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Filosofiska
Notiser Årgång 8, Nr 1, 2021
This is a special issue on modal logic.
Max Cresswell
Prior and Łukasiewicz on Modal Logic
Abstract
A. N. Prior was strongly influenced by the work of Polish logicians,
especially Jan Łukasiewicz. One important consequence is his
adoption of Łukasiewicz's bracket-free notation for logical
formulae, but he also took issue with Łukasiewicz's criticism
of Aristotle's views on possibility. The present paper looks at
the role of I. M. Bochenski in making Prior aware of the Polish
logical tradition.
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Bas C. van Fraassen
Logic of a Self-Transparent Believer
Abstract
Moore's Paradox engendered various proposals for aspects of the
logic of belief, both for believers to avoid falling into its form
of incoherence and for special principles to serve as axioms or
rules for doxastic logic. The proposal here developed is to study
the logic pertaining to believers who are self-transparent in the
sense that, although they may have many false beliefs, they are
right about what their beliefs are. The logic of the language of
factual description of their situation is a normal modal logic KDC4C4,
but is to be distinguished from the internal logic that governs
what follows from their beliefs, on pain of incoherence. The adequacy
and completeness proofs for that logic show it to be, in some respects,
severely non-classical.
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Lloyd Humberstone
Propositional Variables Occurring Exactly Once in Candidate Modal
Axioms
Abstract
One does not often encounter a proposed axiom for extending one
modal logic to another with the following feature: in the axiom
in question some propositional variable (sentence letter) appears
only once. Indeed, for a large range of modal logics L, which includes
all normal modal logics, the sole occurrence of such a sentence
letter can be replaced by a propositional truth or falsity constant,
to give an arguably simpler axiom yielding the same extension of
L, explaining the rarity of such ‘variable-isolating’ axioms in
the literature. But the proof of this simple (and in one form or
another, well-known) result – appearing here as Lemma 2.1 – is sensitive
to the choice of modal primitives. It breaks down, for example,
when, instead of necessity (or possibility), the sole non-Boolean
primitive is taken to be noncontingency (or contingency), the main
topic of Sections 0 and 4, the latter closing with a selection of
the main problems left open. Between these, which we shall have
occasion, inter alia, to observe that the (routine) proof
of the lemma referred to (which is postponed to a final Appendix,
Section 5) is also sensitive to the choice of Boolean primitives
(Section 3).
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Jan Woleński
Propositional Self-Reference and Modalities
Abstract
This paper considers modal self-referential sentences and argues
that they generate semantic paradoxes similar to the Liar. The sources
of related antinomies are similar as in the case of the Liar-sentence,
namely self-referentiality and the T-scheme, additionally
supplemented by some principles connecting modalities and truth.
In the Appendix at the end of the paper, the dual logic is employed
for constructing the Truth-Teller Paradox and its modal counterparts.
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Giovanna Corsi and Eugenio
Orlandelli
FOIL with constant domains revisited
Abstract
FOIL is a family of two-sorted first-order modal logics containing
both object and intensional variables. Intensional variables are
represented by partial functions from worlds to objects and the
abstraction operator λ is used to talk about the object (if
any) denoted by an intension in a given world. This paper answers
a problem left open in Fitting’s [4] by showing that Fitting’s axiomatization
of FOIL augmented with infinitely many inductively defined rules,
CD(k), k ≥ 0, allows for the construction of a canonical model
that is essentially a constant domains model. Moreover, it is shown
that the rules CD(k) are derivable in logics where the symmetry
axiom B holds. Hence, Fitting’s axiomatisation of FOIL is already
complete when the underlying logic imposes symmetric models.
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Andreas Herzig and Elise
Perrotin
True Belief and Mere Belief About a Proposition and the Classification
of Epistemic-Doxastic Situations
Abstract
Starting from standard logics of knowledge and belief with principles
such as introspection of beliefs and ‘knowledge implies belief’,
we study two non-normal modalities of belief: true belief about
a proposition and what we call mere belief about a proposition.
We show that these modalities suffice to define all possible epistemic-doxastic
situations in a combinatorial manner. Furthermore, we show that
two consecutive modalities that are indexed by the same agent can
be reduced for two of the three logics of knowledge and belief that
we consider.
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Valentin Goranko
On relative ignorance
Abstract
I discuss relative ignorance of an agent with respect to
the knowledge or ignorance of other agents. It turns out, not surprisingly,
that even the two-agent case is quite complex and generates a rich
variety of naturally arising non-equivalent operators of relative
ignorance. In this paper I explore these in a more systematic way
and put together several simple, though technically laborious, observations
about their inter-relations. For the technical proofs of these I
employ the software tool MOLTAP, which implements, inter alia, tableaux
for the underlying multi-agent epistemic logic.
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