Logics of Essence and Accident
In the literature, essence is formalized in two different ways, either de dicto, or de re. Following [Marcos:2005], we adopt its de dicto formalization: a formula is essential, if once it is true, it is necessarily true; otherwise, it is accidental. In this article, we study the model theory and axiomatization of the logic of essence and accident, i.e. the logic with essence operator (or accident operator) as the only primitive modality. We show that the logic of essence and accident is less expressive than modal logic on non-reflexive models, but the two logics are equally expressive on reflexive models. We prove that some frame properties are undefinable in the logic of essence and accident, while some are. We propose the suitable bisimulation for this logic, based on which we characterize the expressive power of this logic within modal logic and within first-order logic. We axiomatize this logic over various frame classes, among which the symmetric case is missing, and our method is more suitable than those in the literature. We also find a method to compute certain axioms used to axiomatize this logic over special frames in the literature. As a side effect, we answer some open questions raised in [Marcos:2005].
Keywords: essence, accident, expressivity, frame definability, bisimulation, axiomatization
As far back as Aristotle, like the notions of necessity, possibility and contingency, the notion of essence can also be related either to propositions (de dicto) or to objects (de re). The importance of the notion of essence is argued in [essenceandmodality:1994].
The formalization of essence in terms of de re at least tracks back to Kit Fine. In his writing [Fine:essence], a logic of essence is proposed, where formulas of the form express that is true in virtue of the essence of objects which . A Hilbert-style quantified system E5 is given but without a semantics. In [Fine:semantics], a possible worlds semantics is presented, and a variant of E5 is shown to be sound and complete for the semantics. In [Correia:essence], a propositional version of E5 is established in accompany with an appropriate semantics, and it is shown that the system is sound and complete with respect to the proposed semantics. A new semantics for logics of essence is proposed in [Girodani:2014].
There are also researchers who formalize essence in terms of de dicto. In [Small:2001], in reconstructing Gödel’s ontological argument, accidental truth, i.e. accident is formalized as , i.e. true but not necessarily true. Accordingly, as the negation of accident, essence is formalized as . It is said that the discussions of essential and accidental propositions at least tracks back to the XIX Century, see [Marcos:2005, p. 53]. A logic of essence and accident is introduced in which essence is treated in the metaphysical usage in [Marcos:2005], where a complete axiomatization for the logic is shown with respect to the class of all frames. A simple axiomatization for arbitrary frames and its extensions over various frame classes are proposed in [Steinsvold:2008], but the case for symmetric frames is missing. Even though the completeness proofs thereof are simple, his method has a defect: on one hand, the canonical relation, thus the canonical frame, is automatically provided to be reflexive; on the other hand, the underlying semantics is defined on arbitrary frames, rather than on reflexive frames. This means that there is a non-correspondence between syntax and semantics in the logic of essence and accident. Oblivious to the literature on the logic of essence and accident, in [steinsvold:2008b] the author provides a topological semantics for a logic of unknown truths and shows its completeness over the class of models.
The accident operator has various meanings in different contexts. For instance, in the setting of provability logic, ‘accident’ means ‘true but unprovable’, thus ‘ is accident’ means ‘ is a Gödel sentence’ [Kushida:2010]; in the setting of epistemic logic, ‘accident’ means ‘unknown truths’, thus ‘ is accident’ means ‘ is true but unknown to the agent’ [steinsvold:2008b].
In this article, we will follow the formalization of essence in [Marcos:2005], study the notions of essence and accident from viewpoint of de dicto. We will discuss the model theory of the logic of essence and accident, propose some axiomatizations, whose completeness are shown with a more suitable method than those in the literature, and give an automatic method to compute certain axioms needed to characterize this logic over special frames.
The paper is organized as follows. Section 2 introduces the language of the logic of essence and accident. Section 3 compares the relative expressive power of the logic of essence and accident and modal logic. Section LABEL:sec.framecor explores the frame definability. We propose the bisimulation notion suitable for the logic of essence and accident in Section LABEL:sec.bis, based on which we characterize the expressive power of this logic within modal logic and within first-order logic in Section LABEL:sec.char. Section LABEL:sec.axiomatizations axiomatizes the logic of essence and accident over various frames. In Section LABEL:sec.comparison, we compare our work with the literature on the logic of essence and accident and the modal logic of Gödel sentence. We conclude with some future work in Section LABEL:sec.conclusion.
2 Language and Semantics
First, we introduce the following language with essence operator and necessity operator as modalities, although we will focus on the language of logic of essence and accident.
Definition 1 (Logical language ).
Let P be a set of propositional variables, the logical language is defined as follows:
where . Without the construct , we obtain the language of modal logic ; without the construct , we obtain the language of essence and accident . If , we say is an -formula; if , we say is an -formula; if , we say is an -formula.
Intuitively, is read ‘it is essential that ’, and is read ‘it is necessary that ’. Other operators are defined as usual; in particular, is defined as , read ‘it is accidental that ’. Note that is not the dual of .
Definition 2 (Model).
A frame is a tuple , where is a nonempty set of possible worlds, is a binary relation over . A model is a tuple , where is a valuation function from P to . A pointed model is a model with a designated world in . We always omit the parentheses around whenever convenient. We sometimes write for . We write . We write for the class of reflexive frames.
Definition 3 (Semantics).
Given a pointed model and an -formula , the satisfaction relation is defined as follows:111We here use the notation , respectively, to stand for the metalanguage ‘and’, ‘for all’, ‘if then ’, ‘if and only if’.
If , we say is true, or satisfied at , sometimes we write ; if for all we have , we say is valid on and write ; if for all based on we have , we say is valid on and write ; if for all in a class of frames we have , we say is valid on and write ; if the class of frames in question is arbitrary, then we say is valid and write . We say is satisfiable, if . The case for a set of formula is similarly defined. Given any two pointed models and , if they satisfy the same -formulas, we say they are -equivalent, notation: ; if they satisfy the same -formulas, we say they are -equivalent, notation: .
Under the semantics, it is not hard to show that
Let . Then and .
Proposition 4 is very important. It guides us to find the desired axioms for characterizing over certain frame classes, as we will see in Section LABEL:sec.axiomatization.
In this section, we compare the relative expressivity of and . A related technical definition is introduced as follows.
Definition 5 (Expressivity).
Let logical languages and be interpreted on the same class of models,
is at least as expressive as , notation: , if for any , there exists such that for all , we have .
and are equally expressive, notation: , if and .
is less expressive than , or is more expressive than , notation: , if and .
is less expressive than on the class of -models, -models, -models, -models.
Define a translation from to :
It is clear that is a truth-preserving translation. Therefore is at least as expressive as .
Now consider the following pointed models and , which can be distinguished by an -formula , but cannot be distinguished by any -formulas: