Stability in a Commonsense Ontology of States

Despite their mostly-unchallenged acceptance in a reasoning agent’s ontology, states, as opposed to events, have raised some interesting questions over the years, particularly regarding the representation of their temporal incidence and the striking similarity they have to propositions. I present a family of languages, LogAS, for reasoning about states. A LogAS language is algebraic in the sense that it does not contain sentences, only terms, some of which denoting states. In this paper, I study one particular aspect of the state ontology, namely temporal stability, classifying states according to their pattern of persistence over time. I identify four major intuitive stability classes which partition the domain of states and prove closure properties for each class under the various Boolean operators. One of these classes, the class of atemporal states, corresponds one-to-one to the set of propositions, thus providing a unified treatment of propositions and states. Introduction In discussions of aspectual phenomena, “states” stand as one prominent kind of entity of which temporal incidence is attributable (Vendler 1957; Herweg 1991; Smith 1999). As opposed to “events”, states do not wholly “occur”, but homogeneously “hold” over time. States appear in different guises and under different names in the artificial intelligence and related literature. They are the “states of affairs” of metaphysics (Textor 2012), the “partial states” of (Bennett and Galton 2004), the “properties” of (Allen 1984), and the fluents of the situation calculus (McCarthy and Hayes 1969) and some variants of the event calculus (Shanahan 1999). I have argued elsewhere (Ismail 2008) for the primitiveness of states in the commonsense ontology of an acting agent, being the sole objects of experience from which all of an agent’s beliefs are constructed through inference. As such, a thorough investigation of states is at least insightful. But states are not unanimously accepted as inhabitants of the temporal ontology. As indicated, they do appear as fluents in the situation calculus, and as properties in Allen’s ontology (Allen 1984), for example. But these are all firstorder theories. In tense logics (Prior 1967), for instance, we do not see any notion of states, only propositions which may be true in the present, the past, or the future. It is not clear whether first-order theories which endorse states do that out of conviction or as a side-effect of their attempt to secure the convenience of having state-denoting terms in the logical language, which facilitates tersely axiomatizing temporal properties. In particular, in first-order theories with state-denoting terms, there is always a sense of redundancy when the entire syntax for sentence formation is duplicated to form the set of state terms. (See (Shoham 1987) who criticizes (Allen 1984) particularly for this point. ) The mysterious syntactic division between state terms and sentences, and the corresponding semantic division between states and propositions, is unsettling since it is (i) theoretically suspicious, and (ii) both syntactically and ontologically uneconomical. In this paper, I present LogAS, a family of algebraic logics of states. LogAS is algebraic in the sense that it only contains terms, algebraically constructed from function symbols. No sentences are included in a LogAS language. Instead, there are terms of a distinguished syntactic type that are taken to denote states; some of these terms may, for convenience, be regarded as sentences. LogAS is a variant of LogAB (Ismail 2012), which is, again, an algebraic language for reasoning about propositions and propositional attitudes. The main difference between LogAB and LogAS is that whereas the ontology of the former includes propositions, taken at face value, the ontology of the latter trades propositions for states. In the LogAS ontology, states are structured in a Boolean algebra. This gives us, almost for free, state-counterparts to propositional truth conditions and standard notions of consequence and validity. Here, I investigate one particular aspect of the state ontology, namely temporal stability, classifying states according to their pattern of persistence over time. Four major classes are identified, disjointness between four of the classes, and closure properties of each class under the Boolean operators are proved. As it turns out, the class of atemporal states corresponds one-to-one to the set of propositions. Thus, through its state-based syntax and semantics, LogAS provides a unified, non-redundant treatment of states and propositions. LogAS Languages LogAS is a class of many-sorted languages that share a common core of logical symbols and differ in a signature of nonlogical symbols. In what follows, we identify a sort σ with the set of symbols of sort σ. A LogAS language is a set of terms partitioned into three base syntactic types, σS , σT , and σI . Intuitively, σS is the set of terms denoting states, σT is the set of terms denoting time points, and σI is the set of terms denoting anything else. Syntax As is customary in type-theoretical treatments, an alphabet of LogAS is made up of a set of syncategorematic punctuation symbols and a set of denoting symbols each from a set σ of syntactic types. The set σ is the smallest set containing all of the following types: σS , σT , σI , and ς1 −→ ς2, for ς1 ∈ {σS , σT , σI} and ς2 ∈ σ. Intuitively, ς1 −→ ς2 is the syntactic type of function symbols that take a single argument of type σS , σT , or σI and produce a functional term of type ς2. Given the restriction of the first argument of function symbols to base types, LogAS is, in a sense, a first-order language. A LogAS alphabet is a union of four disjoint sets: Ω∪Ξ∪ Σ ∪ Λ. The set Ω, the signature of the language, is a nonempty set of constant and function symbols. Each symbol in the signature has a designated syntactic type from σ. Ω is what distinguishes one LogAS language from another. The set Ξ = {xi, ti, si}i∈N is a countably infinite set of variables, where xi ∈ σI , ti ∈ σT , and si ∈ σS . Σ is a set of syncategorematic symbols, including the comma, various matching pairs of brackets and parentheses, and the symbol ∀. The set Λ is the set of logical symbols of LogAS, defined as the union of the following sets. 1. ¬ ∈ σS −→ σS 2. {∧,∨} ⊆ σS −→ σS −→ σS 3. HoldsAt ∈ σS −→ σT −→ σS 4. ≺∈ σT −→ σT −→ σS A LogAS language with signature Ω is denoted by LΩ. It is the smallest set of terms formed according to the following rules, where τ and τi (i ∈ N) are terms in LΩ. • Ξ ⊂ LΩ • c ∈ LΩ, where c ∈ Ω is a constant symbol. • f(τ1, . . . , τn) ∈ LΩ, where f ∈ Ω is of type ς1 −→ . −→ ςn −→ ς (n > 0) and τi is of type ςi. • ¬τ ∈ LΩ, where τ ∈ σS . • (τ1 ⊗ τ2) ∈ LΩ, where ⊗ ∈ {∧,∨} and τ1, τ2 ∈ σS . • ∀x(τ) ∈ LΩ, where x ∈ Ξ and τ ∈ σS . • HoldsAT(τ1, τ2) ∈ LΩ, where τ1 ∈ σS and τ2 ∈ σT . • (τ1 ≺ τ2) ∈ LΩ, where τ1, τ1 ∈ σT . As usual, terms involving ⇒, ⇔, and ∃ may be introduced as abbreviations in the standard way. Semantics The basic ingredient of the LogAS semantic apparatus is the notion of a LogAS structure. Definition 1 A LogAS structure is a quadruple S = ⟨D,A, h, <⟩, where • D, the domain of discourse, is a set with two disjoint, nonempty, countable subsets S and T . • A = ⟨S,+, ·,−,⊥,⊤⟩ is a complete (closed under arbitrary products and sums), non-degenerate (⊤ ̸= ⊥) Boolean algebra. • h : S × T −→ S satisfies the following properties, for every Ŝ ⊆ S, s ∈ S and t, t1, t2 ∈ T : 1. h(−s, t) = −h(s, t). 2. h( ∏