Title: The Logic of Knowledge and the Flow of Information

Abstract: In this talk I cover some work still in development which concerns the notions of information and knowledge as exemplified in Fred Dretske’s ‘Knowledge and The Flow of Information’. In particular, I cover (1) some work on the logic of information flow and (2) the issue of developing an epistemic logic which captures Dretske’s notion of knowledge as a semi-penetrating operator.

Making use of Greg Restall’s semantics for LP, he provided a nice translation of LP into the modal logic S5. This got me thinking about converting a K₃-to-modal logic translation I once came across into an LP-to-modal logic translation. Here is what I have so far: A translation of LP into modal logic.

Some of the philosophical conclusions of Allard’s presentation were interesting. Take the following:

LP is the logic of a fragment of the modal logic S5. As a consequence, Priest is wrong, as far as LP’s “theoretical account of negation” is concerned, when he states: “Dialethic logic, unlike modal logic, does [...] provide a genuine rival theory to that provided by classical logic”.

During question time Graham Priest raised a point which I think went something like this. Many logics can be translated into other logics. In this case, although there is a formal translation of LP into S5, he couldn’t see why this should detract from the significance of LP’s many-valued semantics basis.

This seems a fair point. I think that the difference between modal logic and LP relative to classical logic is marked. Obviously, classical logic and modal logic are both bivalent whereas LP is trivalent. But more than this, the standard translation of modal logic into first-order logic involves changing boxes into universal quantifiers and diamonds into existential quantifiers; the syntax changes but the meaning or gist of things is pretty much preserved. The translation of LP into modal logic does not have this same degree of meaning preservation.

Furthermore, although it is shown that “LP is the logic of a fragment of the modal logic S5“, the legitimacy of treating some logic as a fragment of another logic is something to bear in mind. Most logics going around can be translated into first-order logic; that doesn’t mean that the meaning significance of those logics can be dismissed.

It is reasonable to claim that the Logic of Paradox can do perfectly well without true contradictions, since it has a classical, two-valued semantics and therefore need not be interpreted under the assumption of dialetheism. That the formal system of LP can be reduced to a classical system is nothing new.

But going the other way, dialetheism needs LP, or at least has no reason to abandon it in favour of a classical translation. Besides the obvious resonance of the truth value b, meaning `true and false’, another thing that comes to mind is how the many-valued semantics for LP provide the basis for paraconsistent probability functions.

In closing, I wonder how a paraconsistent, LP-based modal logic would fare under this translation.

Plus some introductory material on channel theory

Rohit Parikh proposed a criterion for relevance in belief change, which AGM is not stringent enough to meet. An example given is as follows:

Let p,q be two distinct elementary letters, and put K = Cn(p, q). Then there is an AGM maxichoice contraction that puts [the contraction of K by p ] to be Cn(), thus eliminating not only p but also q from K.

Subsequent expansion of Cn() with ~p or ~q would then result in K = Cn(~p, ~q).

Whilst it is clear to see how some sense of relevance is violated here, I think that such results can be legitimate. Take a Grove sphere modelling of the above example. The possible world where both p and q are true is in the centre. The closest outer shell contains the possible world where both p and q are false. The second outer shell contains the two possible worlds where one is true and one is false. So the ordering is:

This ordering can be motivated in a very reasonable and natural way. Say p stands for ‘Tom went to the party’ and q stands for ‘Harry went to the party’.

A third person, Dick, believes that both Tom and Harry went to the party. He also believes that it is almost certain that Tom and Harry only go to parties if both of them are going. So his second most plausible possible world is the one in which neither go to the party. Therefore, if he is told that Tom did not go to the party, as well as contracting p, q will also be contracted.

In recent work [1] a quantitative account of semantic information based on truthlikeness measures was proposed; statement A yields more information than statement B when A contains more truth or is closer to the whole truth than B. Given a way to measure the information yield of a statement A, an important distinction to make is that between the information yield of A and its informativeness relative to a particular agent. As the simplest of examples, take three true propositions p₁, p₂ and p₃. Although the statement p₁ & p₂ has a greater quantitative information measure than p₃, given an agent that already has p₁ & p₂ in their database, the statement p₃ is going to be more informative than p₁ & p₂ for that agent.

In this paper the idea of agent-relative informativeness is explored. How informative some input statement is for an agent will not only depend upon its (1) information measure, but also on (2) what content the agent already has and also on what they do with the input if they accept it. In order to deal with (2), some framework for belief revision is needed. Thus agent-relative informativeness here involves a combination of truthlikeness/information measures and belief revision. As it so happens, within the last few years there has been some interest in investigating the relationship between the truthlikeness (verisimilitude) and belief revision programs [2, 3]. Continuing on from [1], in this paper the approaches to truthlikeness/information focused on are the Tichy/Oddie and Value Aggregate methods. The belief revision framework employed is predominantly the AGM one, though some alternatives are considered.

Apart from a general outline of the ideas associated with agent-relative informativeness, two further contributions of this paper are:

• Some results on the behaviour of the belief revision operations of expansion, revision and contraction with regards to truthlikeness information measurements.
• Suggestion of some formal frameworks to deal with conflicting sources of input, at least some of which by definition are going to be providing misinformation. This includes (1) combining screened belief revision with estimated information measurement and (2) development of a paraconsistent approach to belief revision.

References
[1] S. D’Alfonso. On quantifying semantic information. Information, 2(1):61 – 101, 2011. URL = <http://www.mdpi.com/2078-2489/2/1/61/>.
[2] Cevolani Gustavo and Franceso Calandra. Approaching the truth via belief change in propositional languages. In EPSA Epistemology and Methodology of Science: Launch of the European Philosophy of Science Association, pages 47 – 62, 2010.
[3] I. Niiniluoto. Theory change, truthlikeness, and belief revision. In EPSA Epistemology and Methodology of Science: Launch of the European Philosophy of Science Association, pages 189 – 199, 2010.