Abstract
We discuss an axiomatic setup as an appropriate account to the intensional structure of epistemic convictions. This includes a resolution of the problem of logical omniscience as well as the individual rendering of knowledge by different persons.
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Notes
- 1.
Our approach reassembles ideas which one also finds in Doyle’s Truth Maintenance Systems (TMSs), [5]. As TMSs were developed in the context of expert systems in Computer Science, they soon fell victim to complexity issues. For us it is, however, just the qualitative setup which matters from a philosophical point of view. The quantitative aspect may go out of control when one tries to explain and store every single step of a derivation.
- 2.
For a fruitful discussion amoung different people it is, however, desirable that one can agree on the same starting points.
- 3.
See the reference to “Vorhalle der Geometrie” for work of Moritz Pasch, one of the founding fathers of modern axiomatics, in [24, p. 80].
- 4.
See [18] for a detailed discussion of AGM in our perspective.
- 5.
Also cited in [1, p. 261].
- 6.
The paragraph on logical omniscience in the article on Epistemic Logic in the Stanford Encyclopedia of Philosophy exposes here a certain helplessness [22, § 4].
- 7.
Of course, one may study an idealized notion of knowability which should be closed under logical omniscience. But, in our view, this form of idealization goes to far to provide a tool to take up challenges concerned with knowledge of a person. Thus, we are explicitely at odds with the first part of the justification of this idealization by Gabbay and Woods [7, p. 158]:
A logic is an idealization of certain sorts of real-life phenomena. By their very nature, idealizations misdescribe the behavior of actual agents. This is to be tolerated when two conditions are met. One is that the actual behavior of actual agents can defensibly be made out to approximate to the behavior of the ideal agents of the logician’s idealization. The other is the idealization’s facilitation of the logician’s discovery and demonstration of deep laws.
.
- 8.
There are other criticisms of the alternative approaches, like, for example, the “ontological overkill” of possible worlds [14], which we cannot discuss here. All these criticisms, of course, do not mean that those approaches do not have their merits; we just like to point to the conceptional difference with our account.
- 9.
Ryle’s distinction of knowing how and knowing that [23] points into another direction. But we share with him, at least, the opposition to a raw knowing that. We like to complement it by knowing why.
- 10.
This sketches the qualitative aspect of our account only; in the presence of a plausibility order of beliefs, for instance, one may revise first the basic convictions with lowest plausibility.
- 11.
The question how different derivations should compared with each other was rised by Hilbert in his 24th problem. This problem was not included in his famous problems lecture at the International Congress of Mathematicians in 1900 in Paris but remained unpublished in his notebook before it was discovered in 2000. Since then, it triggered a lot of a research, including the question of identity of proofs [10]. Our approch is not intended to contribute to a solution of Hilbert’s 24th problem, but rather the other way around: a satisfactory concept for identity of proofs may allow to abstract from the concrete derivations we are relying on.
- 12.
More moderately expressed: unpredictable. The situation is not too different from the question whether you can compute the roulette results of the Monte Carlo casino. Of course, you can do it for those outcomes which already took place in a finite time period; but you will not be able to compute it in advance.
- 13.
For knowledge, we tactically assume here logical reasoning, only. The case of inductive reasoning is—in the case of knowledge—more delicate as it questions the status of knowledge in general.
- 14.
When we subscribe the No False Lemmas condition, it goes without saying that also the “axioms” presupposed for the knowledge need to be true; this is not the case for the example discussed in [11, § 4].
- 15.
Although inductive reasoning, used to justify a universal statement which is supposed to be taken as an axiom, might enter here, rather than as part of the internal aspect of knowledge.
- 16.
Here, we like to express serious doubts that justifications, in terms of derivations, can be “learned” just statistically, in the same way, as it is unlikely that AI could statistically generate a C++ compiler.
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Research supported by the Udo Keller Foundation and by the Portuguese Science Foundation, FCT, through the project UIDB/00297/2020 (Centro de Matemática e Aplicações).
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Kahle, R. (2021). The Intensional Structure of Epistemic Convictions. In: Cleophas, L., Massink, M. (eds) Software Engineering and Formal Methods. SEFM 2020 Collocated Workshops. SEFM 2020. Lecture Notes in Computer Science(), vol 12524. Springer, Cham. https://doi.org/10.1007/978-3-030-67220-1_15
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