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

INTERVAL DESCRIPTION OF CHANGE
Interval Description of Change
Volodymyr Navrotsky
Kharkov Institute of Management

ABSTRACT:

This paper concerns a description of change in the framework of tense interval logic. The main goal is to present an approach to the construction of tense paraconsistent logic. Recent investigations show that moments are inapplicable to the study of the phenomenal continuums: for many kinds of the quality changes, any subdivision of existence time of an object does not separate clearly the state before change from the state after it. It is not possible to determine the last moment of the prior state and the first moment of the posterior one. So, the predicates of natural language are not valued relatively to the moment of time. I consider change as that occupying an interval of time which has fuzzy boundaries. The description of change consists in the conjunction of the descriptions of those states which overlap an interval of change. As a result such description contains inconsistent statements.

MORE PARACONSISTENCY AND NATURAL LANGUAGES
Development of a Paraconsistent Knowledge-Based System

Classical logic predicts that everything (thus nothing useful at all) follows from a contradiction. A paraconsistent logic is a logic where a contradiction does not lead to such an explosion. Since in practice consistency is difficult to achieve there are many potential applications of paraconsistent logics in knowledge-based systems, logical semantics of natural language, etc. A long term aim is the study of a paraconsistent higher order logic and its applications in computer science as well as its use as a classical foundation of mathematics.

The starting point is the following work:

J?rgen Villadsen
Paraconsistent Knowledge Bases and Many-Valued Logic
International Baltic Conference on Databases and Information Systems
H.-M. Haav et al. (Editors), Tallinn, Estonia, 77-90, 2002 (Revised)

The goal is to develope a prototype of a paraconsistent knowledge-based system using the proposed many-valued logic, either by model checking or theorem proving techniques. In particular a case study in the domain of medicine could be considered.

See also:

J?rgen Villadsen
Supra-Logic: Using Transfinite Type Theory with Type Variables for Paraconsistency
WCP-2003 (III World Congress on Paraconsistency, Toulouse, France, 28-31 July 2003)

Natural Language Processing using Lexical and Logical Combinators

Computational linguistics is a research area with many challenges, cf. the recent CONTROL project: CONstraint based Tools for RObust Language processing.

The starting point is the following work:

J?rgen Villadsen
Using Lexical and Logical Combinators in Natural Language Semantics
The Brain and Self Workshop: Towards a Science of Consciousness
Journal of Consciousness Studies, Consciousness Research Abstracts, 51-52, 1997

The goal is to study natural language processing using a categorial grammar and a higher order logic for semantics. In particular relations to the CONTROL project could be persued.

See also:

J?rgen Villadsen
Nabla: A Linguistic System based on Multi-dimensional Type Theory
Ph.D. Thesis: Technical University of Denmark 1995

POSSESSION AND THE DOUBLE OBJECT CONSTRUCTI
Heidi Harley

Abstract:

This paper argues that double-object verbs decompose into two heads,
an external-argument-selecting CAUSE predicate (vCAUSE) and a prepositional element, PHAVE. Two primary types of argument are presented.
First, a consideration of the well-known Oerhle's generalization effects in English motivates such a decomposition, in combination with a consideration of idioms in ditransitive structures. These facts mitigate strongly against a Transform approach to the dative alternation, like that of Larson 1988, and point towards an
Alternative Projection approach, similar in many respects to that of Pesetsky 1995.
Second, the PHAVE prepositional element is identified with the prepositional component of verbal have , treated in the literature by Benveniste 1966; Freeze 1992; Kayne 1993; Gueron 1995. Languages without PHAVE do not allow possessors to c-command possessees, and show no evidence of a double-object construction, in which Goals c-command Themes.
On the current account, these two facts receive the same explanation: PHAVE does not form part of the inventory of morphosyntactic primitives of these languages.

CO-EXISTENCE OF OPPOSITES AND PARACONSISTENCY
An interesting debate is going on in another website.

Both da Costa and Priest in their texts tell us something about the History of Paraconsistent thought and how it became part of modern formal science. They also point to some traces of it before Jean Lukasiewicz and Nicolai I. Vasiliev. For da Costa et al one should trace it back to Aristotlian Logic itself:

`The origins of paraconsistent logics go back to the first systematic studies dealing with the possibility of rejecting or of restricting the law (or principle) of contradiction, which (in one of its possible formulations) says that a formula and its negation cannot be both true. The law of contradiction is one of the basic laws of traditional, or classical (Aristotelian) logic.'

Paraconsistency is something new in Logic and Mathematics though, I sense that it shomehow existed in one or other form in Non-Western cultures in the Ancient World.

One example is the ideal of perfection. Due to our Western traditions we tend to think that perfection is absence of contradiction. But in old eastern cultures perfection was seen as the co-existence of the opposites.

It is not just the Ying and Yang theory of Korean Philosophy. For instance, if you take Mary and the conception of Jesus, you see this idea of opposites co-existing. In that time the general mentality held two values as maximal female virtues: virginity and maternity. The fact that no woman could possess both virtues at the same time was then understood as reflex of human imperfection. When those who wrote the Gospells say that Jesus was born from a virgin mother, they mean that Jesus' conception reflected this sort of perfection, where two opposite maximal virtues co-exist by the action of a being that is above human limitations. I think you can see some paraconsistency in the Gospell.

So the roots of paraconsistency, as the roots of any tree, may have already spread over a very large area of human thought.

Tony Marmo, June 1, 2004 02:18 AM


Hmm, I've only read a little bit about paraconsistency (enough to decide I didn't want to go there :-)

But, isn't the emphasis on the consequences of an inconsistency, rather than whether an inconsistency exists? I thought the general (syntactic) definition of paraconsistency was

T,a,~a ~|- b

(from some theory T and an inconsistency we cannot derive some proposition b).

PS. We need a different topic heading for this discussion.

RdR, June 1, 2004 11:39 AM


But then you go into the formal scientific conception of paraconsistency and into the discussion about dialethism. What I pointed out is that certain ideas already pre-exist in some cultures, traditions or mentalities before they are introduced into some kind of science or formalism. In the example I gave, you can say the co-existence of opposites as a notion of perfection is in a sense a belief in dialethism.

Tony Marmo, June 1, 2004 12:27 PM


Is there a conception of paraconsistency that is not formal? I thought Graham Priest et al coined the term.

In any case, my point is about the emphasis of paraconsistency, as opposed to dialetheism (i.e. true inconsistency). Wider viewpoints on inconsistency seem more related to discussions of dialetheism than paraconsistency. It seems to me if we wish to compare other philosophies with paraconsistency, we have to say something about the consequences (or truths) that those philosophies arive at, from a perception of inconsistency.

BTW, I'm not sure that the co-existence of opposites is necessarily inconsistent. Heat and cold can co-exist as warmth. A glass can be half full and half empty. Inconsistency to me seems to require more than that.

RdR, June 1, 2004 02:24 PM


[1] Well, all formal concepts may be tracked back to informal notions. One of the merits of a scientist or philosopher is precisely his capacity to bring notions into formalisation. Formal concepts cannot be completely strange to human cognition before they are created. That is why one valid way of making a valid reasoning consists of departing from commonly accepted ideas. Another example of this informal-to-formal transition is that you need atomic entities or term that you do not define to make formal constructions. You do not define them, but people know what you mean because they have an informal notion of what you say. But, of course, something remains an informal notion until someone decides to make into a formal concept and even give it a proper name.

But mind you that I am not a philosopher of science.

[2] In the example I gave, the case of Mary the virgin mother, the possession of such opposite virtues is a contradiction. At least it is contradiction from the perspective of the `science' of that time, since, before genetic manipulation, it was unthinkable that a human female could bear a child without having had intercourse. The point is just how to write it so that you get an a and ?a forms. Of course, the opposite properties or predicates alone are not contradictions. But when you talk about a property of a certain individual, or predicate something of him/her, so that Px and ?Px you have a contradiction.

If x gave birth to a child, x had sex before (Sx). Mary (m) gave birth to a child, so Mary had sex before= T(a=Sm).

If x is virgin, x had not sex before (?Sx). Mary is virgin, so Mary had not sex fore = T(?a).

Then you get T(a) and T(?a), which is a contradiction. And this contradiction was regarded as token of perfection in the Gospell.

Tony Marmo, June 1, 2004 07:28 PM


PS2 Alternatively:

(I) If x is (still) a virgin (a), then x has not given birth (?b).

(II) If x given birth (b), then x is not a virgin (anymore) (?a).

Now:

(M)T(x is a virgin)=(a) and T(x has given birth)=(b).

But b->?a, so

(M) T(x is a virgin)=(a) and T(x is not a virgin) (?a).

Tony Marmo, June 1, 2004 09:00 PM


I like your last formulisation better. However, I still don't think I made my point clear. One last attempt and then I'll call it quits.

Let's say we accept your example of immaculate birth as an inconsistency. The "theory" is Christianity. To show a likeness to paraconsistency, I think you have to show that from Christianity and the immaculate birth some thing(s) do not follow.

From my lay viewpoint, I am not clear that this can be shown. The reason is that the inconsistency arose through an act of God, and I don't know to what extent Christianity argues that some things are not possible through acts of God.

To me, the likeness to paraconsistency is only present to the extent that some theory argues that not everything follows from an inconsistency. If it can be shown that Christianity had that kind of argument before Priest et al, then fine, there is a precedent for paraconsistency.

RdR, June 2, 2004 06:37 AM


Of course, Christians will not argue that there are things God will not be able to do. This remark of yours is right.

But mind that the authors of the Gospells intended to send a message to the humans, a set of (moral, spiritual or religious) teachings to them. They did not want to unvail the essence of God, but to prescribe patterns of behaviour and systems of values. Accordingly, they give us this contradiction of the idea of a virgin mother as a form or manifestation of an ideal of perfection: the conciliation of opposite virtues. When the authors of the Gospell wrote this passage, they wanted the readers to extract some conclusions and not others. They do not mean that any consequence is deriveable from the virgin-mother contradiction.

For instance, the sorts of consequences that they admit include the idea that Jesus is the saviour of human kind and is perfect, that human imperfection requires repentance and forgiveness, that you should not stone prostitutes, etc. The recurrent idea is that humans cannot reconcile opposite virtues and even if they try to do things right to the best of their ability their actions will yield imperfections. These are things the authors of the Gospell want to be consequences of the kind(s) of contradiction(s) they admit as true. These are opposed to other moral teachings the authors of the Gospells do not want to be extracted therefrom. Thus, the passage from Dan. ix. 9. should not be derived from the Gospell:

(1) To the Lord our God belong mercies and forgivenesses.

But if (1) is somehow incorporate into the set of teachings, then it must NOT be interpreted as (2):

(2) Only God can forgive.

The Christians then have to revise (1) and infer (3):

(3) Given that even God forgives people, humans should forgive one another.

Of course, nothing of this comes in any formal language or sort of formalism in the Gospells. But their authors certainly intuitively worked with the non-explicit notion that the consequences of the things they tried to teach were somehow controlled.

But you can say the same about other cases.

Tony Marmo, June 2, 2004 01:05 PM


To continue the debate go here.

POINT OF VIEW ROLES
Configurational Properties of Point of View Roles
Carol Tenny and Peggy Speas

The pragmatic force of a sentence and the pragmatic roles of discourse
participants have traditionally been considered to be peripheral to the
syntactic component of Grammar. Recently, there have been a variety
of proposals for syntactic projections that encode information relevant
to the interface between syntax and pragmatics. (Rizzi (1997),
Cinque (1999), Ambar (1999), among others). At the same time, linguists
have been exploring the various notions of pragmatic prominence or point
of view that are relevant to that interface. (Sells 1987, Zribi-Hertz 1989,
Tenny 1998, Speas 2000 etc.) Studies of this sort naturally raise questions
about the extent to which pragmatic information is syntactically represented.
After all, the idea that syntax encodes extensive pragmatic information
was rejected as being too unconstrained in the 1970s. On a separate track,
linguists have observed that sentience (also variously described as
animacy, subjectivity or experiencer-hood) plays an interesting role in the
grammar. (Kuno and Kaburaki 1977, Stirling 1993, Smith 2000)
However, these phenomena have been treated as involving pragmatics or
Discourse Representation; syntactic representation of sentience has been
largely limited to treatments such as associating lexical features for animacy
or logphoricity with individual lexical items. Our proposal will unify both
tracks: representation of sentience and representation of pragmatic
properties, under one syntactic approach.
We will argue that basic syntactic principles constrain projections of
pragmatic force as well as the inventory of grammatically relevant pragmatic
roles. We take our inspiration from the work of Hale and Keyser (1993,1998,
1999), Di Sciullo (1999, 1996), Travis (2000), Borer (1998), among others,
who have explored constraints on the mapping from Lexical Conceptual
Structure (LCS) to syntactic structure. Although there are interesting differences
among the proposals made by these authors, they seem to be converging
on two points: syntactic principles impose constraints on possible lexical
items and their projections, and semantic roles are not primitive, but are
determined within these basic asymmetric projections. We will argue that the
same basic structural principles that constrain lexical primitives and the
lexicon-syntax interface also operate on primitives of a Sentience Domain,
and restrict the pragmatics-syntax interface.
The above authors have offered theories of what can count as a
"grammatically relevant" thematic property. Our goal is to use their
insights to restrict what will count as a "grammatically-relevant" pragmatic
property. We will not be proposing a new theory of the specific structural
restrictions on the lexiconsyntax interface, and we don't offer much insight
into how one might choose among the existing theories. What we will
do is use Hale and Keyser's theory as a point of departure, and show
how the constraints they propose mediate the interaction between
syntax and pragmatics.