Claudio E.A. Pizzi

The paper aims at identifying the modal fragments of systems of contingency logic whose language includes a prepositional constant r. It turns out that the language of such systems allows defining two necessity operators of different strength, □ and O. It is proved that in the weakest contingency system KΔτw the τ-free fragment containing □-wffs is K and that an analogous result holds for O-wffs; that in KΔτ the fragment for both is KD; that in KTΔτ the fragment containing □-wffs is KT and the one containing O-wffs is KD. Such results are derived from the central result of the paper, which concerns the bimodal fragments of KΔτw, KΔτ, KTΔτ, here called K□O, KD□O and KT□O. The axiomatic bases for such systems consist in the unions of the axioms of the related monomodal fragments extended with the bridge axiom □p ⊃ Op. © 2014 Elsevier B.V., All rights reserved.

The central result of the paper is an alternative axiomatization of the conditional system VC which does not make use of Conditional Modus Ponens: (A > B) ⊃ (A ⊃ B) and of the axiom-schema CS: (A ∧ B) ⊃ (A > B). Essential use is made of two schemata, i.e. X1: (A ∧ ♢A) ⊃ (♢A > ♢B). A hierarchy of extensions of the basic system V called VInt, VInt1, VInt1T is then construed and submitted to a semantic analysis. In Section 3 VInt1T is shown to be deductively equivalent to VC. Section 4 shows that in VC the thesis X1 is equivalent to X1∨: (♢A > < ¬A), so that VC is also equivalent to a variant of VInt1T here called VInt1To. In Section 6 both X1 and X1∨ offer the basis for a discussion on systems containing CS, in which it is argued that they cannot avoid various kinds of partial or full trivialization of some non truth-functional operators.

An operator ∗→ is said to be Boethian in a logic L iff it is either primitive or defined inLand (i)Lcontains the so-called Boethius’ Thesis A ∗→B ⊃ ¬(A ∗→¬B); (ii) the wffs (A ⊃ B) ⊃ A ∗→B and A ∗→B ⊃ B ∗→A are not L-theorems. The aim of the paper is to show how a Boethian operator may be generated by suitable definitions from non-Boethian operators such as strict implication (J) and Stalnaker-Lewis conditional (>). The schemata of the two proposed definitions are A⇝B =df A J B & δ and A +→B =df A > B & δ where δ is a wff in the two sets Ant = {A, ¬A,□A,□¬A, ♢A, ♢¬A} or Cons = {B, ¬B,□B,□¬B, ♢B, ♢¬B}. It is proved that, if the background system for strict implication is the modal system KD, the only modal instance of δ which grants the properties of a Boethian operator is the wff ♢A. A drawback of this definition of ⇝ is that it does not allows proving Identity, i.e. A⇝A. In order to justify Identity a more refined analysis requires that the values of δ be chosen not only in Ant and Cons but also in the set of disjunctive wffs {B ∨ ♢A, ¬B ∨ ♢A, ♢B ∨ ♢A, ♢¬B ∨ ♢A,□¬B ∨ ♢A,□B ∨ ♢A}. It turns out that in this set the only two values of δ which yield Boethian operators are □¬B ∨ ♢A and □B ∨ ♢A. The former turns out to be equivalent to a consequential operator axiomatized in Pizzi [7], the latter is equivalent to another one which has been axiomatized in Lowe [4]. It is proved in section 4.6 that the former is the only value of δ that grants Identity. In the second part of the paper the same schema of analysis is applied to Lewis’ conditional operator >, while the background system is the conditional system VW−. It turns out that the modal values of δ in Ant and Cons which allow defining a Boethian operator +→ are not only ♢A but also □A and □B. Given that all them fail to grant Identity, the values of δ must again be chosen in a set of disjunctive statements, which however is wider than the one for ⇝. For both ⇝and +→ an investigation is performed in sections 4.7 and 4.8 aiming to see which special cases of ⇝ and +→ satisfy the following properties: Aristotle’s Thesis, Simplification, Secondary Boethius, Transitivity, Contraposition, Monotonicity.

The paper proposes a first approach to systems whose language includes two primitives (>+ and >-) as symbols for factual and counterfactual conditionals which are explicit, i.e. that are stated jointly with the truth or falsity of the antecedent clause. In systems based on this language, here called 2-conditional, the standard corner operator may be defined by (Def>) A > B := (A >+ B)∨(A >- B), while in classical conditional systems one could introduce the two symbols for explicit conditionals by the definitions (Def>+) A >+ B := A∧(A>B) and (Def>-) A>- B := ¬A∧(A>B). Two 2-conditional systems, V² and VW², are axiomatized and proved to be definitionally equivalent to the monoconditional systems V and VW. A third system VWTr² is characterized by an axiom stating the transitivity of factual conditionals and is shown to be distinct from V², from VW² and from the 2-conditional version of Lewis’ well-known system VC, here named VC². The same may be said for a fourth system VW²◊ ±, based on an axiom inderivable in VC²: ◊(A >+ B) ⊃ ◊( ¬A >- ¬B). VC² contains what is here called a “semi-collapse” of the operator >+ and it is argued that it is inadequate as a logic for both factual and counterfactual conditionals. The last section shows that several different definitions of the corner operators in terms of >+ and >- may be introduced as an alternative to (Def>).

This paper compares two logical conditionals which are strengthenings of the strict conditional and avoid the paradoxes of strict implication. The logics of both may be viewed as extensions of KT, and the two conditionals are interdefinable in KT. The implicative conditional requires that its antecedent and consequent be both contingent. The consequential conditional may be viewed as a weakening of the implicative conditional, insofar as it also admits the case in which the antecedent and the consequent are strictly equivalent (either both necessary or both impossible). The two conditionals share a number of properties, among them Transitivity, Contraposition, Aristotle’s Thesis, Weak Boethius’ Thesis and Aristotle’s Second Thesis. They also share some restricted principles such as Possibilistic Monotonicity, Possibilistic Simplification and Possibilistic Right Weakening. They differ in relation to Identity, which is validated by consequential implication, while the implicative conditional only validates the restricted principle of Possibilistic Identity. The relations between the two conditionals are represented by two Aristotelian cubes of opposition, one involving the contrariety between If A, then B and If A, then ¬B, according to Weak Boethius’ Thesis, and the other the contrariety between If A, then B and If ¬A, then B, according to Aristotle’s Second Thesis. We also explore the relations between the two logical conditionals and natural language conditionals, emphasizing the dependence of the latter on the context, and the need to distinguish natural language conditionals which may be viewed as consequential or implicative, on one side, and concessive and some other types of conditionals, on the other.

The objects called cubes of opposition have been presented in the literature in discordant ways. The aim of the paper is to offer a survey of such various kinds of cubes and evaluate their relation with an object, here called “Aristotelian cube”, which consists of two Aristotelian squares and four squares which are semiaristotelian, i.e. are such that their vertices are linked by some so-called Aristotelian relation. Two paradigm cases of Aristotelian squares are provided by propositions written in the language of the logic of consequential implication, whose distinctive feature is the validity of two formulas, A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ (A $$\rightarrow $$ → $$\lnot $$ ¬ B) and A $$\rightarrow $$ → B $$\supset \lnot $$ ⊃ ¬ ($$\lnot $$ ¬ A $$\rightarrow $$ → B), expressing two different forms of contrariety. Part of section 1 is devoted to define the notions of rotation and of r-Aristotelian square, i.e. a square resulting from some rotation of an Aristotelian square. In section 2 this notion is extended to the one of a r-Aristotelian cube, i.e. of a cube resulting from some rotation of some square of an Aristotelian cube. This notion is used in the sequel to analyze various cubes of oppositions which can be found in the literature: (1) the one used by W. Lenzen to reconstruct Caramuel’s Octagon; (2) the one used by D. Luzeaux to represent the implicative relation among S5-modalities; (3) the one introduced by D. Dubois to represent the relations between quantified propositions containing positive predicates and their negations; (4) the one called Moretti cube. None of such cubes is strictly speaking Aristotelian but each of them may be proved to be r-Aristotelian. Section 5 discusses the assertion that Dubois cube was anticipated in a paper published by Reichenbach in 1952. Actually Dubois’ construction was anticipated by the so-called Johnson–Keynes cube, while the Reichenbach cube, unlike Dubois cube, is an instance of an Aristotelian cube in the sense defined in this paper. The dominance of such notion is confirmed by J.F. Nilsson’s cube, representing relations between propositions with nested quantifiers, and also by a cube introduced by S. Read to treat quantifiers with existential import. A cube similar to Read’s cube, introduced by Chatti and Schang, is shown to be r-Aristotelian. In section 6 the author remarks that the logic of the formulas occurring in the cubes of Chatti–Schang and Read have the drawback of not satsfying the law of Identity. He then proposes a definition of non-standard quantifiers which satisfies Identity, are independent of existential assumptions and such that their interrelations are represented by an Aristotelian cube.

The basis of the paper is a logic of analytical consequential implication, CI.0, which is known to be equivalent to the well-known modal system KT thanks to the definition A → B = df A ⥽ B ∧ Ξ (Α, Β), Ξ (Α, Β) being a symbol for what is called here Equimodality Property: (□A ≡ □B) ∧ (◊A ≡ ◊B). Extending CI.0 (=KT) with axioms and rules for the so-called circumstantial operator symbolized by *, one obtains a system CI.0*Eq in whose language one can define an operator ↠ suitable to formalize context-dependent conditionals (so to counterfactual conditionals) via the definition (Def ↠) A ↠ B = df *A ⥽ B ∧ Ξ (Α, Β).. The central problem of the paper is to identify inside CI.0*Eq + Def ↠ a set of axioms yielding the fragment consisting of all and only theorems in which only truth-functional operators and the two operators → and ↠ occur. This system, here called CI.0, is introduced in §3. In view of the intended purpose, it is introduced a complete and tableau-decidable system KTw, which is an extension of KT with axioms for the so-called quasi-variables w(A), w(B)… Three translation functions among the three languages used in the paper are then introduced. The first, Tr, maps every wff of form *A into wffs of form w(A) ∧ A and translates theorems of CI.0*Eq into theorems of KTw. The second, t°, translates theorems of CI.0 into theorems of CI.0*Eq by applying Def↠. A third one, t, translates theorems of CI.0 into theorems of KTw. As a consequence, it follows that t°A = TrtA. In §4 it is proved that CI.0 is complete w.r.t. the class of CI.0-models of form where f is a selection function and R an access relation. It is then proved (i) that CI.0 models may be converted into KTw-models; (ii) that the truth-value of a proposition in a world of a CI.0-model is preserved in the same world of the derived KTw-model; (iii) that A is a CI.0-thesis iff its translation tA is KTw-thesis. It follows that, if A is not a CI.0-thesis, tA is not a KTw-thesis, so also that TrtA is not a CI.0*Eq-thesis. Since t°A = TrtA and t°A is a function built on Def↠, this proves that every t°-translation of a CI.0-wff that is a CI.0*Eq-theorem is a CI.0-theorem. Since the converse proposition is proved in §2, their conjunction establishes the desired result. In the final section, it is proved that CI.0 is tableau-decidable, that ↠ is not a trivial operator and that ↠ enjoys the positive and negative properties required in §1 for context-dependent conditionals. Some final remarks suggest that studying the relations between CI.0 and systems of classical conditional logic may be a promising line of investigation.

In the present work we provide a logical analysis of normatively determined and non-determined propositions. The normative status of these propositions depends on their relation with another proposition, here named reference proposition. Using a formal language that includes a monadic operator of obligation, we define eight dyadic operators that represent various notions of “being normatively (non-)determined”; then, we group them into two families, each forming an Aristotelian square of opposition. Finally, we show how the two resulting squares can be combined to form an Aristotelian cube of opposition.

The paper aims at developing the idea that the standard operator of noncontingency, usually symbolized by Δ, is a special case of a more general operator of dyadic noncontingency Δ(−, −). Such a notion may be modally defined in different ways. The one examined in the paper is __Δ__(B, A) = df ◊B ∧ (A ⥽ B ∨ A ⥽ ¬B), where ⥽ stands for strict implication. The operator of dyadic contingency __∇__(B, A) is defined as the negation of __Δ__(B, A). Possibility (◊A) may be then defined as __Δ__(A, A), necessity (□A) as __∇__(¬A, ¬A) and standard monadic noncontingency (__Δ__A) as __Δ__( \({\textsf{T}}\), A). In the second section it is proved that the deontic system KD is translationally equivalent to an axiomatic system of dyadic noncontingency named KDΔ 2, and that the minimal system KΔ of monadic contingency is a fragment of KDΔ 2. The last section suggests lines for further inquiries.

The first part of the paper contains a probabilistic axiomatic extension of the conditional system WV, here named WVPr. This system is extended with the axiom (Pr4): PrA = 1 ⊃ □A. The resulting system, named WVPr∗, is proved to be consistent and non-trivial, in the sense that it does not contain the wff (Triv): A ≡□A. Extending WVPr∗ with the so-called Generalized Stalnaker’s Thesis (GST) yields the (first) Lewis’s Triviality Result (LTriv) in the form (◊(A ∧ B) ∧◊(A ∧ ¬B)) ⊃ PrB|A = PrB. In §4 it is shown that a consequence of this theorem is the thesis (CT1): ¬A ⊃ (A > B ⊃ A ⥽ B). It is then proven that (CT1) subjoined to the conditional system WVPr∗ yields the collapse formula (Triv). The final result is that WVPr∗+(GST) is equivalent to WVPr∗+(Triv). In the last section a discussion is opened about the intuitive and philosophical plausibility of axiom (Pr4) and its role in the derivation of (Triv).

The paper introduces a contingential language extended with a propositional constant τ axiomatized in a system named KΔτ , which receives a semantical analysis via relational models. A definition of the necessity operator in terms of Δ and τ allows proving (i) that KΔτ is equivalent to a modal system named K□τ (ii) that both KΔτ and K□τ are tableau-decidable and complete with respect to the defined relational semantics (iii) that the modal τ -free fragment of KΔτ is exactly the deontic system KD. In §4 it is proved that the modal τ -free fragment of a system KΔτw weaker than KΔτ is exactly the minimal normal system K.

The paper studies the relation between systems of modal logic and systems of consequential implication, a non-material form of implication satisfying "Aristotle's Thesis" (p does not imply not p) and "Weak Boethius' Thesis" (if p implies q, then p does not imply not q). Definitions are given of consequential implication in terms of modal operators and of modal operators in terms of consequential implication. The modal equivalent of "Strong Boethius' Thesis" (that p implies q implies that p does not imply not q) is identified

It is natural to ask under what conditions negating a conditional is equivalent to negating its consequent. Given a bivalent background logic, this is equivalent to asking about the conjunction of Conditional Excluded Middle (CEM, opposite conditionals are not both false) and Weak Boethius' Thesis (WBT, opposite conditionals are not both true). In the system CI.0 of consequential implication, which is intertranslatable with the modal logic KT, WBT is a theorem, so it is natural to ask which instances of CEM are derivable. We also investigate the systems CIw and CI of consequential implication, corresponding to the modal logics K and KD respectively, with occasional remarks about stronger systems. While unrestricted CEM produces modal collapse in all these systems, CEM restricted to contingent formulas yields the Alt2 axiom (semantically, each world can see at most two worlds), which corresponds to the symmetry of consequential implication. It is proved that in all the main systems considered, a given instance of CEM is derivable if and only if the result of replacing consequential implication by the material biconditional in one or other of its disjuncts is provable. Several related results are also proved. The methods of the paper are those of propositional modal logic as applied to a special sort of conditional.