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Friday, 20 October 2006

The Concept of Mathematical Elucidation:
theory and problems

By José Seoane

There is a contrast between concepts which may be treated in accordance with the criteria of mathematical rigor and concepts which are not susceptible of such a treatment. We will call theoretical concepts to the former and pre-theoretical to the latter. In mathematical world, sentences which relate theoretical and pre-theoretical concepts (in a determinated way) are denominated thesis; sentences which relate only theoretical concepts (in a determinated way) are denominated theorems. Elucidatory processes in mathematics have thesis as their principal output. I intend to establish in this paper that the introduction of the concept of mathematical elucidation has an important theoretical value to the effects of studying a certain special type of intended conceptual relation and a certain type of justificatory argumentation for it. The analysis of the contrast between thesis and theorems will allow us to construct a context where the interest for those conceptual relations and their supporting justificatory mechanisms arises naturally. Then, I will attempt to offer some structural features of mathematical elucidation qua conceptual relation and their impact on the strategies of elucidatory justification. This is what I grandiloquently call theory in the heading. I will suggest also a classification of the problems which a theoretical reflexion as the one proposed may contribute to clarify and I will make some brief observations on some paradigmatic examples of each one of the categories of the classification constructed. This work should be considered as a modest definition of a sort of research program.

Source: CLE e-prints


Posted by Tony Marmo on 18:22 BST | post your comment (0) | link to this post
Wednesday, 21 July 2004
Here I share some pieces of the current draft version of one of my articles on opacity in natural languages. I shall post one excerpt or two by day. Hope you like it. Comments are wellcome.


1. Prelude

1.1 General Considerations

A. The Issue

In this work I shall examine some aspects of the semantic phenomenon called opacity from the perspective of human languages in their common usage (rather than artificial languages or usages created by Logicians), relating it to the manner such languages, as computational systems, equip their users with the tools and the techniques to handle (pseudo-) paradoxes and explosions caused by contradictions. Although I resort to the work of Logicians and Philosophers, the principles and theoretic notions herein proposed to formalise such phenomena are primarily hypotheses respecting the inherent machinery of human languages, rather than merely invented solutions to approach the issues in question.
The latu sensu notion of opacity can be initially figuratively characterised as the phenomenon of a sentential context not allowing the light of a semantic/logic principle to pass through, i.e., a certain context is opaque because a certain (mode of) inference is not visibly valid therein.
There have been some more specific and/or stronger hypotheses trying to define actual instantiations of this notion occur and/or to predict when and to explain why they occur. Indeed, as far as I know, there have been at least two basic approaches on opacity.
The first basic approach on opacity, which is herein called classic or traditional, and which will be questioned in Section 2, assumes the definition of opaque context given in (1) below or variants thereof: Although (1) has never been accepted by important academic factions, like the Russellian philosophers among others, it is still the most spread conception in the literature:

(1) Opaque context (classic version)
A sentential context C containing an occurrence of a term t is opaque, if the substitution of co-referential terms is an invalid mode of inference with respect to this occurrence. (See Mckinsey 1998, Quine 1956)

The second basic view, which will be approached in Section 4, revolves around the idea of non-symmetry of accessibility relations. This second view has more adepts among linguists.
The alternative view I shall sketch here attempts to determine how fundamental opacity is and to relate it to issues of consistency and non-paradoxical interpretation in the common usage of natural languages.

Posted by Tony Marmo on 14:01 BST | post your comment (0) | link to this post
Sunday, 11 July 2004
Purver and Ginzburg shed some light on the Semantics of Noun Phrases, from the perspective of the HPSG school, which I both respect and dissent from:

Clarifying Noun Phrase Semantics

Matthew Purver and Jonathan Ginzburg

Reprise questions are a common dialogue device allowing a conversational participant to request clarification of the meaning intended by a speaker when uttering a word or phrase. As such they can act as semantic probes, providing us with information about what meaning can be associated with word and phrase types and thus helping to sharpen the principle of compositionality. This paper discusses the evidence provided by reprise questions concerning the meaning of nouns, noun phrases and determiners. Our central claim is that reprise questions strongly suggest that quantified noun phrases denote (situation-dependent) individuals-or sets of individuals-rather than sets of sets, or properties of properties. We outline a resulting analysis within the HPSG framework, and discuss its extension to such phenomena as quantifier scope, anaphora and monotone decreasing quantifiers.

Download link


Yoad Winter on Choice Functions

Winter's page has many papers, and his concerns include computational linguistics. It is worthy to check it. One of his recent works, More...

Posted by Tony Marmo on 07:06 BST | post your comment (0) | link to this post
Wednesday, 7 July 2004

Jackendoff talk: semantics must be generative

by nick on Sunday, June 06, 2004 6:54 AM

On Friday (4th) I heard Ray Jackendoff give the keynote lecture at a conference organised by the UCL Centre for Human Communication which my department ( UCL Phonetics and Linguistics ) is part of (in some way I don't understand).

What he said may not be news to anyone else, but I hadn't heard it, not having read any of his recent stuff, except the bits about music.

Broadly, he thinks that mainstream - ie Chomskyan - linguistics is on the wrong track by supposing that syntax is the only generative component needed in the grammar, so that phonology and semantics need only interpret the output from syntax.

On the phonology side, he thinks that auto-segmental phonology is on the right track, since it is a separate generative system with a 'dirty connection' to syntax, in the sense that there is no one-to-one mapping between phonological and syntactic structure.

Jackendoff wants to treat semantics in the same way, as an independent generative component, linked with syntax, and - separately - with phonology, by constraints, not isomorphic mappings.

This has a number of interesting consequences, one of which is that words become constraints across components of the language faculty, which seems like a neat idea. Another is that in the evolution of language, syntax might have come last to mediate a previously direct link between rich conceptual structure and phonology, rather than first as Hauser, Chomsky and Fitch (2002) seem to propose.

I'm with Jackendoff on the last point - it doesn't make sense to me that recursion would develop first in syntax to link a conceptual system with a sound system, because a conceptual system without recursion wouldn't be worth linking with anything. If you couldn't think, you wouldn't have anything to say.

But I don't think agreeing with this commits anyone to Jackendoff's position. It seems to me that he assumes that anything (or, presumably, anything propositional) that you can think must be representable in linguistic semantics. But that doesn't seem right. All that you can really say is that anything (propositional) you can think must be representable in the Language of Thought (assuming there is such a thing). The semantics of sentences, on the other hand, are notoriously underspecified relative to the Language of Thought, and if Sperber and Wilson's recent work is right, the same goes for the semantics of lexical items. (Since the contribution a lexical item ends up making to the propostion expressed by an utterance will generally be an ad hoc concept reached by narrowing and/or broadening the concept encoded by the lexical item.)

So it's perfectly consistent to think that we need to have a richly structured conceptual system for propositional thought, and that some parts of this must predate a structurally rich linguistic syntax (or there would have been nothing for the syntax to express) - without thinking that linguistic semantics needs to be generative in its own right. It could just be read off LF, which seems the simplest assumption.

I'll give one example to try to make this clearer. Jackendoff mentioned Pustejofsky 's work on lexical semantics. The basic idea is something like this: You interpret 'begin' differently in 'I began the book' and 'I began the beer' or, say, 'The goat began the book'.
It's clear (I think) that something like this is the case at the level of the thought formed in interpreting utterances of these sentences, but it's a huge (and apparently unjustified) step from there to say that those differences in the meaning of 'begin' should be encoded in the lexical semantics.

Permanent link.

Posted by Tony Marmo on 06:05 BST | post your comment (0) | link to this post
Wednesday, 30 June 2004


Knowledge and Stability

by Joe Shieber
June 08, 2004

Marc Moffett has been considering some interesting questions concerning knowledge and stable belief and justification at Close Range. In response to some probing questions, he submitted a follow-up post, including the following example :

The other day I was going out of town and was supposed to call some friends when I got into the airport. My wife wrote their number down and I glanced over it. As I was leaving, she reminded me to take the number. I said, 'I know it' and proceeded to recite it from memory. Knowing that the number was still fresh in my mind her response was, 'Do you really know it?'

Marc suggests that the example shows that knowledge sometimes requires not simply reliably-produced true belief (let's grant that the short-term memorial faculty allowing Marc to rattle off the number correctly is reliable), but stable belief, or stably justified belief. Marc claims that we have an intuitive grasp of stability and instability to which he can appeal in making this suggestion. However, and without meaning to be difficult, I still don't know what stability is; nevertheless, let's leave this problem aside.

What I want to do here is suggest an alternate diagnosis for Marc's example.Continue

Knowledge Discourses and Interaction Technology

by Carsten S?rensen & Masao Kakihara

Research within knowledge management tends to either overemphasize or underestimate the role of Information and Communication Technology (ICT). Furthermore, much of the ICT support debate has been shaped by the data-information- knowledge trichotomy and too focused on repository-based approaches.
We wish to engage in a principled debate concerning the character and role of knowledge technologies in contemporary organizational settings. The aim of this paper is to apply four perspectives on the management of knowledge to highlight four perspectives on technological options. The paper presents, based on four knowledge discourses --four interrelated perspectives on the management of knowledge-- four perspectives on ICT support for the management of knowledge each reviewing relevant literature and revealing a facet of how we can conceptualize the role of technology for knowledge management.
The four technology discourses focus on the: Production and distribution of information; interpretation and navigation of information; codification and embedding of collaboration; and establishment and maintenance of connections.Continue

The Relationship Between Knowledge and Understanding

by Michelle Jenkins

I've been thinking a lot lately about the relationship between knowledge and understanding. Knowledge and understanding, I think, are quite different sorts of things. My general grasp of the nature of understanding is influenced largely by the Ancients. One understands something if she 1) is able to provide a comprehensive explanation 2) has a systematic grasp of all of the information and 3) can defend her explanation against any questions or criticisms. First, in order to understand something I must be able to provide a comprehensive explanation of it. A physicist, for example, who understands the theory of relativity must be able to provide an explanation about why the theory of relativity is as it is, how it works, how it affects a variety of other physical laws and observations, and so forth. Second, to understand something, one must be able to `see' the relationship between different bits of information in the whole of the field to which the bit of information belongs. You must have a systematic grasp of the information relating to the matter at hand such that you see that the information, and the relationships that the different bits of information have with one another, forms an almost organic whole. Thus, a car mechanic who understands why the part of the car is making the sound that it is, has this understanding because he has a systematic grasp of the whole of the vehicle. He knows how the different parts relate to each other and how and in what ways certain conditions will affect both the different parts of the vehicle and the vehicle as a whole. This ties closely into the need for a comprehensive explanation. The physicist (or car mechanic) is able to provide a comprehensive explanation about the thing that she understands because she understands and `sees' the thing as a whole, as part of a complete system. Finally, in order to understand something, one must be able to defend her claim against any criticisms that are leveled against it. This defense must itself be explanatory. One cannot defend her view by pointing to the words of another, but must defend it by demonstrating an ability to look at the issue in a variety of ways and as part of a systematic whole. She is not proving her certainty with regard to an issue, but is demonstrating her understanding of the issue. In defending her view successfully, she demonstrates a reliability and stability within her account. Apparent in this account of understanding (I hope!) is that one must have a rather large web of information about the matter which one claims to understand. In order to develop and defend a suitably comprehensive explanation, one must be able to employ a huge number (and variety) of facts and bits of knowledge that relate to the thing she claims to understand. And, as the systematicity requirement shows, that web of information must be structured in a systematic manner.Continue

Not Every Truth Can Be Known:
at least, not all at once

According to the knowability thesis , every truth is knowable. Fitch's paradox refutes the knowability thesis by showing that if we are not omniscient, then not only are some truths not known, but there are some truths that are not knowable. In this paper, I propose a weakening of the knowability thesis (which I call the "conjunctive knowability thesis") to the effect that for every truth pthere is a collection of truths such that
(i) each of them is knowable and
(ii) their conjunction is equivalent to p.

I show that the conjunctive knowability thesis avoids triviality arguments against it, and that it fares very differently depending on one other issue connecting knowledge and possibility. If some things are knowable but false , then the conjunctive knowability thesis is trivially true. On the other hand, if knowability entails truth, the conjunctive knowability thesis is coherent, but only if the logic of possibility is quite weak.Continue

Some Thoughts About the Relationship Between Information and Understanding

Michael O. Luke
Paper to be presented at the American Society for Information Science Conference, San Diego, CA, May 20-22, 1996

That there is a relationship between information and understanding seems intuitively obvious. If we try to express this relationship mathematically, however, it soon becomes clear that the relationship is complex and mysterious. Knowing more about the connection, however, is important, not the least because we need more understanding as our world becomes faster paced and increasingly complex. The influence of increasing the amount of information, increasing the effectiveness of information mining tools and ways of organizing information to aid the cognitive process are briefly discussed.Continue

Posted by Tony Marmo on 02:22 BST | post your comment (0) | link to this post
Monday, 21 June 2004

Due to some technical problems, I am moving this blog. For ne posts, please go to the new address of this blog.

Posted by Tony Marmo on 20:20 BST | post your comment (0) | link to this post
Tuesday, 15 June 2004
When it comes about subjunctives, the first meaningful problem that comes to my mind is the famous deflationist claim that the Tarkian sentence below:


[S] The sentence `the snow is white' is true iff the snow is white in a certain world.

merely involves a disquotation device, does not apply if the sentence is produced in a Romance language. Accordingly, there is an important and large semantic difference clearly captured by a tense distinction between the quoted and non-quoted statement:


[S'] A frase "a neve ? branca" ? verdadeira se e somente se a neve for branca num certo mundo. (Subjunctive Future for=will be)

[S"] A frase "a neve ? branca" ? verdadeira desde que a neve seja branca num certo mundo. (Subjunctive Present seja=that it be)

[S"'] **A frase "a neve ? branca" ? verdadeira se e somente se a neve ? branca num certo mundo. (Indicative Present ?=is)

(The * sign marks ungrammaticality.)

There are more issues involved beyond this one, as pointed out by many linguists. Here comes an interesting paper on the Subjunctive issue:

The Lazy (French)man's Approach to the Subjunctive

(Reference to Worlds, Presuppositions and Semantic Defaults in the Analysis of Mood: Some Preliminary Remarks)

by Philippe Schlenker

First draft, UCLA & IJN (last modified: June 10, 2004; new title, no other modifications for the moment).


It has proven rather difficult to provide a unified semantics for the French subjunctive (the difficulty applies more generally to Romance, but we concentrate on French). In this preliminary note, we suggest that this is because the French subjunctive is a semantic default, to be used just in case the indicative would have triggered a presupposition failure. Thus the environments in which the subjunctive appears do not form a natural class, although they are the complement of a natural class. Once this is established, a large part of the question becomes: what is the semantic contribution of the indicative? Modifying minimally the analysis of Stalnaker 1975 (which was concerned with English), we suggest that the indicative triggers a presupposition on the value of a world term w, of the form w{CS}, indicating that the world denoted by w lies in the Context Set of individual x' at time t' in world w' (x', t', and w' may be left free -- if the context provides them with a salient value -- or they may be bound). This derives indirectly the intuition, found both in traditional grammar and in recent research (e.g. Farkas 2003), that the indicative marks an assertive act on somebody's part, though this person need not be the speaker. We also discuss an extension of this theory to the German Konjunktiv I, which we analyze in essence as a reportive indicative, in line with the intuitions though not with the implementation of Fabricius-Hansen & Saeb?? 2004. If correct, the theory we sketch makes it possible to analyze mood by analogy with person and tense as introducing a presupposition on the value of word-denoting terms, and in particular on world-denoting variables.

If you want to comment on this paper, you may also go to Kai von Fintel's blog.


I would like to hear from the German native speakers whether they agree or not to the following statements from Schlenker's paper:

There is a further piece to this puzzle. As noted in Schlenker 2003a and Fabricius-Hansen & Saebo
2004, the Konjunktiv I cannot be used when the thought or assertion is attributed to the speaker at the
time and in the world of utterance (the following is from Schlenker 2003a):

(63) a. *Ich glaube, da? Maria krank sei

I believe that Maria sick is-KONJ1

b. Ich glaubte, da? Maria krank sei

I believed that Maria sick is-KONJ1

`I believed that Maria was sick'

c. Peter glaubt, da? Maria krank sei

Peter believes that Maria sick is-KONJ1

`Peter believes that Maria is sick'

d. Peter glaubte, da? Maria krank sei

Peter believed that Maria sick is-KONJ1

`Peter believes that Maria is sick'

This suggests that the Konjunktiv 1 is -despite its name- an indicative, though with the special
requirement that the Context Set it refers to should not be that of the actual speaker at the time and in the
world of his utterance. We also obtain in this way the observation that the Konjunktiv I cannot occur in
conditionals, since the Context Set which is relevant for conditionals is always that of the speaker at the
time and in the world of utterance.

Do you agree with the indicative analysis of the Konjuctiv 1?

Posted by Tony Marmo on 13:17 BST | post your comment (0) | link to this post
Updated: Friday, 18 June 2004 03:39 BST

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

Posted by Tony Marmo on 12:50 BST | post your comment (0) | link to this post
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


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.

Posted by Tony Marmo on 12:41 BST | post your comment (0) | link to this post
Monday, 14 June 2004


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.

Posted by Tony Marmo on 02:01 BST | post your comment (0) | link to this post
Updated: Monday, 14 June 2004 02:02 BST

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