the same thing. If Priest says that we should accept and deny the strengthened liar statement, then I don't see how logical consequence gets any grip on evaluating states of belief or acceptance.

Or that's how I see it, anyway.

Posted by: Greg Restall at June 10, 2004 12:14 PM

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OK. I've updated the paper with a little new section on another of Priest's arguments in favour of gluts over gaps. It's all about the law of the excluded middle and the law of non-contradiction. I've also added an expository section on multiple conclusion consequence, so everyone can get up to speed.

The paper is now up to version 0.7.

(After AAP2004 I'll let you know what Graham thinks of it.)
Posted by: Greg Restall at June 10, 2004 04:02 PM

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If you want to discuss it, go here.

NON-DISTRIBUTIVE LOGICS
The Geometry of Non-Distributive Logics

By Restall & Paoli

Francesco Paoli and Greg Restall have finished their paper 'The Geometry of Non-Distributive Logics', on which they have been working since the middle of 2001. The authors welcome comments.

PARACONSISTENCY AS COINCIDENTIA OPPOSITORUM

Paraconsistency and dialectics as
coincidentia oppositorum
in the philosophy of Nicholas of Cusa

by Marko Ursic, University of Ljubljana, Slovenia

[Note: I thank Stefano Ulliana for this reference]

Philosophy is a collection of big mistakes, but mistakes so seemingly close to an aspect of truth, that they require serious consideration as premises, at least until their consequences and revelations become temporarily exhausted.
(Florencio Asenjo, 1985)


There is an obvious conceptual connection between the modern concept of paraconsistency and the traditional term coincidentia oppositorum (coincidence of opposites) as the corner stone in the philosophy of Nicholas of Cusa (or Cusanus, 1401-64). When I was considering this connection, my attention was attracted by a couple of passages concerning Cusanus from some seminal recent books on paraconsistency. I have in mind especially the following three works: 1. Graham Priest, In Contradiction (1987), 2. Paraconsistent Logic , eds. G. Priest, R. Routley and J. Norman (1989), 3. Graham Priest, Beyond the Limits of Thought (1995). In (1) Cusanus is only mentioned among "the number of philosophers who have consciously believed explicit contradictions"; in (2) he is included into the Christian tradition of Neo-Platonism, and his coincidentia oppositorum is represented with a famous passage from Cusanus' major work De docta ignorantia ("Of Learned Ignorance", 1440) where the coincidence between maximum and minimum is stated:

...in no way do they [distinctions] exist in the absolute maximum [the One]... The absolute maximum... is all things and, whilst being all, it is none of them; in other words, it is at once maximum and minimum of being ( Of Learned Ignorance , I, 4).

In the third (3) of the mentioned books, a whole section of the first chapter (1.8) is devoted to Cusanus' thought, considered from the point of limits of _expression and (in)comprehensibility of God. Priest states that in Cusanus' philosophy we have a paradoxical, and _as he argues _also a "dialetheic" situation (Priest defines "dialetheia" as a true contradiction), since Cusanus "accepts this contradiction about God [i.e. incomprehensibility vs. comprehensibility] as true"; Priest points out that also in this case, as in many other philosophical cases, both contradictory claims, named by him "Transcendence" and "Closure", are true:

Moreover, even to claim that God is incomprehensible [Transcendence] is to express a certain fact about God. Hence we have Closure.

Cusanus, then, unlike Aristotle, not only perceives the contradictions at the limits of the expressible, but endorses them.

In general, I agree with Priest's conclusions _however, I think something more (or, maybe better to say, less ) should be said concerning the "dialetheism" of Cusanus, so the main object of this paper is to put forward this distinction. In the following discussion I prefer to use the traditional term dialectic(s) , adv. dialectical , because I think that Priest's term "dialetheism" has, at least from the epistemological point of view which I am concentrated on, almost the same or very close meaning as the historical concept of (Hegelian) dialectic: the contemporary "dialetheism" is supposed to be a logical reconstruction of classical philosophical dialectics, revival of dialectical methods of thinking and formalization of them by means of modern nonclassical logics.

One more introductory remark has to be put here: in recent literature of paraconsistency there is no quite unanimous, among paraconsistent logicians generally accepted distinction between paraconsistent and dialectical logical systems. Following Priest, we will say that a logical system is paraconsistent , if and only if its relation of logical consequence is not "explosive", i.e., iff it is not the case that for every formula Pand Q,Pand not- Pentails Q; and we will say a system is dialectical , iff it is paraconsistent and yields (or "endorses") true contradictions , named "dialetheias" (I take over this term from Priest, because it has no adequate classical equivalent). A paraconsistent system enables to model theories which in spite of being (classically) inconsistent are not trivial, while a dialectical system goes further, since it permits dialetheias, namely contradictions as true propositions. Still following Priest, semantics of dialectical systems provide truth-value gluts (its worlds or set-ups are overdetermined); however, truth-value gaps (opened by worlds or set-ups which are underdetermined) are considered by Priest to be irrelevant or even improper for dialectical systems. Beside that, sometimes the distinction is drawn between weak and strong paraconsistency, the latter considered as equivalent with dialectics. A reader of recent literature in this field may have an impression that dialectics as strong paraconsistency is more a question of ontology than of logic itself, namely that it states the existence of "inconsistent facts" (in our actual world) which should verify dialetheias. But it remains an open question whether, for example, semantical paradoxes express any "inconsistent facts".

Now let us go to Nicholas of Cusa. The question is: can we claim that Cusanus is a dialectical philosopher, can we say that his coincidentia oppositorum is a precursor of Hegelian dialectic and eo ipso of contemporary dialectical logic, formally (re)constructed by Priest and other paraconsistent and/or dialectical logicians? In the following discussion I am arguing that the epistemological attitude of Cusanus, expressed by himself as docta ignorantia , precludes any simple (or "categorical") affirmation of contradictions, as well as, of course, Cusanus does not accept the simple negation of them in the manner of the classical (Aristotelian) logic. This point can be expressed also in this way: docta ignorantia does not affirm contradictions just simpliciter , but ambigue - namely, Cusanus' opposita , forming an "endorsed" contradiction, are both true or both false, depending on how we understand them. The term "dialetheia", when applied to Cusanus, should be taken - differently from Priest - in a double sense, applied not only to truth-value gluts, but also to truth-value gaps: a contradiction as coniunctio oppositorum is true not only if its opposites are both true, but also if they are both false. (Formally, this revised concept of dialetheia means that the rejection of the Law of Non-Contradiction entails the rejection of the Law of Excluded Middle.) Indeed, a typical Cusanus' dialetheia, for example the conjunction of "Transcendence" ( P) and "Closure" (not- P), mentioned above, has always two sides, like Janus' head: from its "positive side" (leading to the "positive way", traditionally called via positiva ), its opposites are both true (i.e., the propositional conjunction ' Pand not- P' is true); but if we consider dialetheia from its "negative side" (leading to the "negative way", traditionally called via negativa ), its opposites are both false (i.e., the propositional binegation 'neither Pnor not- P' is true). My point here is that just this ambiguity of dialetheias is essential for understanding the "middle way" of Cusanus - the way directed by his basic epistemological insight and maxim: docta ignorantia . We will return to this point later.

We always meet difficulties when we try to interpret an ancient informal wisdom with our modern formal means. Cusanus' coincidentia oppositorum has not been written in the formal language, even less it presented a well-defined logical system. So it is certainly difficult to determine its "underlying" logic, since "it is only in contemporary times that a clear conception of a formal or semantical system has developed." Nevertheless, we can surely claim that the underlying logic of Cusanus' philosophy is not Aristotelian, but (at least) paraconsistent _in the sense, outlined above, namely that the relation of logical consequence (albeit informal one) in Cusanus' philosophical thought is not "explosive": docta ignorantia surely admits a philosophical theory which is inconsistent and non-trivial - such a theory is Cusanus' philosophical "system" itself. Let us call it (the system of) Docta Ignorantia (DI) and ask: is (DI), being paraconsistent, also dialectical? The answer is not so obvious as it seems from Priest's passages concerning Cusanus. In order to see the problem more clearly, we have to examine some relevant passages from Cusanus' great work De docta ignorantia.

When we try to understand Cusanus' philosophy from the point of view of modern logic(s), we must not forget the following: God, named as maximum , is, by coincidentia oppositorum , also minimum , however, this concidentia is incomprehensible for human reason ( ratio ), for our discursive, logical thinking _yet it is in an unthinkable transcendent way present to our mind ( mens ,intellectus ), namely by an intellectual intuition, philosophical contemplation. The incomprehensibility of coincidentia oppositorum for human reason (for our logical, even dialectical thinking) is considered by Cusanus to be essential for his philosophy. Here are two relevant passages:

Maximum absolutum incomprehensibiliter intelligitur, cum quo minimum coincidit. (De docta ignorantia , Book I, Chapter 4)

Supra omnem igitur rationis discursum incomprehensibiliter absolutam maximitatem videmus infinitam esse, cui nihil opponitur, cum qua minimum coincidit. (Ibid.)

From the point of view of Cusanus it would be a mistake to think "positively" (or simpliciter ) the coincidence of opposites _since reason, using the principle of non-contradiction, actually cannot think coincidentia oppositorum which is supra omnem rationis discursum (i.e., "beyond the limits of thought"); and that is why it cannot be rationally decided whether opposites are both true or both false. This point is very important for understanding Cusanus' docta ignorantia .

However, on the other hand, Cusanus is not a mystic, he is a great philosophical thinker who _like his brothers in spirit: Plotin, Eriugena, Kant, Wittgenstein, Nagarjuna and others _"manages to say a good deal about what cannot be said". How does Cusanus manage to do it?

In his last work De apice theoriae ("Of the Summit of Contemplation", 1464), as well as many times before, Cusanus wrote:

Posse igitur videre mentis excellit posse comprendere. ( De ap. th. , ch. 10)

However, what does it mean _videre mentis ? It is easier to say what it does not mean as what it actually means. (Needless to remark, this is one of the most difficult classical philosophical questions.) For Cusanus, "to see by mind" means neither a rational cognitive act nor just sitting and contemplating in silence. Mens (and/or intellectus , the distinction between them is not sharply outlined in Cusanus' works) by contemplating "sees" symbols which "transfer" mind from their positive, finite meaning (being immanent in the world, articulated in language) to infinite transcendence, beyond any positive meaning and distinction.

And here Cusanus is especially interesting: for him, the most important philosophical "symbols" are provided by mathematics (mostly by geometry as the dominant mathematical discipline in those times). Cusanus based his metaphysical "intuitions" on geometrical symbolic models. Of course, he considered mathematics in its ancient (Platonic and Pythagorean) sense, namely as the clearest reflection of the universal order, of the World of Forms, _nevertheless, his idea that in the mirror of mathematics as "symbolic thinking" metaphysical and/or theological truths can be "seen" by the intellectual intuition, is new in the pre-Renaissance philosophy, and it is inspiring nowadays as well. Cusanus wrote:

Consensere omnes sapientissimi nostri et divinissimi doctores visilibia veraciter invisibilium imagines esse atque creatorem ita cognoscibiliter a creaturis videri posse quasi in speculo et in aenigmate. Hoc autem, quod spiritualia per se a nobis inattingibilia symbolice investigentur, radicem habet ex his, quae superius dicta sunt, quoniam omnia ad se invicem quandam nobis tamen occultam et incomprehensibilem habent proportionem, ut ex omnibus unum exsurgat universum et omnia in uno maximo ipsum unum. (De docta ignorantia , I, 11).

And in this symbolic way of contemplating God's incomprehensible and infinite being mathematics play a very important role:

...si finitis uti pro exemplo voluerimus ad maximum simpliciter ascendendi, primo necesse est figuras mathematicas finitas considerare cum suis passionibus et rationibus, et ipsas rationes correspondenter ad infinitas tales figuras transferre... (Ibid., 12).

One of the most famous mathematical "figures" of Cusanus which he used for symbolic representation of coincidentia oppositorum is the coincidence of ("the maximal") circle and a straight line (tangent); this coincidence is the "incomprehensible" limit of the sequence of larger and larger circles. Let's quote Cusanus' comment to this "figure":

...quare linea recta AB erit arcus maximi circuli, qui maior esse non potest. Et ita videtur quomodo maxima et infinita linea necessario est rectissima, cui curvitas non opponitur, immo curvitas in ipsa maxima linea est rectitudo. Et hoc est primum probandum. (Docta ignorantia , I, 13).

This model ("symbol") of coincidentia oppositorum can be advanced by including triangles: the Triangle with "the maximal angle" coincides with the straight line and with the Circle; this is supposed to be a reductio ad perfectionem of geometrical objects, since: Circulus est figura perfecta unitatis et simplicitatis. (Doc. ign., ch. 21; "The circle is a perfect figure of unity and simplicity.", op. cit. , p. 46), and just in the "infinite circle" the coincidence of opposites reveals itself in the most manifest, although still "symbolic" way:

Haec omnia ostendit circulus infinitus sine principio et fine aeternus, indivisibiliter unissimus atque capacissimus. ... Patet ergo centrum, diametrum et circumferentiam idem esse. Ex quo docetur ignorantia nostra incomprehensibile maximum esse, cui minimum non opponitur. Sed centrum est in ipso circumferentia. (Ibid.)

We could go on with Cusanus in his geometrical symbolism by introducing the infinite Sphere instead of the Circle: "...centrum maximae sphaerae aequatur diametro et circumferentia..." ( Doc. ign., ch. 23), but for our purpose the Circle will do. Let us denote this "maximal" Circle whose centrum est in ipso circumferentia with Greek capital letter Omega , and - making a sort of thought experiment _suppose that Omega can be an object of thought (an idea in the Lockean sense, without any heavy ontological commitment); then we put a pair of Kantian questions which lead to an antinomy, similar to Kant's first antinomy:

(Q) Is Omega finite?

Answer: It seems reasonable to assert YES, since every circle is finite, even "the maximal"; it is irrelevant if its center coincides with its circumference.

(Q') Is Omega infinite?

Answer: Again it seems reasonable to assert YES, since how could it be finite if its circumference is nowhere and its center everywhere ? Therefore (by reductio ): if ?is not finite, then it is infinite.

continue

100TH ISSUE OF PESQUISA FAPESP
The Sao Paulo State Foundation for the Support to Research (Fapesp) has published its monthly magazine in Portuguese, English and Spanish for many years.
In June of this year, there comes the one hundredth print edition of Pesquisa Fapesp in Portuguese, a volume of 174 colourful and fully illustrated pages in Portuguese with many interesting articles in the several ramifications of human knowledge.

It celebrates the scientific achievements and discoveries of the past ten years.
It brings an interview with the vice-CEO of Alellyx Applied Genomics and molecular biologist Professor F. Reinach about the foretold scientific revolution of the XXIst Century (page 38).

It has an article about Sodre Jr. and de Oliveira's discovery of four stellar nurseries located outside of any Galaxy (page 106).

It contains a retrospect of the fieldwork on tropical diseases done by pioneering biologists and physicians, like
Lutz, Ribas, Chagas and Cruz (page 82).

It also remembers the 70th anniversary of the consolidation of the formerly independent Colleges into the University of Sao Paulo (page 108).

LINGUISTICS: NEW PAPER BY FOX & PESETSKY
Cyclic Linearization of Syntactic Structure
Danny Fox & David Pesetsky (2004)
[to appear, Theoretical Linguistics , special issue on Object Shift in Scandinavian; Katalin E. Kiss, ed.]

Note: this paper contains about 1/3 of the material that will form part of a monograph, in prep. For the remaining 2/3, the best current source is the lengthy handout below.

Cyclic Linearization and the Typology of Movement (handout by the same authors)

Related work:
Cyclic Linearization and Asymmetry in Scrambling
by Heejeong Ko
Pseudo-gapping and Cyclic Linearization
by Shoichi Takahashi