Tag: representation

Modal logic: developments & applications in IT research areas

This post contains preliminary and very general research into recent developments in nonclassical (i.e. modal) logics and information technology and other relevant areas of study (namely, knowledge representation, computer programming, decision theory, artificial intelligence, verificationism)

A modal logic framework for multi-agent belief fusion

Liau, C. 2005. A modal logic framework for multi-agent belief fusion. ACM Trans. Comput. Logic 6, 1 (Jan. 2005), 124-174. DOI= http://doi.acm.org/10.1145/1042038.1042043

Keywords:
Epistemic logic, belief fusion, belief revision, database merging, multi-agent systems, multi-sources reasoning

ABSTRACT

This article provides a modal logic framework for reasoning about multi-agent belief and its fusion. We propose logics for reasoning about cautiously merged agent beliefs that have different degrees of reliability. These logics are obtained by combining the multi-agent epistemic logic and multi-source reasoning systems. The fusion is cautious in the sense that if an agent’s belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first, called level cutting fusion, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously. In the second, called level skipping fusion, only the level at which the inconsistency occurs is skipped. We present the formal semantics and axiomatic systems for these two strategies and discuss some applications of the proposed logical systems. We also develop a tableau proof system for the logics and prove the complexity result for the satisfiability and validity problems of these logics.

 

An internal semantics for modal logic

Fagin, R. and Vardi, M. Y. 1985. An internal semantics for modal logic. In Proceedings of the Seventeenth Annual ACM Symposium on theory of Computing (Providence, Rhode Island, United States, May 06 – 08, 1985). STOC ’85. ACM, New York, NY, 305-315. DOI= http://doi.acm.org/10.1145/22145.22179

ABSTRACT

In Kripke semantics for modal logic, “possible worlds” and the possibility relation are both primitive notions. This has both technical and conceptual shortcomings. From a technical point of view, the mathematics associated with Kripke semantics is often quite complicated. From a conceptual point of view, it is not clear how to use Kripke structures to model knowledge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the “internal semantics” of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, using modal structures. Furthermore, modal structures offer a much more intuitive approach to modelling knowledge and belief.

First-order classical modal logic: applications in logics of knowledge and probability

Arló-Costa, H. and Pacuit, E. 2005. First-order classical modal logic: applications in logics of knowledge and probability. In Proceedings of the 10th Conference on theoretical Aspects of Rationality and Knowledge (Singapore, June 10 – 12, 2005). R. van der Meyden, Ed. Theoretical Aspects Of Rationality And Knowledge. National University of Singapore, Singapore, 262-278.

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. We conclude by offering a general completeness result for the entire family of first order classical modal logics (encompassing both normal and non-normal systems).

Modal logic: developments & applications in IT research areas

This post contains preliminary and very general research into recent developments in nonclassical (i.e. modal) logics and information technology and other relevant areas of study (namely, knowledge representation, computer programming, decision theory, artificial intelligence, verificationism)

A modal logic framework for multi-agent belief fusion

Liau, C. 2005. A modal logic framework for multi-agent belief fusion. ACM Trans. Comput. Logic 6, 1 (Jan. 2005), 124-174. DOI= http://doi.acm.org/10.1145/1042038.1042043

Keywords:
Epistemic logic, belief fusion, belief revision, database merging, multi-agent systems, multi-sources reasoning

ABSTRACT

This article provides a modal logic framework for reasoning about multi-agent belief and its fusion. We propose logics for reasoning about cautiously merged agent beliefs that have different degrees of reliability. These logics are obtained by combining the multi-agent epistemic logic and multi-source reasoning systems. The fusion is cautious in the sense that if an agent’s belief is in conflict with those of higher priorities, then his belief is completely discarded from the merged result. We consider two strategies for the cautious merging of beliefs. In the first, called level cutting fusion, if inconsistency occurs at some level, then all beliefs at the lower levels are discarded simultaneously. In the second, called level skipping fusion, only the level at which the inconsistency occurs is skipped. We present the formal semantics and axiomatic systems for these two strategies and discuss some applications of the proposed logical systems. We also develop a tableau proof system for the logics and prove the complexity result for the satisfiability and validity problems of these logics.

 

An internal semantics for modal logic

Fagin, R. and Vardi, M. Y. 1985. An internal semantics for modal logic. In Proceedings of the Seventeenth Annual ACM Symposium on theory of Computing (Providence, Rhode Island, United States, May 06 – 08, 1985). STOC ’85. ACM, New York, NY, 305-315. DOI= http://doi.acm.org/10.1145/22145.22179

ABSTRACT

In Kripke semantics for modal logic, “possible worlds” and the possibility relation are both primitive notions. This has both technical and conceptual shortcomings. From a technical point of view, the mathematics associated with Kripke semantics is often quite complicated. From a conceptual point of view, it is not clear how to use Kripke structures to model knowledge and belief, where one wants a clearer understanding of the notions that are primitive in Kripke semantics. We introduce modal structures as models for modal logic. We use the idea of possible worlds, but by directly describing the “internal semantics” of each possible world. It is much easier to study the standard logical questions, such as completeness, decidability, and compactness, using modal structures. Furthermore, modal structures offer a much more intuitive approach to modelling knowledge and belief.

First-order classical modal logic: applications in logics of knowledge and probability

Arló-Costa, H. and Pacuit, E. 2005. First-order classical modal logic: applications in logics of knowledge and probability. In Proceedings of the 10th Conference on theoretical Aspects of Rationality and Knowledge (Singapore, June 10 – 12, 2005). R. van der Meyden, Ed. Theoretical Aspects Of Rationality And Knowledge. National University of Singapore, Singapore, 262-278.

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K) in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. We conclude by offering a general completeness result for the entire family of first order classical modal logics (encompassing both normal and non-normal systems).

‘Language as representational’ on my mind

Is language representational?  The question might be more carefully phrased as, need all languages be representational languages?

That question has been on my mind a lot.  Wittgenstein’s theories (if you can call them that) regarding language are close to my own, in some important ways.  Having said that, it’s hard to really say what Wittgenstein thought regarding just how, or when, language ought to be considered as a representational system.  I’ll leave that issue aside since my primary concern in this post is only to introduce what I take to be a very insightful presentation on the matters as I see them.

The author of the Nedcricology blog introduces the problems of representationalism in a very simple but intriguing way.

I. The representational interpretation of language:

In his Tractatus Logico-Philosophicus, Wittgenstein claims that all expressions susceptible to the ascription of truth or falsity are propositions. Propositions share the same structure and logic as states of affairs, hence their suitability for one another. Wittgenstein encourages thinking of a proposition as a picture, that we can just as easily communicate in pictoral form – and, of course, events lend themselves to being captured in picutres. Propositions represent various possible states of affairs, and true propositions represent actual states of affairs. We can accurately represent in propositional form whatever actually occurs. And added to that is Wittgenstein’s famous dictum: Whatever can be said, can be said clearly.

A few problems pop up:
a) We don’t always need “clear” pictures; not all pictures are representational or portraits – what about abstract expressionism?

b) Thinking of propositions as pictures does not entail that all meaningful sentences are pictures – what about flipping someone the bird?

c) Besides, how do propositions “match up” with states of affairs anyway?

These three problems suggest that the representational interpretation of language is either somewhat narrow or just downright incorrect. First of all, to capture someone, we (well at least I) don’t merely take portraits of them. I take a variety of pictures, and in fact sometimes even encourage them to take a few of their own pictures with my camera so I can see things from their point of view. Second, many of the ways in which we communicate involve little to no “representation,” such as when I say, “I gotta go!” and run towards the toilet. And finally, whatever connections there are between our more “representational” propositions and the world, they are not metaphysically necessary but are conventionally (and humanely, I might add!) important.

I take the third problem as the most pressing for theorists who support the idea, or rather can’t help but to presume it, that meaningful expressions necessary represent the thing(s) or states of affairs, or objects, they are ‘about’.  On the one hand, it is said that any p is true if it accurately represents the R which it is about; on the other hand, if p represents R, in virtue of what does the ‘representing’ obtain? Is it the relation of p and R, or is it it some property of one or the other only, such that it might be said that p inherently is able to represent R or R-type things?

How about propositions about mathematical entities.  ‘I think it’s a number.’ In what sense does my thought make ‘it’–the thing I am thinking of–a number or in what sense does my thought represent it as a number?  Doesn’t a number represent itself as itself without my thinking about it?  Or is its identity as a number contingent on a thought to express it as such?

Do we learn to use language representationally such that its function as a representional system is somehow more basic to its other possible functions–for instance, as ‘capable of emotive expression’ or ‘as a metaphoric system’ ?  To see how how misplaced this idea is, consider the following exchange:

Billy pointed his finger at the apple and said it look rotten.

OR

Billy: <points finger at the apple> It’s rotten!

OR

Billy says, as he points his finger at the apple, “That’s rotten”

Do all three expressions refer equally to the same state of affairs and if so, just what is that state?  On some level, it might appear that yes, the three expression do equally refer to the same state of affairs–namely, the state of affairs containing the individual Billy, who points to a particular apple and exclaims that it is rotten.

Then again it isn’t clear, is it, that in each case, the order of events is always the same.  For instance, in the third expression, the simultaneity of Billy’s act of saying and his act of pointing is emphasized whereas the matter isn’t completely settled in the first instance.  Does that mean that the first expression is comparatively lacking in descriptive value?

Perhaps the first expression is uttered in a different circumstance than the second.  The second looks as it if it belongs in a play, or in some sort of written dialogue.  The third looks more appropriate to a novel.  The first looks hard to place.  But then maybe they each represent different states of affairs, but if that’s the case, then how could we justifiably say that they mean more or less the same thing?

In any event, please do check out the post on the nedricology blog since it presents the case against ‘language as representational’ in a simple but sophisticated way.

Response to de Villiers’ Language for Thought: Coming to Understand False Belief

The following is a short response I wrote to de Villiers and de Villiers’ Language for Thought: Coming to Understand False Belief. (de Villiers, J.G., and P. A. de Villiers. Language for Thought: coming to understand False Beliefs. Chapter prepared for Whither Whorf? (in press)) You can view a version of it here, although I’m not sure it is the final version.

 

de Villiers and de Villiers, in Language for Thought, articulate the view that language is prerequisite to thought and not merely an effect of it. They focus exclusively on the issue of false belief and our ability to reason and form explanations about them. Specifically, the acquisition of language is a necessary condition for the ability to describe not the content of false beliefs (others’ false beliefs).

 

De Villiers and de Villiers offset their hypothesis that language is prerequisite for thought with the following dilemma: any appropriate experimental design results in either triviality or incoherence, depending on the criterion for acceptable results and/or the encouraging of participants (children, in this case) to use intentional language capable of describing false beliefs. (351) To resolve this tension, de Villieres and de Villiers propose two solutions:

 

1) Select tasks that do not require the explicit use of “linguistic complements”—the propositional content of an intentional expression—and thus accept responses that fail to denote ‘what about the belief is false’.

2) (a). Select tasks that require very little regarding the understanding of linguistic complements, so in effect children would merely be required to imitate (i.e. “repeat”)—and not grasp–the false intensional expressions they hear. (b) Inquire as to whether children have “mastered complements with nonmental verbs, such as verbs of communication that require precisely the same complement structures syntactically and semantically as mental verbs, but with none of the reference to invisible mental events.” (352)

 

I want to focus on the latter half of the second proposed solution. The authors seem to imply a sort of dualism concerning mental predicates such that so-called folk psychological states—i.e., intensional verbs—necessarily denote a state with content that cannot be confirmed in an empirical sense; hence de Villiers and de Villiers use of “invisible mental events”. This is the hallmark of 20th century theories of mental content-intentional states like to believe, to think, to remember, and to wish, are understood as states having objects that do not refer to anything physical and/or confirmable; at least not in the sense that “The ball is front of the desk” is.

 

Without getting into the matter of how best to think about the meaning of such expressions, it should be acknowledged that anyone, let alone children, need not be using intensional verbs in such a Cartesian way (‘Cartesian’ because such verbs are taken to denote mental, ‘invisible’ things). In many circumstances one might be disposed to say that his or her use of the predicate ‘to think that’ ought not be thought of as denoting a mental state but rather as merely ‘directing the audiences’ attention’. Here the meaning of intensional verbs becomes less mysterious and more socially embedded. Thus, the use of intensional verbs might be merely for emphasizing what follows the intentional verb. Compare “I think that the Patriots are too good” with “The Patriots are too good”: with regard to syntax alone, the latter expression would not fall under de Villiers and de Villiers’ notion of complement structures since it lacks an intensional verb conjoined with a corresponding ‘mental’ or representational content. In a room of crowded people, someone who uses ‘I think’ or ‘I wish’ might be more realistically be thought of as an attention-grabber. I suppose the use of intensional verbs might be looked at in both ways simultaneously, and certainly I don’t think that the two are incompatible.

 

That said, if it’s the case that, on many occasions, an individual might not use intensional verbs in the strict sense that the authors require in order to resolve the alleged dilemma, then they need to rethink just how pressing the tension is in the first place.

———-

Note: After reviewing this rather hasty response, I need to qualify my critique, to an extent. Yes, the description of intensional predicates as ‘invisible’ sounds or seems to imply a sort of Cartesian dualism-the fact is, the authors do not require a separate ontological category of “mental substance”, so its not entirely (that is, ontologically) dualistic.