Concluding Remarks

According to the account of epistemic justification developed in this book, one has propositional justification for p just in case one is in no position to know that one is in no position to know p; and one has doxastic justification for p just in case one is in no position to know that one does not know p. The account gives internalists much of what they want from a theory of justification—in particular, a notion of justification according to which propositional justification is non-factive and luminous, underwrites principles of positive and negative introspection, and remains available to the victims of systematic deception. All the while, that notion is explained in terms of other notions that clearly belong to the knowledge-firsters’ toolkit, and coheres with an externalist account of the grounds for justification.

Outline of a theory of justification

Core theses of the novel account of justification to be developed are first stated: one has propositional justification for p just in case one is in no position to know that one is in no position to know p; and one has doxastic justification for p just in case one is in no position to know that one does not know p. Unlike other theories that conceive of justification in terms of the metaphysical possibility of knowing, the present account thus construes it as a distinctive kind of epistemic possibility. It treats propositional justification as non-factive, both its presence and its absence as luminous conditions, and by assuming a weak non-normal modal logic for knowledge and being in a position to know, validates principles of positive and negative introspection for it. The account thereby attributes features to justification that internalists care about. But it does so without construing justification as an internal condition. The account allows one to systematically distinguish between the condition of being justified and the metaphysical grounds for its obtaining, thereby heeding externalist insights into the difference between the good cases and the bad cases envisaged by radical scepticism. Lines of argument that show the account’s potential, e.g. in dealing with the preface and lottery paradoxes, are previewed, and so are lines of defence against challenges and objections, including prominent anti-luminosity arguments.

Group belief

While logical formalizations of group notions of knowledge such as common and distributed knowledge have received considerable attention in the literature, most approaches being based on modal logic, group notions of belief have received much less attention. In this paper we systematically study standard notions of group belief under different assumptions about the properties of belief. In particular, we map out (lack of) preservation of belief properties against different standard definitions of group belief. It turns out that what is called group belief most often is not actually belief, i.e. does not have the properties of belief. In fact, even what is called group knowledge is sometimes not actually knowledge either. For example, under the common assumption that belief has the KD45 properties, neither common belief (does not satisfy the negative introspection axiom 5) nor distributed belief (does not satisfy the consistency axiom D) are not actually belief. There has been some confusion in the literature regarding soundness of proposed axiomatizations of logics with distributed knowledge, related to the mentioned lack of preservation. In this paper we also present detailed completeness proofs of sound and complete axiomatizations of KD45 with distributed belief, both with and without common belief.

Evidence of Evidence in Epistemic Logic

The slogan ‘Evidence of evidence is evidence’ is obscure. It has been applied to connect evidence in one situation to evidence in another. The link may be diachronic or interpersonal. Is present evidence of past or future evidence for p present evidence for p? Is evidence for me of evidence for you for p evidence for me for p? The chapter discusses intra-perspectival evidential links. Is present evidence for me of present evidence for me for p present evidence for me for p? Unless the connection holds between a perspective and itself, it is unlikely to hold between distinct perspectives. Evidence will be understood probabilistically, using formal models from epistemic logic. Bridge principles between first-level and higher-level epistemic conditions often imply versions of controversial principles, such as positive and negative introspection. Formalizations of intra-perspectival principles that evidence of evidence is evidence have similarly implausible connections.

SMALL INFINITARY EPISTEMIC LOGICS

We develop a series of small infinitary epistemic logics to study deductive inference involving intra-/interpersonal beliefs/knowledge such as common knowledge, common beliefs, and infinite regress of beliefs. Specifically, propositional epistemic logics GL (Lα) are presented for ordinal α up to a given αo (αo ≥ ω) so that GL(L0) is finitary KDn with n agents and GL(Lα) (α ≥ 1) allows conjunctions of certain countably infinite formulae. GL(Lα) is small in that the language is countable and can be constructive. The set of formulae Lα is increasing up to α = ω but stops at ω We present Kripke-completeness for GL(Lα) for each α ≤ ω, which is proved using the Rasiowa–Sikorski lemma and Tanaka–Ono lemma. GL(Lα) has a sufficient expressive power to discuss intra-/interpersonal beliefs with infinite lengths. As applications, we discuss the explicit definability of Axioms T (truthfulness), 4 (positive introspection), 5 (negative introspection), and of common knowledge in GL(Lα) Also, we discuss the rationalizability concept in game theory in our framework. We evaluate where these discussions are done in the series GL(Lα), α ≤ ω.

Scholastická logika „vědění“ I.

Fourteenth-century logic gave rise, among others, to the genre De scire et dubitare, which offered a unified framework for discussing different forms of epistemic sophisms by utilising the underlying systems of epistemic logic. One of the problems introduced in this context already by the founding father of this genre, William Heytesbury, was the so-called axiom of positive introspection, i.e., the principle that an agent who knows that something is the case, knows that she knows that it is the case. Owing to Heytesbury’s enormous popularity in the subsequent centuries, discussion of this problem became relatively widespread. This debate was addressed already in Boh’s seminal Epistemic Logic in the Later Middle Ages, which, despite its limitations acknowledged by its author, is a standard source. The present study elaborates on Boh by extending the corpus of his works (both in the sense of including new authors and of utilising manuscripts along with printed editions) and drawing new connections based on that. The core of the survey consists of an analysis of the positions of William Heytesbury and John Wyclif (both pertaining to the context of Merton College), their Italian reception by Peter of Mantua, and the “continental” reception of Heytesbury by John of Holland. The main goals of this study are to formalise the key arguments, which makes it possible to address the underlying systems of epistemic logic and their respective “strength”, and to articulate the conceptual background of those arguments and systems (the concepts of evidence, attention, and order of cognitive operations). The gist of the debate is, on one of the sides, an attempt to prove that it is impossible to doubt whether one knows that something is the case by employing whether the principles of positive introspection and of distribution of knowledge over implication, or the principles of positive and negative introspection combined.

Scholastická logika „vědění“ II.

Further development of the research on the fourteenth-century logic of iterated modalities (Heytesbury, Wyclif, and Peter of Mantua) leads to further exploration in fifteenth- and sixteenth-century Italian scholasticism, in particular, the contributions of Paul of Venice and his followers (including Paul of Pergula, Cajetan of Thiene, and Domenico Bianchelli). The research confirms the well-established notion of “British logic in Italy”, as the major logical strategies used in the analysed works can be traced back to earlier British authors. Logically speaking, the problem of iterated epistemic modalities (such as knowledge and doubt) was framed as debate on the consistency of the hypothesis that an agent doubts whether she knows φ and the hypothesis that an agent knows φ and doubts whether she knows φ, in which the principles of positive and negative introspection play a major part. Philosophically speaking, the debate on the possibility of doubting one’s own knowledge utilised theories of evidence and scientific proof and philosophy of the mind (including the problems of direct and reflexive mental acts and of propositional attitudes).

Decidability for some justification logics with negative introspection

Justification logics are modal logics that include justifications for the agent's knowledge. So far, there are no decidability results available for justification logics with negative introspection. In this paper, we develop a novel model construction for such logics and show that justification logics with negative introspection are decidable for finite constant specifications.