Logical omniscience, a key theorem of normal epistemic logics, states that the knowledge set of ordinary rational agents is closed for its logical consequences. Although epistemic logicians in general consider logical omniscience unrealistic, there is no clear consensus on how it should be restrained. The challenge is most of all conceptual: we must find adequate criteria for separating obvious logical consequences (i.e., consequences for which epistemic closure certainly holds) from non-obvious ones. Non-classical game-theoretic semantics has been employed in this discussion with relative success. On the one hand, based on urn semantics [5], an expressive fragment of classical game semantics that weakens the dependence relations between quantifiers occurring in a formula, we can formalize, for a broad array of examples, epistemic scenarios in which an individual ignores the validity of a given first order argument or sentence. On the other hand, urn semantics offers a disproportionate restriction of logical omniscience. Therefore, an improvement of this system is required to obtain a more accurate solution of the problem. In this paper, I propose one such improvement based on two claims. First, to avoid the difficulties faced by accounts of logical obviousness in terms of easy provability [e.g., 2, 3, 4], I argue that we should rather conceive logical knowledge in terms of a default and challenge model of justification [1, 6]. Secondly, I sustain that our linguistic competence in using quantifiers requires a sort of basic hypothetical logical knowledge that can be roughly formulated as follows: (R ∀) when inquiring on the truth-value of a sentence of the form ∀x p, an individual might be unaware of all substitutional instances that this sentence accepts, but at least she must know that, if an element a is given, then ∀x p holds only if p(a/x) is true. Both claims accept game-theoretic formalization in terms of a refinement of urn semantics. I maintain that the system so obtained (US+R ∀) affords an improved solution of the logical omniscience problem. To do this, I prove that it is complete for a special class of urn models and, subsequently, I characterize first order theoremhood in this logic. Based on this characterization, we will be able to see that the classical first order validities which are not preserved in US + R ∀ form a class of formulae such that we have semantic reasons to affirm that an individual can present perfect linguistic competence and still ignore their logical truth.
References
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[2] D’Agostino, M.”Tractable depth-bounded logics and the problem of logical omniscience”, pages 245–275,in H.Hosni and F. Montagna (eds.), Probability, Uncertainty and Rationality, Dordrecht: Springer, 2010.
[3] Jago, M. “Logical information and epistemic space”, Synthese, 167, 2 (2009): 327–341.
[4] Jago, M.”The content of deduction”, Journal of Philosophical Logic, 42,2(2009):317–334.
[5] Rantala,V. “Urn models: a new kind of non-standard model for first-order logic”, pages 347–366, in E. Saarinen (ed.) Game-Theoretical Semantics, Dordrecht: Springer, 1979
[6] Williams, M., Problems of Knowledge, Oxford: Oxford University Press, 2001.