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1 the Idea of First-Degree Entailment FIRST-DEGREE ENTAILMENT AND BINARY CONSEQUENCE SYSTEMS YAROSLAV SHRAMKO Department of Philosophy, Kryvyi Rih State Pedagogical University, Kryvyi Rih, 50086, Ukraine [email protected] Abstract This paper presents an account of Belnap and Dunn’s logic of first-degree entail- ment and some related logics based on a proof-theoretic machinery of binary (FMLA- FMLA) consequence systems. It is shown how the logic of first-degree entailment can be represented by various deductively equivalent systems, up to a purely structural sys- tem with transitivity as the only inference rule. A family of possible extensions of this later system is represented in a systematic manner. This is a review article, which reca- pitulates certain recent advances in investigating the Belnap-Dunn logic, and organizes the corresponding material from a genuinely first-degree entailment perspective. Keywords: First-degree entailment, consequence system, binary consequence, logical frameworks, structural reasoning, super-Belnap logics 1 The idea of first-degree entailment The logic of first-degree entailment occupies an important place among modern non-classi- cal logics. As Hitoshi Omori and Heinrich Wansing put it: There is a continuum of nonclassical logics, but some systems have emerged as particularly interesting and useful. Among these distinguished nonclassical logics is a system of propositional logic that has become well-known as Belnap and Dunn’s useful four-valued logic or first-degree entailment logic, FDE. [25, p. 1021] The notion of first-degree entailment has been introduced by Nuel Belnap in his doctoral dissertation [5] (see also [25, p. 1021]), and put into circulation in a short abstract of his talk Vol. 7 No. 6 2020 Journal of Applied Logics — IfCoLog Journal of Logics and their Applications SHRAMKO at the twenty-fourth annual meeting of the Association for Symbolic Logic held on Monday, December 28, 1959 at Columbia University in New York [6]. This notion was elaborated then in detail in [2] as a tool for “finding plausible criteria” for valid entailments of the form ϕ ψ, where ϕ and ψ are purely truth-functional. → To this effect Belnap defines: “ϕ ψ is a first degree entailment iff ϕ and ψ are → both written solely in terms of propositional variables, , , and (other truth-functional ∧ ∨ ∼ connectives being treated as defined by these)” [6, notation adjusted]. Belnap explains that a first-degree entailment is valid if and only if it is tautological, which means that it is of the form ϕ ... ϕ ψ ... ψ (or reducible to them by special replacement 1 ∨ ∨ m → 1 ∧ ∧ n rules), where every ϕ ψ is an “explicitly tautological entailment”. A first-degree i → j entailment is explicitly tautological iff it is of the form χ1 ... χm ξ1 ... ξn, where ∧ ∧ → ∨ ∨ 1 χ1, . , χm, ξ1, . , ξn are all atoms (i.e. propositional variables or the negates thereof), and with some atom χi being the same as some atom ξj. Belnap remarks that tautological entailmenthood is effectively decidable, and observes strong equality between the set of first-degree theorems of the system E (of entailment) and the set of tautological entailments, see also [2]. J. Michael Dunn in [10] initiated a highly innovative research program for semantic justification of the first-degree entailments, culminating in his paper [11]. The main point of the program consists in allowing underdetermined and overdetermined valuations that can in certain situations falsify logical laws or verify contradictions, see [31]. One way to achieve this is to treat valuation as a function from the set of sentences of a language to the subsets of classical truth values t, f , cf. [11, p. 156]. { } Consider sentential language defined as follows:2 CDN ϕ ::= p ϕ ϕ ϕ ϕ ϕ. | ∧ | ∨ | ∼ A generalized valuation is a map v from the set of sentential variables into ( t, f ). P { } This valuation is extended to the whole language by the following conditions: Definition 1. (1) t v(ϕ ψ) t v(ϕ) and t v(ψ), f v(ϕ ψ) f v(ϕ) or f v(ψ); ∈ ∧ ⇔ ∈ ∈ ∈ ∧ ⇔ ∈ ∈ (2) t v(ϕ ψ) t v(ϕ) or t v(ψ), f v(ϕ ψ) f v(ϕ) and f v(ψ); ∈ ∨ ⇔ ∈ ∈ ∈ ∨ ⇔ ∈ ∈ (3) t v( ϕ) f v(ϕ), f v( ϕ) t v(ϕ). ∈ ∼ ⇔ ∈ ∈ ∼ ⇔ ∈ A truth-value function, so defined, produces exactly four possible assignments that can be ascribed to a formula ϕ: 1“Atom” is the original term used by Belnap for what nowadays is usually called “literal”. 2 stands for “conjunction, disjunction, negation”. CDN 1224 FIRST-DEGREE ENTAILMENT AND BINARY CONSEQUENCE SYSTEMS 1. v(ϕ) = t , i.e., t v(ϕ) and f / v(ϕ); { } ∈ ∈ 2. v(ϕ) = f , i.e., t / v(ϕ) and f v(ϕ); { } ∈ ∈ 3. v(ϕ) = t, f , i.e., t v(ϕ) and f v(ϕ); { } ∈ ∈ 4. v(ϕ) = , i.e., t / v(ϕ) and f / v(ϕ). {} ∈ ∈ In this way one arrives at a certain reinterpretation of classical truth and falsity on a semantic level. Indeed, if we consider a classical valuation vc to be a usual map from the set of formulas into the set t, f , then “ϕ is true” is explicated as vc(ϕ) = t, and “ϕ is false” { } is explicated as vc(ϕ) = f. In Dunn’s semantics alternatively “ϕ is true” is explicated as t v(ϕ), and “ϕ is false” is explicated as f v(ϕ). As a result, the following principles ∈ ∈ (for any ϕ): ϕ is true, or ϕ is false (Bivalence) ϕ is not true, or ϕ is not false (Univocality) do not generally hold in Dunn’s semantics, as they do in classical. Belnap in his seminal papers [7] and [8] (reproduced in [4] as 81, see also [26]) fa- § mously explicated assignments in Dunn’s semantics as new truth values (which can be called “generalized truth values”, see [34, p. 763]): N = ∅, F = f , T = t and { } { } B = f, t , thus obtaining a “useful four-valued logic” for a “computer-based reasoning”. { } Truth values for compound formulas are determined by the following matrices: TBNF TBNF ∼ ∧ ∨ T F T TBNF T TTTT B B B BBFF B TBTB N N N NFNF N TTNN F T F FFFF F TBNF Moreover, Belnap’s four truth values (being ascribed to a sentence ϕ) can be explicated as follows by Dunn’s generalized valuation: v(ϕ) = T t v(ϕ) and f / v(ϕ): ϕ is true only; ⇔ ∈ ∈ v(ϕ) = F t / v(ϕ) and f v(ϕ): ϕ is false only; ⇔ ∈ ∈ v(ϕ) = B t v(ϕ) and f v(ϕ): ϕ is both true and false; ⇔ ∈ ∈ v(ϕ) = N t / v(ϕ) and f / v(ϕ): ϕ is neither true nor false. ⇔ ∈ ∈ 1225 SHRAMKO If a sentence has the truth value T , it is said to be exactly true; if it has one of the values T or B, it can be viewed as at least true, and analogously for falsehood. By defining entailment as a relation between sentences, one may rely on a basic understanding that valid inference always preserves truth as well as non-falsity—from a premise to the conclusion. Belnap implements this understanding in such a way that if the premise is at least true, so is the conclusion, and if the conclusion is at least false, so is the premise (cf. [4, p. 519]). We have thus the following definition: Definition 2. ϕ ψ =df v : t v(ϕ) t v(ψ). FDE ∀ ∈ ⇒ ∈ Observation 3. The entailment relation as defined by Definition 2, explicitly takes FDE T,B as the set of designated truth values. That is, Definition 2 can be reformulated as { } follows: ϕ ψ =df v : v(ϕ) T,B v(ψ) T,B . FDE ∀ ∈ { } ⇒ ∈ { } Remarkably, within Dunn’s setting, the first half of Belnap’s understanding of entail- ment implies the second, and vice versa. Indeed, it can be shown (see, e.g., Proposition 4 in [14]), that the following holds: Lemma 4. ϕ ψ v : f v(ψ) f v(ϕ). FDE ⇔ ∀ ∈ ⇒ ∈ Observation 5. Lemma 4 can be taken as a definition of entailment relation, in which case Definition 2 becomes provable as a lemma. The set of designated truth values is then T,N , and definition of entailment can be reformulated as follows: { } ϕ ψ =df v : v(ϕ) T,N v(ψ) T,N . FDE ∀ ∈ { } ⇒ ∈ { } It should also be observed, that the set of valid entailments under Definition 2 (and Lemma 4) is exactly the set of tautological entailments in Belnap’s sense. In this way, Anderson and Belnap’s approach to finding criteria “for picking out from among first-degree entailments . those that are valid” [2, p. 9] is enshrined in a semantic framework of Dunn’s generalized valuation and Belnap’s generalized truth values. As already said, the distinctive feature of this approach consists in explicating entail- ment as a relation between truth-functional formulas, i.e. between single formulas, which do not include expressions of entailment in any form. Such a relation can serve then as a general base and starting point for further expansions and generalizations, by consider- ing, e.g., entailments between sets of formulas, entailments between entailments, etc. For a comprehensive overview of various extensions (in the same vocabulary) and expansions of the logic of first-degree entailment see a survey paper [25]. Remarkably, by all these ex- pansions the core relation of first-degree entailment remains basic, providing thus a sound general framework for the whole conception of entailment. It is noteworthy, that this core relation, as originally introduced by Belnap and Dunn, can itself be extended in different directions, thus giving rise to the first-degree entailment 1226 FIRST-DEGREE ENTAILMENT AND BINARY CONSEQUENCE SYSTEMS fragments of various non-classical logics, such as Kleene’s strong three-valued logic or Priest’s logic of paradox.
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