The first section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisfied by at least...

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Two Kinds of Consequential Implication

Abstract. The ﬁrst section of the paper establishes the minimal properties of so-called consequential implication and shows that they are satisﬁed by at least two diﬀerent operators of decreasing strength (symbolized by → and ⇒). Only the former has been analyzed in recent literature, so the paper focuses essentially on the latter. Both operators may be axiomatized in systems which are shown to be translatable into standard systems of normal modal logic. The central result of the paper is that the minimal consequential system for ⇒, CI⇒, is deﬁnitionally equivalent to the deontic system KD and is intertranslatable with the minimal consequential system for →, CI. The main drawback ot the weaker operator ⇒ is that it lacks unrestricted contraposition, but the ﬁnal section of the paper argues that ⇒ has some properties which make it a valuable alternative to →, turning out especially plausible as a basis for the deﬁnition of operators representing synthetic (i.e. context-dependent) conditionals. Keywords: Connexivity, Necessity, Normal modal logics, Contraposition, Conditionals.

§0. The basic features of so-called consequential implication are the same which were originally at the root of connexive logic. However, some special properties of the logic of consequential implication make it an independent branch of non-classical logic, which actually may be seen as providing a bridge between connexive logic and standard monadic modal logic.1 Propositional systems of consequential implication are extensions of the standard propositional truth-functional calculus PC in which the operator representing the relation of consequential implication, symbolyzed by CI , satisﬁes the following minimal positive and negative properties (where ⊃, ∧, ∨, ¬ have the usual meaning, T is any truthfunctional tautology and ⊥ is ¬T). Positive properties: A CI A

(i)

Identity

1

For a survey on connexive logic and consequential implication see Pizzi [18]. The name “consequential implication” appeared for the ﬁrst time in Pizzi [13], but the ﬁrst system of consequential implication was introduced as a “weak connexive logic” in Parry [12]. For other papers on consequential implication see Pizzi’s titles reported in the References. Presented by Heinrich Wansing; Received October 11, 2016

Studia Logica DOI: 10.1007/s11225-017-9749-5

c Springer Science+Business Media B.V. 2017

C. E. A. Pizzi

(ii) A CI B entails ¬(A CI ¬B)

Strawson’s Thesis or (Weak) Boethius’Thesis (BT)

(iii) ¬(A CI ¬A)

Aristotle’s Thesis (AT)

(iv) ¬(¬A CI A)

Secondary Aristotle’s Thesis

(v) ⊥ CI A is logically equivalent to A CI ⊥

Non-Explosion

Negative properties: (vi) A CI B is not entailed by A ⊃ B

Non-triviality

(vii) (A ∧ B) CI A is not logically valid

Failure of Simpliﬁcation

(viii) A CI (A ∨ B) is not logically valid

Failure of Addition

(ix) A CI B does not entail (A ∧ R) CI (B ∧ R)

Non-monotonicity

In the characterization provided by the mentioned set of properties it is not speciﬁed if CI is a context-dependent operator or is not such, i.e. if the truth of A CI B depends or does not depend essentially on some unexpressed contingent proposition playing the role of an additional premise. Recovering an old terminology, context-dependent conditionals will be called here synthetic conditionals, while conditionals lacking this quality will be called analytic conditionals. This analytic–synthetic distinction can be drawn within a suitably rich object-language in which (a)

if CI is intended to represent analytic consequential implication, it satisﬁes, beyond (i)–(v), the following positive property:

(x)

A CI B entails T CI (A ⊃ B)

(b)

a second consequential operator CI s is introduced, either by deﬁnition or as a primitive, satisfying the two following positive properties (xi) and (xii):

(xi) A CI B entails A CI s B (xii) A CI s B does not entail T CI (A ⊃ B) Note that a consequence of (x) and (xii) is that A CI s B does not entail A CI B: if this were true, by (x) A CI s B would entail by transitivity T CI A ⊃ B, contrary to what required in (xii). This fact grants that the two operators CI and CI s are distinct and have diﬀerent properties. In what follows, the logical investigation will be restricted to the analytic CI , in the sense that the operators which will be the focus of the discussion will have the minimal properties required for CI (i.e. (i)–(x)). However, we cannot avoid noticing that so-called “subjunctive conditionals” and “counterfactual conditionals” belong to the class of synthetic conditionals, so that

Two Kinds of Consequential Implication

according to the preceding assumption their properties are not coincident with the properties of analytic conditionals. This fact, however, appears to be neglected in the ﬁrst paper published on connexive logic (Angell [1]), which bears the title A propositional logic with Subjunctive Conditionals) and in all the subsequent literature on connexive implication, whose properties are akin to the ones of analytic consequential implication. Looking at the list of positive properties one could notice that Aristotle’s Thesis follows from Boethius’ Thesis thanks to the identity law A CI A and Modus Ponens, while Secondary Aristotle’s Thesis follows from Aristotle’s Thesis provided that the reference system has among its rules Uniform Substitution, Replacement of proved material equivalents and the PC-law A ≡ ¬¬A. One could be willing to require that, just as we have a Primary and Secondary Aristotle’s Thesis, we should have among the basic consequential laws both a Primary and Secondary Boethius’ Thesis, the latter being the expression of this further property: (xiii) A CI B entails ¬(¬A CI B)

Secondary Boethius’ Thesis

Principle (xiii) is called Strawson2 by MCCALL [8] (p. 91). Two remarks however are in order. The ﬁrst is that while the two Aristotle’s Theses are interdeducible thanks to the rules of PC, this cannot be said for the two Boethius’Theses: in order to derive the Secondary from the Primary Boethius, or viceversa, we need rules and principles which go beyond the resources of truth-functional propositional logic. The second remark concerns the soundness of Secondary Boethius, which is less intuitive than the ﬁrst (on this point see later, p. 26). The principle qualiﬁed as of Subjunctive Contrariety in the Angell’system named PA1 and named (Strong) Boethius’ Thesis is formulated in this way in KNEALE [5] (p. 416): (SBT) (A → B) → ¬(A → ¬B) In all connexive systems containing the law (A → B) ⊃ (A ⊃ B) SBT obviously implies Weak Boethius (BT); but any system containing Strong Boethius and not Weak Boethius cannot be said, strictly speaking, a system of consequential implication. However, in the literature one meets systems of connexive logic in which only Weak Boethius appears as an axiom. For instance, let us look at the class logic which S. McCall called connexive algebra CA (see [6]) where the primitive symbol ⊂ stands for class inclusion. As McCall observes, only two axioms of the systems are non Boolean:

C. E. A. Pizzi

(I) (0 ⊂ a) ⊃ (a ⊂ 0) (II) ¬ (a ⊂ -a). Interpreting the class symbols a, b, c . . . as propositional variables, the inclusion symbol ⊂ as an implicative relation and the other set-theoretical symbols as the truthfunctional operators, in the propositional version of CA (let us call it CAp) we ﬁnd that (I) corresponds to property (v) of the list of page 1 (Non-Explosion) and (II) to property (iii) (Aristotle’s Thesis). In the set of CAp’s axioms Weak Boethius is lacking, but via transitivity and contraposition (which are the translations of two axioms of CAp) it is straightforward to reach it via Aristotle’s Thesis. In fact, if transitivity is expressed by ((A → B) ∧ (B → ¬A)) ⊃ (A → ¬A), by PC we have ¬(A → ¬A) ⊃ ¬((A → B) ∧ (B → ¬A)). Since → is contrapositive, B → ¬A is equivalent to A → ¬B. Thus we obtain via PC ¬(A → ¬A) ⊃ ((A → B) ⊃ ¬(A → ¬B)); and by Aristotle and Modus Ponens (A → B) ⊃ ¬(A → ¬B). It turns out then that it is indiﬀerent to have Weak Boethius or Aristotle as axioms, provided the background systems contains contraposition and transitivity of the implicative operator. Another example of a connexive system which contains BT but not SBT is provided by McCall’s system CFL published in [7]. Furthermore, in both CFL and CAp Modus Ponens is introduced in terms of ⊃. So why not to classify systems like CAp and CFL as systems of consequential implication? Unfortunately we cannot make this step since all known systems of connexive logic contain the so-called “Factor Law”, i.e. the monotonicity of the arrow: (FL) (A → B) ⊃ ((A ∧ C) → (B ∧ C)), 2 which in the stronger version is provided by (FL→) (A → B) → ((A ∧ C) → (B ∧ C)). The reason why Factor cannot be qualiﬁed as a consequentialist theorem is the following. Suppose by Reductio that it is such and take ¬A as a value of C. Then from A → B one derives by Modus Ponens (A ∧¬A) → (B ∧¬A). But suppose that A → B is an analytic but non logical truth, e.g. (m) a is a cat → a is an animal. Under this premise we derive the truth of a consequent that says (n) a is a cat and not a cat → a is an animal but not a cat. Notice that if Monotonicity were deﬁned as (A → B) ⊃ ((A ∧ C) → B), as frequently is done, we would have, by substituting A to B and Modus Ponens, (A ∧ C) → A, i.e. Simpliﬁcation, which is obviously a non-connexive and non-consequential theorem. 2

Two Kinds of Consequential Implication

But in (n) a contradiction entails a noncontradictory statement, which conﬂicts with what is required by the property (v) formulated at the beginning. Of course one could object to the preceding argument that a meaning postulate or an analytic truth such as (m) is not a logical truth, so one cannot apply Modus Ponens to reach (n) as a theorem. But no one can deny that (m) is a necessary truth by virtue of its meaning, so we have the counterintuitive result that a necessary truth like (m) implies a wﬀ like (n), which is not a necessary true if the arrow has the properties required for CI . This remark suggests that a possible interpretation of the consequential analytic arrow is that it represents an implication which is valid ex vi terminorum, i.e. by virtue of the meaning of the logical and non logical terms, not simply by virtue of its formal properties.3 It may be of some interest to see that consequential implication may receive a deﬁnition in terms of a theory which has also the roots in Middle Ages, W.E. Johnson’s theory of determination. As is well-known, Johnson outlined a theory of predication according to which a predicate can be either a determinable (as “coloured”) or a determinate under some determinable (as “red”) ([4], vol. I, chapter XI, p. 173). It is less known that in the same book Johnson gives room to a weaker kind of determination, according to which a predicate X may be a superdeterminate of some predicate Y which is said to be a subdeterminate of X. Let us quote Johnson: “When, in considering diﬀerent degrees of determinatess, the predication of one adjective is found to imply another, but not conversely, then the former we shall call

3

The distinction and the relation between two kinds of strong (non Philonian) implication was well-known in the Middle Ages (See W. and M. Kneale’s [4] in the part of Chapter 5 devoted to Pseudo Scotus). Consequentia formalis depends on the meaning of the syncategorematic terms (i.e. of what we call connectives) and may be seen to be represented by strict implication. Consequentia materialis, as Scotus explains, may be thought as an enthymematic form of consequentia formalis since its validity depends on some tacit premise which may be either necessary (bona simpliciter) or contingent (bona ut nunc). “Consequentia vera simpliciter est illa quae potest reduci ad formalem per assumptionem unius propositionis necessariae” (e.g. “Homo currit, igitur animal currit”) et reducitur ad formalem per istam necessaria: “omnis homo est animal”— where the necessary premise may be seen as a meaning postulate. This kind of consequentia may be identiﬁed with analytic consequential implication. “Consequentia materialis bona ut nunc est illa quae potest reduci ad formalem per assumptionem alicuius propositionis contingentis verae” (e.g. “Socrates currit, igitur album currit”, which presupposes “Socrates est album”). Such second kind of consequentia corresponds to synthetic consequential implication.

C. E. A. Pizzi

a superdeterminate of the latter and the latter a subdeterminate of the former” ([3] vol. I, p. 177). To give one of his examples, “greater than 3” is a subdeterminate of “7” or, what is the same, “7” is a superdeterminate of “greater than 3”. Another example could be provided by the predicate “having a non-red colour”: it is not a determinate of “coloured” since “nonred” is not a colour but it is a superdeterminate (i.e. is more speciﬁc) of coloured; it is a also a subdeterminate of “green-or-yellow”, which in its turn is a superdeterminate of “coloured”. We make also the assumption that a subdeterminate predicate is equivalent to the (possibly inﬁnite) disjunction of all its superdeterminate predicates. Now we may try to develop this theory at a propositional level, giving sense to the idea that compound propositions may be seen as superdeterminates or subdeterminates of other propositions. Interpreting inside the object language the notion of theoremhood by means of the necessity operator, symbolized as usual by ◻, deducibility will be rendered by strict implication (≺), consistency by the possibility operator (♦), provable equivalence by the strict equivalence operator (=). Supdet will be the object-language representation of superdetermination. The background system is an arbitrary normal modal system as strong as the deontic KD. According to what has been said, A supdet B implies at least the following statements: (i)

A≺B

(ii)

¬(B ≺ A)

(iii)

B = Vi Ai for every Ai in Vi Ai such that Ai ≺ B and ¬(B ≺ Ai )

A straightforward consequence of the preceding conditions is that A supdet B implies ♦B and also ♦¬A since both follow from ¬(B ≺ A), i.e. from ♦(B ∧ ¬A). A however does not follow from (i) and (ii) but from (iii). Suppose in fact by Reductio that A ≺ B, A = ⊥, that C is a second distinct superdeterminate of B and that B is, say, equivalent to A ∨ C. Then we would have B = (⊥ ∨ C), so B = C, contrary to the condition (ii) which requires the asymmetry between B and C. Then A cannot be ⊥, so ♦A is true. However, since condition (iii) is not expressible in propositional modal language, a minimal deﬁnition of supdet which is possible to formulate in such language is (Def supdet) A supdet B = df A ≺ B ∧ ¬(B ≺ A) ∧ ♦A

Two Kinds of Consequential Implication

Now we may deﬁne CI not only by using supdet but by also considering that A and B may be related by CI simply for being strictly equivalent:4 in other words A CI B should be expressed by the disjunction (A = B) ∨ A supdet B. This means that a possible deﬁnition of CI could be based on the equivalence (1) A CI B if and only if A = B ∨ (A ≺ B ∧ ¬(B ≺ A) ∧ ♦A) Luckily enough, the conjunct ¬(B ≺ A) turns out to be redundant, as we can prove in what follows. Proposition 1. In any normal modal system the wﬀ (z1)A = B ∨ (A ≺ B ∧ ♦A) is logically equivalent to the wﬀ (z2)A = B ∨ (A ≺ B ∧ ¬(B ≺ A) ∧ ♦A). Proof. From right to left the implication is obvious. In the other direction, the ﬁrst remark is that A ≺ B is logically equivalent to (A ≺ B ∧ B ≺ A) ∨ (A ≺ B ∧ ¬(B ≺ A)), so by replacing this disjunction in (z1) and by the deﬁnition of = we obtain this wﬀ equivalent to (z1): (2) A = B ∨ ((A = B ∨ (A ≺ B ∧ ¬(B ≺ A)))∧ ♦A). (2) is in its turn equivalent to the disjunction of (2.1): (2) ∧ ♦A and (2.2): (2) ∧ ¬♦A. From (2.1) we have by PC and Simpliﬁcation ♦A ⊃ (A = B∨(A = B ∨((A ≺ B) ∧ ¬(B ≺ A)))). But given the PC-equivalence C ∨ (C ∨ D) ≡ C ∨ D and the PC-law (C ⊃ (D ∨ R)) ⊃ (C ⊃ (D ∨ (R ∧ C))) we reach ♦A ⊃ (A = B ∨ ((A ≺ B ∧ ¬(B ≺ A)) ∧ ♦A)), so ♦A ⊃ (z2). Suppose now (2.2), i.e. ¬♦A ∧ (A = B ∨ ((A = B ∨ (A ≺ B ∧ ¬(B ≺ A))) ∧ ♦A)). From this formula we have by PC (¬♦A ⊃ A = B) ∨ (¬♦A ⊃ (A = B∨(A ≺ B∧¬(B ≺ A)))∧♦A). Looking at the second disjunct, thanks to the PC-theorem (¬C ⊃ (D ∨ (E ∧ C))) ⊃ (¬C ⊃ D), by Modus Ponens we conclude that ¬♦A implies A = B, so a fortiori the disjunction (z2). The ﬁrst disjunct leads obviously to the same conclusion. So it turns out that (z2) is implied by both (2.1) and (2.2). But the disjunction of (2.1) and (2.2) is equivalent to (2) which is equivalent to (z1), so (z2) follows from (z1). The preceding metatheorem suggests that a minimal deﬁnition of CI might be simply provided by 4

For sake of simplicity it is understood here that strict equivalence and consequential equivalence have the same properties. This fact will be object of a rigorous proof in what follows (see p. 16).

C. E. A. Pizzi

(Def CI ) A CI B = df A = B ∨ (A ≺ B ∧ ♦A) which, let us recall, is equivalent to A = B ∨ (A ≺ B ∧ ¬(B ≺ A) ∧ ♦A). An important remark concerning (Def CI ) is however as follows. Given that ♦A and ♦¬A follow from the deﬁnition of supdet, this implies that any superdeterminate A of B is contingent (possibly true and possibly false): but this condition does not follow for the subdeterminate B (i.e. for the consequent B of A CI B). Suppose in fact that the consequent B of A is necessary, e.g. for being equivalent to some elementary tautology T. Since A ≺ T is a theorem of KD, the disjunction A = T ∨ (A ≺ T ∧ ♦A) turns out to be true under the given set of suppositions, so by (Def CI ) we conclude that A consequentially implies T. Are we ready to accept that, excluding the trivial case of equivalence, every contingent A consequentially implies every tautology? As a a matter of fact, it seems that we have diﬀerent intuitions about the determination relation when the subdeterminate proposition is a logical truth. On the one hand it is clear that, if A and ¬A are both possible, both are more determinate than T, for the same reason why every contingent predicate is more speciﬁc than any tautological predicate. On the other hand, if both A and ¬A are more determinate not only of A ∨ ¬A, but of every equivalent tautology such as B ∨ ¬B, C ≺ C etc. it seems that we lose any intuitive relation of relevance or connection between the clauses5 . If someone is puzzled by such a diﬃculty and is willing to deny that tautologies make sense as subdeterminate of contingent propositions, we need a stronger sense of superdetermination. This is grasped by the following deﬁnition: (Def supdet*) A supdet* B = df A supdet B ∧ ¬ ◻B. So, as it happens, we have two notions of superdetermination, a stronger and a weaker. Consequently, we will have two notions of CI of diﬀerent strength, CI 1 and CI 2 . (Def CI 1 ) A CI 1 B = df A = B ∨ (A ≺ B ∧ ♦A ∧ ¬ ◻B) (Def CI 2 ) A CI 2 B = df A = B ∨ (A ≺ B ∧ ♦A) The normality of the background system leads to the simple consideration that (Def CI 1 ) and (Def CI 2 ) are equivalent to the more complicated deﬁnitions 5 In Pizzi [17] it is argued that consequential implication grasps a special relation of relevance which is diﬀerent from the one worked out in so-called relevant logics. On this point see also see p. 13.

Two Kinds of Consequential Implication

(Def CI 1 +) A CI 1 B = df A = B ∨ (A ≺ B ∧ ♦A ∧ ♦¬B ∧ ♦B ∧ ♦¬A)6 (Def CI 2+ ) A CI 2 B = df A = B ∨ (A ≺ B ∧ ♦A ∧ ♦B) Our problem is now to show that both these relations satisfy the positive and negative properties which has been listed at page 1. This will be done by working out two decidable diﬀerent systems for CI 1 and CI 2 and to reach the result by applying the decision procedures which will be devised for such systems. §1. Let us begin with a formal analysis of the stronger operator of consequential implication which have been devised in the preceding section, CI 1 . We will represent CI 1 by the arrow →. The intuitive meaning of CI 1 may be redescribed as follows. A ≺ B may be or not be symmetric. In the ﬁrst case the clauses A and B are strictly equivalent, so they have the same modal status inside the Aristotelian square of modalities. If A ≺ B is asymmetric, i.e. if it is true that ¬(B ≺ A) and ♦(B ∧ ¬A), the deﬁnition (Def CI 1+ ) implies that A and B are both contingent (possible and not-necessary). Now, if contingency is considered as a modal status7 , in both cases the two clauses of the implication turn out to have the same modal status. But if contingency is not considered a modal status, a further step can be 6

In some papers written beginning from 1982 an Italian historian of ancient logic, Mauro Nasti, proposed a deﬁnition of the notion of connection (sunarthesis, which according to his analysis goes back to Chrysippus) in terms of contingency and non- contingency (see [10] and [11]). Using the conventional deﬁnition of contingency and non-contingency symbols (i.e. ∇A =df ♦A ∧ ♦¬A and ΔA = ¬∇A the deﬁnition he proposed was (Def → N ) A →N B =df (A ≺ B ∧ ∇A ∧ ∇B) ∨ (A = B ∧ ΔA ∧ ΔB) It may be proved that this deﬁnition is equivalent to the above proposed deﬁnition of CI 1 . To begin with, it is easy to show that (Def →N) is equivalent to the simpler (Def → n ) A →n B =df A = B ∨ (A ≺ B ∧ ∇A ∧ ∇B) The proof may be sketched as follows. It is obvious that A →N B entails A →n B by Simpliﬁcation. In the converse direction the proof is as follows. Let F = ∇A ∧ ∇B. Then A=B is equivalent to (A = B ∧ F) ∨ (A = B ∧ ¬F). So the deﬁnition of A → n B becomes (A = B ∧ F) ∨ (A = B ∧ ¬F) ∨ (A ≺ B ∧ F). But since two disjuncts of the formula are identical, the definiens amounts to (A = B ∧ (ΔA ∨ ΔB)) ∨ (A ≺ B ∧ ∇A ∧ ∇B). Now it is easy to see that A = B ∧ (ΔA ∨ ΔB) is equivalent to A = B ∧ ΔA ∧ ΔB, which is the second disjunct of (Def →N ). Thus A →n B implies and is implied by A →N B, so the Nasti deﬁnition boils down to A = B ∨ (A ≺ B ∧ ∇A ∧ ∇B)), which is the deﬁniens of (def CI 1+ ). 7 Contingency in the sense of bilateral possibility (see preceding note) strictly speaking is not part of the Aristotelian square, even it is part of an extended Aristotelian square, named Blanch`e’s hexaghon, in which contingency and non - contingency are placed on two opposite corners of the hexaghon.

C. E. A. Pizzi

made. Given the mentioned premises, it is not diﬃcult to ﬁnd a proof which identiﬁes Def CI 1 with another deﬁnition proposed in Pizzi [13]. According to this deﬁnition A → B amounts to saying that A ≺ B is true and that the modal status of the two clauses is coincident. The deﬁnition is (Def →) A → B =df ◻(A ⊃ B) ∧ (◻B ⊃ ◻A) ∧ (♦B ⊃ ♦A) Notice that Def → is equivalent in any normal system to (3) A → B = ◻(A ⊃ B) ∧ (◻B ≡ ◻A) ∧ (♦B ≡ ♦A) ∧ (¬◻B ≡ ¬ ◻A) ∧ (¬♦B ≡ ¬♦A). The four-elements conjunction at the right of the formula will be said to describe the Equimodality Property of the implication relation between A and B. The connection (or relevance, as one could be willing to say by giving a special sense to the notion of relevance) between the clauses may then be identiﬁed with the fact that they share a common modal status. Thanks to the following proposition we are now able to prove that in all interesting modal systems the deﬁnition of CI 1 and the Equimodality Property of the strict implication between A and B make the same assertion. Proposition 2. In KD Def CI 1 and Def → are equivalent deﬁnitions. Proof. It is enough to prove A CI 1 B ≡ A → B in KD+Def CI 1 +Def →8 (1) A ≺ B ∧ ♦A ∧ ♦¬B

Hyp., Def CI 1

(2) ♦A ⊃ (♦B ⊃ ♦A)

PC

(3) ♦¬B ⊃ (◻B ⊃ ◻A)

PC

(4) (A ≺ B ∧ ♦A ∧ ♦¬B) ⊃ (A ≺ B ∧ (♦B ⊃ ♦A) ∧ (◻B ⊃ ◻ A))

1, 2, 3, PC,MP

(5) A = B ⊃ (A ≺ B ∧ (♦B ⊃ ♦A) ∧ (◻B ⊃ ◻A)) (6) A CI 1 B ⊃ A → B

KD 4, 5, PC, Def →

In the other direction: suppose by Reductio the conjunction of A → B and not-(A CI 1 B) (1) A ≺ B ∧ (♦B ≡ ♦A) ∧ (◻B ≡ ◻A) ∧ ¬(A = B) ∧ ¬(A ≺ B ∧ ♦A ∧ ♦¬B) Hyp., Def → 8

The proof will be given, here and in other Propositions of the papers, by using Conditional Proof. The reader can check that here and elsewhere another proof can be performed without using this device.

Two Kinds of Consequential Implication

(2) A ≺ B∧(A ≺ B ⊃ ¬(B ≺ A)) PC

1, ¬(A = B) ≡ (A ≺ B ⊃ ¬(B ≺ A)),

(3) A ≺ B ∧ ♦(B ∧ ¬A) ∧ (◻B ≡ ◻A) ∧ (♦B ≡ ♦A)

2, MP,1

(4) A ≺ B ∧ (♦B ≡ ♦A) ∧ ♦B ∧ ♦¬A ∧ (◻B ≡ ◻A) ∧ (♦B ≡ ♦A) (5) A ≺ B ∧ ♦A ∧ ♦¬A ∧ (◻B ≡ ◻A)

4, (♦B ∧ (♦B ≡ ♦A)) ⊃ ♦A,

(6) A ≺ B ∧ ♦A ∧ ♦¬A ∧ (♦¬B ≡ ♦¬A) (7) A ≺ B ∧ ♦A ∧ ♦¬B

3, KD PC

5, KD

((♦¬B ≡ ♦¬A) ∧ ♦¬A) ⊃ ♦¬B

(8) ¬(A ≺ B ∧ ♦A ∧ ♦¬B)

1, PC

(9) ⊥

7,8

(10) A → B ⊃ A CI 1 B

1,9

§2. In passing to the axiomatic systems for →, we begin by axiomatizing an ultraweak system named CIw. Primitive functors are ⊥, ⊃, →. The formation rules are standard. The auxiliary symbols ¬, T, ∧ are deﬁned as usual and the deﬁnition of wﬀ is standard. Parentheses will be omitted outside the →-formulas when no ambiguity arises. Auxiliary operators: (Def ◻) ◻A = df T → A (Def ♦) ♦A = df ¬ ◻ ¬A (Def ↔) A ↔ B = df A → B ∧ B → A CIw may be axiomatized as follows. (PC) All truthfunctional tautologies (a) (b) (c) (d) (e) (f) (g)

(p → q ∧ q → r) ⊃ p → r (T → (p ⊃ q) ∧ ¬(T → ¬p) ∧ ¬(T → q)) ⊃ p → q ¬(T → ¬(p ∧ r)) ⊃ (p → q ⊃ (p ∧ r) → (q ∧ r)) ¬p → ¬q ⊃ q → p p→⊥⊃⊥→p ⊥→p⊃p→⊥ p→p

Rules: Uniform Substitution (US), Modus Ponens for ⊃ (MP), Replacement of proved material equivalents (Eq). CIw does not contain neither Aristotle nor Boethius. Strictly speaking, it is not a system satisfying the minimal requirements for CI 1 and we will call it a pre-consequential system. Note that among the axioms we have unrestricted contraposition (d) and transitivity (a). CIw extended with

C. E. A. Pizzi

(h) p → q ⊃ ¬(p → ¬q) (i.e. Weak Boethius) is named CI and is a full-blooded system of consequential implication. CI.0 is the name of CIw extended with (i) p → q ⊃ (p ⊃ q) It is easy to see that in CI and CIw the arrow satisﬁes all the positive properties required for CI 1 , but in order to know if it lacks the negative properties we need a decision procedure. This is obtained by translating CI and other systems of the same family into some decidable modal systems. The main result in this direction lies in proving that the three mentioned systems may be translated, thanks to a one-one embedding, into the three basic systems of normal modal logic without reduction axioms: K, KD and KT. The result is reached by deﬁning inductively two mappings. The ﬁrst, ϕ, goes from the language of CIw to the language of the propositional modal logic K; the second, ψ, goes in the reverse direction. 1a. ϕ(p) = p 2a. ϕ(⊥) = ⊥ 3a. ϕ(A ⊃ B) = ϕ(A) ⊃ ϕ(B) 4a. ϕ(A → B) =◻(ϕ(A) ⊃ ϕ(B)) ∧ (◻ϕ(B) ⊃ ◻ϕ(A)) ∧ (♦ϕ(B) ⊃ ♦ϕ(A)) 1b. ψ(p) = p 2b. ψ(⊥) = ⊥ 3b. ψ(A ⊃ B) = ψ(A) ⊃ ψ(B) 4b. ψ(◻A) = T → ψ(A) The systems CIw and K may be proved to be deﬁnitionally equivalent by proving the following three metatheorems: T1. CIw A only if K ϕ(A) T2. K A only if CIw ψ(A) T3. K A ≡ ϕψ(A) and CIw A ≡ ψϕ(A). The ﬁrst two results T1 and T2 are proved by induction on the length of the proofs, and the two equivalential formulas in T3 are proved by an induction on the complexity of the wﬀs. The proofs may be found in Pizzi [14] and Pizzi-Williamson [18]. Notice that the equivalence (TrEq) A → B ≡ ◻(A ⊃ B) ∧ (◻B ⊃ ◻A) ∧ (♦B ⊃ ♦A)

Two Kinds of Consequential Implication

is a CIw –theorem for every A. Thus the condition (x) of p.2 is satisﬁed since A → B ⊃ ◻(A ⊃ B) and A → B ⊃ T → (A ⊃ B) are both CIw-theorems thanks to the deﬁnition of ◻. Since T → A is equivalent to A → T via contraposition and (e) and (f), we have also the equivalence ◻A ≡ A → T. An analogous result may be proved for KD and KT, i.e. for K+ ◻ p ⊃ ♦p and K+ ◻ p ⊃ p. KD may be proved to be equivalent to CI and KT to CI.0. Since it is easy to prove that the rules of inference preserve the translation, in the light of T1-T3, in order to prove an equivalence between modal and consequential systems it is enough to show that the characteristic axioms of the relevant systems are intertranslatable9 . Note that ◻A = T ≺ A is a KD-theorem, so that T → A and T ≺ A are equivalent assertions. At this point it is straightforward to see that the positive properties required for CI 1 are satisﬁed by →. And thanks to the decision procedure for KD we are able to show that the arrow satisﬁes the negative properties required for CI 1 : Simpliﬁcation, Factor and Addition are underivable in CIw, and obviously A ⊃ B does not entail A → B. But other interesting properties may be devised. The ﬁrst concerns “Secondary Boethius”, i.e. (A → B) ⊃ ¬(¬A → B) (property (xiii) of p. 6), which turns out to be a theorem. Suppose not: then A → B and ¬A → B would both be true, so such would also A ≺ B and ¬A ≺ B, so also ◻B. Then by the Equimodality Property applied to A ≺ B and ¬A ≺ B, both ◻A and ◻¬A should also be true, so also ◻(A ∧ ¬A), which is absurd. The devised decision procedure helps to establishing variants of the negative properties. Simpliﬁcation and Addition hold only in weakened form: (wS) (♦(p ∧ q) ∧ ¬ ◻ p) ⊃ (p ∧ q) → q (wA) (♦p ∧ ¬ ◻ (p ∨ q)) ⊃ q → (p ∨ q) The problem concerning Factor is of some interest. To begin with, Strong Factor, i.e. (SF) (p → q) → ((p ∧ r) → (q ∧ r)) is inconsistent with CI. In fact, if it were a theorem of CI we would have: 9

In the case of KD the proof may be simpliﬁed. It is easy, for instance, to see that Aristotle’s Thesis ¬(B → ¬B) is translated into the deontic axiom D: ◻B ⊃ ♦B or into the equivalent wﬀ ♦(Bv¬B); and we already proved (see p. 7) that in every system containing transitivity and contraposition AT is equivalent to BT. This is enough to establish that the ϕ-translation of axiom (h) is a KD-theorem. The converse ψ-translation from ♦(Bv¬B) to Aristotle’s Thesis is also straightforward.

C. E. A. Pizzi

(1) (p → ¬p) → ((p ∧ ¬p) → (¬p ∧ ¬p))

SF, ¬p/q¬p/r

(2) (p → ¬p) → ((p ∧ ¬p) → ¬p)

1, PC

(3) ⊥→ ((p ∧ ¬p) → ¬p)

⊥≡ (p → ¬p), 2, Eq

(4) ⊥→ (p → T)

3, (d), PC

(5) ⊥→ (T → T)

4, PC, p/T

The formula in line (5) is equivalent to ⊥→ T, which is incompatible with Aristotle’s Thesis. On the contrary, Equivalential Factor, i.e. (EqF) p ↔ q ⊃ ((p ∧ r) → (q ∧ r)) is a CI-theorem. This fact is easily established since strict equivalence and consequential equivalence are equivalent: A ↔ B ≡ A = B may be proved simply because A = B logically implies ◻ (A ≡ B)∧(◻ A ≡ ◻B)∧(♦A ≡ ♦B), so A ↔ B, and viceversa. It may be proved that if Factor were a theorem we would reach as a theorem the formula which we call Symmetry: (S) p → q ≡ q → p Proof. (1) p → q ⊃ ((p ∧ r) → (q ∧ r))

Factor

(2) p → q ⊃ ((p ∧ ¬p) → (q ∧ ¬p))

1, ¬p/r

(3) p → q ⊃ ((p ∨ ¬p) → (q ∧ ¬p))

2, Axioms (e),(f), (d), PC

(4) p → q ⊃ (q ≡ p)

3, Def ◻, p → q ⊃ ◻(p ⊃ q)

(5) ◻(q ≡ p) ⊃ (◻(q ≡ p) ∧ (◻q ≡ ◻p) ∧ (♦q ≡ ♦p)) (6) p → q ⊃ q → p

KD 4,5, PC

Note that (6) is equivalent to p → q ≡ q ↔ p and to p → q ≡ p = q That Symm implies Factor is obvious: from Equivalential Factor (which is a theorem) we derive implicational Factor thanks to Symm. So Symmetry and Factor are equivalent if added as axioms to any consequential system. Symm and Factor are consistent with CIw but underivable in all three systems CIw, CI, CI.0, as shown by the following proposition. Proposition 3. Factor is not a CI.0-theorem Proof. It is enough to show that Symm is underivable in CI.0. Recalling that CI.0 is translated into KT, a formal refutation of Symm by a semantic argument is as follows. The translation of Symm, (◻(p ⊃ q) ∧ (♦q ⊃ ♦p) ∧ (◻q ⊃ ◻p)) ⊃ ((◻(q ⊃ p) ∧ (♦q ≡ ♦p) ∧ (◻ q ≡ ◻p)) is refuted in a KT-model

Two Kinds of Consequential Implication

and q is true at y. In such a world y, q ⊃ p turns out to be false, so the consequent of the implication is also false since ◻(q ⊃ p) is false at x due to the reﬂexivity of R. Now suppose that at x p and q are both true. Then ◻(p ⊃ q), ♦p and ¬ ◻ q are all true at x, so that the antecedent of Symm is true at x by PC. No contradiction follows from the proposed assignment to the atomic variables. The formula Symm A → B ⊃ A ↔ B should not be confused with a property of the system which is essentially diﬀerent: A → B implies

A ≡ B. In fact it is possible to prove what follows: Proposition 4. In CIw, CI, CI.0 A → B implies A ≡ B (the proof is in Pizzi and Williamson [18], pp. 581–582). It is also easy to prove the converse of the preceding proposition, i.e. that A ≡ B implies

A → B, so that A ≡ B and A → B are equivalent statements. A by-product of Proposition 4 is that Strong Boethius’ Thesis (see page 2) is not derivable as a theorem of the three mentioned systems. Suppose in fact by Reductio that it is derivable: then its substitution instance (T → B) → ¬(T → ¬B) yields by Proposition 4 (T → B) ≡ ¬(T → ¬B) and ♦B ≡ ◻B, which obviously is not a theorem of KT; so SBT is not a theorem of CI.0. The positive information we receive from the preceding result is that both Factor and Strong Boethius’ are consistent with the systems CIw, CI, CI.0, but a critical feature of this class of systems is that the arrow is symmetric when it is in main position. A consequence of Proposition 4 is in fact Proposition 5. In CIw, CI, CI.0 A → B implies B → A. §3. We are now ready to examine the second kind of consequential implication. We recall that the deﬁnition of CI 2 was (Def CI 2 ) A CI 2 B = df A = B ∨ (A ≺ B ∧ ♦A) Let us now deﬁne a weaker variant of → (let us call it a truncated variant) in this way: (Def ⇒) A ⇒ B = df A ≺ B ∧ (♦B ⊃ ♦A) Now we may prove what follows: Proposition 6. In CIw A CI 2 B iﬀ A ⇒ B Proof. In (Def CI 2 ) the disjunct A ≺ B∧♦A implies by PC A ≺ B∧(♦B ⊃ ♦A). The second disjunct of (Def CI 2 ) is A=B, which implies A ≺ B, B ≺ A and ♦B ⊃ ♦A. So both the disjuncts imply A ⇒ B. In the other direction

C. E. A. Pizzi

let us move from A ⇒ B, i.e. A ≺ B ∧ (♦B ⊃ ♦A), and let us suppose ♦(B ∧ ¬A), from which ♦B. Now ♦B ∧ (♦B ⊃ ♦A) implies ♦A. So from A ⇒ B and ♦(B ∧ ¬A) it follows (A ≺ B ∧ ♦A) ∨ A = B, i.e. A CI 2 B. And from A ⇒ B and ¬♦(B ∧ ¬A), i.e. B ≺ A, it follows again A = B and (A ≺ B ∧ ♦A) ∨ A = B, so A CI 2 B. A possible criticism of ⇒ is that its modal deﬁnition seems to imply a deviance from the Equimodality Property, given that it lacks the component ◻ A ≡ ◻B. One could remark, however, that the Equimodality Property of strict implication might be formulated in a weaker variant which yields the following deﬁnition: (Def2 ⇒) A ⇒ B =df A ≺ B ∧ ((♦A∨ ◻A) ≡ (♦B∨ ◻ B)) As a matter of fact, the equivalence ♦A∨ ◻A ≡ ♦B∨ ◻B, in KD (but not in K) entails ♦A ≡ ♦B, and the converse implication is also sound. So (Def2 ⇒) and (def ⇒) are actually equivalent. We know that the peculiarity of ⇒ is that a tautology may be considered a subdeterminate of every consistent antecedent, so among the CI-theorems for ⇒ we have the following theorems of KD+Def ⇒ (1) (p ≺ q ∧ ♦p) ⊃ p ⇒ q

Def. CI 2

(2) (p ≺ T ∧ ♦p) ⊃ p ⇒ T

from (1) (T/p)

(3) ♦p ⊃ p ⇒ T

p ≺ T, 2, MP

The wﬀ in line (3) will be called semiparadox of truncated consequential implication. This theorem has no counterpart in the logic of the standard arrow →. It is useful to remark again that the main diﬀerence between CI 1 and CI 2 concerns contraposition, which holds for →, as seen in Axiom (d) of p. 7. Consider now the semiparadox ♦p ⊃ (p ⇒ T). If ⇒ were contrapositive we would have by contraposition ♦p ⊃ (⊥ ⇒ ¬p). But this would mean, by (Def ⇒), ♦p ⊃ (⊥ ≺ ¬p∧(♦¬p ⊃ ♦⊥)). Given that ⊥ ≺ ¬p is a valid formula for strict implication, the last formula boils down to ♦p ⊃ (♦¬p ⊃ ⊥), i.e. ♦p ⊃ ◻ p, which obviously is a non- theorem in every non-trivial normal system. So contraposition cannot hold for ⇒ in unrestricted form. It is fair to realize that contraposition, however, holds in non-restricted form for the equivalence relation for the truncated arrow, which is deﬁned as follows: (Def ⇔) A ⇔ B =df A ⇒ B ∧ B ⇒ A As a matter of fact, strict equivalence = and ⇔-equivalence are provably equivalent, due to the fact that A = B implies A = B ∧ XA ≡ XB,

Two Kinds of Consequential Implication

where X is any modal operator belonging to the Aristotelian square of modalities. The failure of contraposition marks an important diﬀerence in what follows: even if we have among the theorems of CI⇒ (i)⊥ ⇒ p ≡ p ⇒ ⊥, we lack (ii) T ⇒ p ≡ p ⇒ T. In fact, if (ii) were valid, given the semiparadox ♦p ⊃ (p ⇒ T) we would have as a consequence ♦p ⊃ (T ⇒ p). But T → A (i.e. ◻ A) is the same as T ⇒ A ∧ (◻ A ⊃ ◻ T), where the second conjunct is an obvious theorem, so by ♦p ⊃ (T ⇒ p) and Eq we would have again the unwanted ♦p ⊃ ◻ p. It is clear then that not only contraposition cannot hold, but that the equivalence (ii) cannot hold for the truncated arrow ⇒. Via KD-tableaux it is possible to establish, however, that contraposition holds for ⇒ in a weakened form which is the following: (WC1) ♦¬p ⊃ (q ⇒ p ⊃ (¬p ⇒ ¬q)) (WC1) allows applying contraposition to any contingent p which is the consequent of q ⇒ p. Furthermore, contraposition holds for ⇒ also when the antecedent of q ⇒ p is necessary. (WC2) ◻ q ⊃ (q ⇒ p ⊃ (¬p ⇒ ¬q)) The deﬁnition of ◻ which was presupposed until now was in terms of →, i.e. ◻A = T → A. The deﬁnition of modal operators in terms of ⇒ does not yield any diﬀerence with the deﬁnition of the same operators in terms of →. If we deﬁne a new box in terms of ⇒ as (Def ◻◦ ) ◻◦A =df T ⇒ A, which is the same as ◻◦ A =df T ≺ A ∧ (♦A ⊃ ♦T), it is clear that in CIw ◻B (i.e. T → B) implies ◻◦ B. But the converse implication is also true. In fact from ◻◦ B, i.e. ◻(T ⊃ B) ∧ (♦B ⊃ ♦T), considering that ♦B ⊃ ♦T and ◻B ⊃ ◻T are logical truths, it is straightforward to obtain ◻B. So in CI + Def → + Def ⇒ and in every system containing CI, ◻A ≡ ◻◦A is a theorem. Up to now ⇒ has been treated as an auxiliary symbol of CIw, but we are interested in giving it an independent axiomatization. The axiomatization of the fragment of CIw containing ⇒-formulas is parallel to the one for →, with the exception of some relevant diﬀerences in two axioms, (b) and (d). The preconsequential system CIw⇒ is axiomatized in this way: (PC) All truthfunctional tautologies (a) (p ⇒ q ∧ q ⇒ r) ⊃ (p ⇒ r) (b) (T ⇒ (p ⊃ q) ∧ ¬(T ⇒ ¬p)) ⊃ p ⇒ q

C. E. A. Pizzi

(c) (d) (e) (f) (g)

¬(T ⇒ ¬(p ∧ r)) ⊃ (p ⇒ q ⊃ ((p ∧ r) ⇒ (q ∧ r)) (¬(T ⇒ ¬q) ∨ T ⇒ ¬p) ⊃ (¬p ⇒ ¬q ⊃ q ⇒ p) p⇒⊥⊃⊥⇒p ⊥⇒p⊃p⇒⊥ p⇒p

Rules: Modus Ponens for ⊃, Uniform Substitution, Eq To CIw⇒ one could add the proper axiom of the consequential system CI⇒ (h) p ⇒ q ⊃ ¬(p ⇒ ¬q) and the axiom of the stronger system CI.0⇒ (i) p ⇒ q ⊃ (p ⊃ q) Now we are in conditions to prove that the consequential system CI⇒ is equivalent to KD. The proof parallels the one which is already known for → but is complicated by the fact that contraposition can be applied only with some restriction. As before, we deﬁne two translation functions, f and g , which recall the preceding ϕ and ψ. The former are coincident with the latter as far as truth-functional operators are concerned, but are diﬀerent for the following clauses: (i) f(A ⇒ B) =◻◦(fA ⇒ fB) ∧ (♦◦ fB ⊃ ♦◦ fA) (ii) g (◻ ◦A) = T ⇒ g A We restrict our interest to the consequential system CI⇒ and have then to prove the following three metatheorems 7, 8, 9. Proposition 7. For every A such that A is a KD-theorem, g(A) is a CI⇒ -theorem Proof. By induction on the length of the proofs. The application of the function g to the axioms of KD gives the following result: (1) g (◻◦p ⊃ ♦◦ p) = (T ⇒ g p) ⊃ ¬(T ⇒ g (¬p)) = (T ⇒ p) ⊃ ¬(T ⇒ ¬p) (2) g (◻◦(p ⊃ q) ⊃ (◻◦p ⊃ ◻◦q)) = (T ⇒ g (p ⊃ q)) ⊃ (T ⇒ g p ⊃ (T ⇒ g q) = (T ⇒ (p ⊃ q)) ⊃ (T ⇒ p ⊃ T ⇒ q) The two output wﬀs may be proved to be CI ⇒-theorems. The former is an instance of axiom (h). The latter is proved as follows: (1) (T ⇒ ((p ∧ q) ⊃ q)) ∧ ¬(T ⇒ ¬(p ∧ q))) ⊃ (p ∧ q) ⇒ q (2) T ⇒ ((p ∧ q) ⊃ q)

Ax(b), p ∧ q/p

T ≡ ((p ∧ q) ⊃ q), Ax(g), Eq

Two Kinds of Consequential Implication

(3) ¬(T ⇒ ¬(p ∧ q)) ⊃ (p ∧ q) ⇒ q (4) (T ⇒ (p ∧ q)) ⊃ (p ∧ q) ⇒ q

1, 2, MP 3, Ax (h), T/p, p ∧ q/q, PC

(5) (T ⇒ (p ∧ q)∧(p ∧ q) ⇒ q) ⊃ T ⇒ q

Ax(a), T/p, p∧q/q, q/r

(6) (T ⇒ (p ∧ q)) ⊃ T ⇒ q

4,5, PC

(7) ¬(T ⇒ ¬p) ⊃ (T ⇒ q ⊃ (p ⇒ (p ∧ q)) Ax(c), T/p, p/r, T ∧ p ≡ p, Eq (8) (p ⇒ (p ∧ q) ∧ T ⇒ p) ⊃ T ⇒ (p ∧ q)

Ax(a),T/p, p ∧ q/r, p/q

(9) ¬(T ⇒ ¬p) ⊃ ((T ⇒ p ∧ T ⇒ q) ⊃ T ⇒ (p ∧ q)) (10) (T ⇒ ¬p) ⊃ ((T ⇒ p ∧ T ⇒ q) ⊃ T ⇒ (p ∧ q)) ¬(T ⇒ p) (11) (T ⇒ p ∧ T ⇒ q) ⊃ T ⇒ (p ∧ q)

7, 8, PC 9, T ⇒ ¬p ⊃ Ax(h), PC 9, 10, PC

(12) (T ⇒ p ∧ T ⇒ (p ⊃ q)) ⊃ T ⇒ (p ∧ (p ⊃ q))

11, p ⊃ q/q

(13) (T ⇒ p ∧ T ⇒ (p ⊃ q)) ⊃ T ⇒ (p ∧ q)

12, PC, Eq

(14) (T ⇒ p ∧ T ⇒ (p ⊃ q)) ⊃ T ⇒ q

13, 6, PC

It is then to be proved that the rules MP, US, Eq preserve the translation. The proof is routine. Proposition 8. For every A such that A is a CI⇒- theorem, f(A) is a KD-theorem Proof. By induction on the length of proofs. The translation of the CI⇒axioms may be tested with the well-known tableaux method for KD, even if quicker arguments may be viable. –

The proof concerning axioms Ax (a)–(c) is trivial and is left to the reader.

–

As Axiom (d) (weakened contraposition) is concerned, we have two steps to do in order to take care of the disjunctive premise ¬(T ⇒ ¬q) ∨ (T ⇒ ¬p), considering that both disjuncts imply the consequent by Simpliﬁcation of Disjunctive Antecedents. (i) We remark that ¬(T ⇒ ¬q) ⊃ (¬p ⇒ ¬q ⊃ q ⇒ p) is translated into ¬(◻◦ (T ⊃ ¬q) ∧ (♦◦ ¬q ⊃ ♦◦ T)) ⊃ (◻◦ (¬p ⊃ ¬q) ∧ (♦◦ ¬q ⊃ ♦◦ ¬p)) ⊃ (◻◦ (q ⊃ p) ∧ (♦◦ p ⊃ ♦◦ q)). The wﬀ ¬(T ⇒ ¬q) is translated into the disjunction ♦◦ (T ∧ q) ⊃ (♦◦ ¬q ∧ ¬♦◦ T), where the second disjunct equals ⊥ and the wﬀ ♦◦ (T ∧ q) amounts to ♦◦ q and implies (♦◦ p ⊃ ♦◦ q). ◻◦(q ⊃ p) is implied by ◻◦(¬p ⊃ ¬q), so the conclusion q ⇒ p is implied by the two joint premises.

C. E. A. Pizzi

(ii) As far as the second disjunct T ⇒ ¬p is concerned: T ⇒ ¬p and ¬p ⇒ ¬q imply T ⇒ ¬q. But ◻◦ ¬q jointly with ◻◦ ¬p implies in KD q ≺ p ∧ (◻◦¬p ≡ ◻◦¬q), so q ≺ p ∧ (♦◦ p ≡ ♦◦ q), so q ⇒ p. –

Ax(e) and Ax(f). Their conjunction is the equivalence p ⇒ ⊥ ≡ ⊥ ⇒ p. We have simply to remark that f(p ⇒ ⊥) is ◻◦ (p ⊃ ⊥) ∧ (♦◦ p ⊃ ♦◦ ⊥) and, given that ⊥ ≡ ♦◦ ⊥, ¬♦◦ p. But f(⊥ ⇒ p) is ◻◦(⊥ ⊃ p) ∧ (♦◦ ⊥ ⊃ ♦◦ p), so ◻◦¬p, which is again equivalent to ¬♦◦ p. So f(p ⇒ ⊥) = f(⊥ ⇒ p) is a KD-theorem.

–

Ax(h). f(p ⇒ q ⊃ ¬(p ⇒ ¬q)) = (◻◦(p ⊃ q) ∧ (♦◦ q ⊃ ♦◦ p)) ⊃ ¬(◻◦(p ⊃ ¬q) ∧ (♦◦ ¬q ⊃ ♦◦ p)), where the consequent is equivalent to ♦◦ (p ∧ q) ∨ (♦◦ ¬q ∧ ¬♦◦ p). We have two cases to consider: (i)

(ii)

suppose ♦◦ p. From ◻◦ (p ⊃ q) and ♦◦ p in all normal modal systems it follows ◻◦ q ⊃ ♦◦ (q ∧ p) and ♦◦ p ⊃ ♦◦ (p ∧ q): so, by PC, (♦◦ ¬q ∧ ¬♦◦ p) ∨ ♦◦ (p ∧ q), i.e. the translation of ¬(p ⇒ ¬q). suppose ¬♦◦ p. By f(p ⇒ q) and contraposition of ⊃ we obtain ¬♦◦ p ⊃ ¬♦◦ q, so by ¬♦◦ p and Modus Ponens ¬♦◦ q, so ¬♦◦ p∧♦◦ ¬q. This conjunction implies ¬♦◦ p∧¬◻◦q and a fortiori ♦◦ (p ∧ q) ∨ (¬♦◦ p ∧ ¬◻◦q), i.e. the f -translation of ¬(p ⇒ ¬q).

The proof that the inference rules preserve the KD-theoremhood of the translations is routine. We have now to prove the following propositions: Proposition 9. For every A written in the language of CI ⇒, CI ⇒ A ≡ gfA Proof. By induction on the complexity of A. The proof is trivial for atomic wﬀs, ⊥ and A ⊃ B. The non- trivial case is A ⇒ B. Let us suppose by Induction Hypothesis A ≡ gfA and B ≡ gf B, from which by Eq we have A ⇒ B ≡ gfA ⇒ gf B. Then it is enough to prove in CI⇒ the equivalence gf (A ⇒ B) ≡ gfA ⇒ gf B. But gf(A ⇒ B) is the same as T ⇒ (gfA ⊃ gf B) ∧ (¬(T ⇒ ¬gf B) ⊃ ¬(T ⇒ ¬gfA). Hence it is enough to prove in CI⇒ the equivalence p ⇒ q ≡ (T ⇒ (p ⊃ q)) ∧ (¬(T ⇒ ¬q) ⊃ ¬(T ⇒ ¬p)), since the required result follows by US. (i) From right to left we have to prove a formula which is equivalent to (T ⇒ (p ⊃ q) ∧ (T ⇒ ¬q ∨ ¬(T ⇒ ¬p)) ⊃ p ⇒ q As a matter of fact, this amounts to proving the following two wﬀs: (A) (T ⇒ (p ⊃ q) ∧ ¬(T ⇒ ¬p)) ⊃ p ⇒ q (B) (T ⇒ (p ⊃ q) ∧ (T ⇒ ¬q)) ⊃ p ⇒ q

Two Kinds of Consequential Implication

But (A) is axiom (b) of CI⇒, so we have simply to prove (B). We know from Proposition 7 that the g -translation of any KD-proposition belongs to CI⇒, and this fact may be used to simplify the proof. (1) T ⇒ ¬q

Hyp.1

(2) T ⇒ (p ⊃ q)

Hyp.2

(3) T ⇒ ¬p ⊃ T ⇒ ¬q

Hyp.1, PC

(4) (T ⇒ ¬q ∧ T ⇒ (p ⊃ q)) ⊃ (T ⇒ ¬q ⊃ T ⇒ ¬p) KD(◻◦¬q ∧ ◻◦ (p ⊃ q)) ⊃ ◻◦¬p, PC (5) T ⇒ ¬p ≡ T ⇒ ¬q

1, 2, 4,MP, 3, PC

(6) ¬(T ⇒ q) ⊃ ¬q ⇒ T

semiparadox (line 3 of p.16)

(7) ¬q ⇒ T

1, Ax (h), 6

(8) T ⇒ ¬q ≡ T ⇔ ¬q

1,7, PC, Def ⇔

(9) T ⇒ ¬p ≡ T ⇔ ¬q

8, 5

(10) (T ⇔ ¬q ∧ ¬p ⇔ T) ⊃ ¬q ⇔ ¬p

ax a), PC (Theor. Praeclarum)

(11) (T ⇒ ¬q ∧ ¬p ⇔ T) ⊃ ¬q ⇔ ¬p

8, 10, PC

(12) (T ⇒ ¬p ∧ (¬(T ⇒ p) ⊃ ¬p ⇒ T)) ⊃ (T ⇒ ¬p ⊃ ¬p ⇔ T) PC, ax(h) (13) T ⇒ ¬p ⊃ ¬p ⇔ T

12, ¬(T ⇒ p) ⊃ ¬p ⇒ T (semiparadox)

(14) (T ⇒ ¬q ∧ T ⇒ ¬p) ⊃ ¬q ⇔ ¬p

8,13,10

(12) T ⇒ ¬q ⊃ q ⇔ p

5, 11, PC q ⇔ p ≡ ¬q ⇔ ¬p

(13) T ⇒ ¬q ⊃ p ⇒ q

12, def ⇔, PC

(14) (T ⇒ (p ⊃ q) ∧ T ⇒ ¬q) ⊃ p ⇒ q

1, 2, 13, PC

Proposition 10. For all A written in the language of KD, KD A ≡ fgA Proof. Induction hypothesis: for any arbitrary A, let us suppose KDA ≡ fg A. Avoiding the trivial steps, one should prove KD ◻◦A ≡ ◻◦fg A. It is enough to prove ◻◦A ≡ (◻◦(T ⊃ fg A)∧(♦◦ fg A ⊃ ♦◦ T), where the right side boils down by Eq to ◻◦(T ⊃ A) ∧ (♦◦ A ⊃ ♦◦ T), i.e. to ◻◦A. The remaining part of the proof is routine. The one-one translation between KD and CI⇒ may be extended to stronger systems: in particular, it is not diﬃcult to prove that KT and CI.0⇒ are deﬁnitionally equivalent systems. The translation provides a decision procedure for CI⇒ and allows seeing whether the negative properties

C. E. A. Pizzi

required for CI 2 are satisﬁed. Looking at the non-theorems it is easy to establish via the decision procedure for KD what follows: (i)

A ⇒ B is not implied by A ⊃ B

(ii) (A ∧ B) ⇒ A is not KD-valid (iii) A ⇒ (A ∨ B) is not KD-valid (iv) A ⇒ B does not entail in KD (A ∧ R) ⇒ (B ∧ R). With reference to (ii), we notice that line 3 of the proof of Proposition 7 contains Weakened Simpliﬁcation in the form ♦◦ (p ∧ q) ⊃ ((p ∧ q) ⇒ p). With reference to (iii), we observe that the decision procedure grants that a simple theorem of CI⇒ is ♦◦ p ⊃ (p ⇒ (p ∨ q)). With reference to (iv), it was proved in Pizzi [16] (p.323-324) that ⇒ -Factor in CI yields the underivable trivializing formula ♦A ≡ ◻A. This makes a diﬀerence with → -Factor which, as proved at p. 17, is equivalent to Symmetry. On the other hand, since A ↔ B is equivalent to A ⇔ B, Equivalential Factor holds for both ↔ and ⇔, while Strong Factor for ⇒ yields ⊥⇒ (T ⇒ T) along the lines given for → at p. 17, so it is inconsistent with CI⇒. The result of the analysis developed in Sects. 2 and 3 is as follows: it turns out that we have two systems, CI and CI⇒, that are both equivalent to the same system KD. So for every wﬀ A written in ⇒-language we may ﬁnd not only a modal translation f(A) but a ψ-translation of f(A), ψf(A), which is a translation of A in terms of →. The same double translation holds, mutatis mutandis, in the direction from →-formulas to ⇒-formulas. This suggests that to CI⇒ we could add the following deﬁnition of →: (Def → ◦ ) A → B = df A ⇒ B ∧ (T ⇒ B ⊃ T ⇒ A) But, given that ¬B ⇒ ¬A is equivalent to ◻◦(¬B ⊃ ¬A) ∧ (♦◦ ¬A ⊃ ♦◦ ¬B), i.e. to ◻◦ (A ⊃ B) ∧ (◻◦ B ⊃ ◻◦ A), another equivalent alternative deﬁnition could be: (Def → ◦◦ ) A → B =df A ⇒ B ∧ ¬B ⇒ ¬A In the other direction we could deﬁne A⇒ B in terms of → by extending CI with the following deﬁnition: (Def ⇒’) A ⇒ B =df (T → (A ⊃ B)) ∧ (¬(T → ¬B) ⊃ ¬(T → ¬A)). So the two systems CI and CI⇒ are in principle intertranslatable. We have now to evaluate the peculiarities of the double arrow that are not directly recorded in the axiomatic basis (the main of which, as we remind, is the diﬀerent treatment of contraposition).

Two Kinds of Consequential Implication

It turns out that, while Secondary Boethius is a CI -theorem (as proved at page 8), this does not hold for the truncated arrow. The more direct proof of this fact is as follows. Suppose by a contradiction that (p ⇒ q) ⊃ ¬(¬p ⇒ q) is a CI⇒-theorem. If T is substituted for q in such formulas we obtain (p ⇒ T) ⊃ ¬(¬p ⇒ T). So ◻◦(p ⊃ T) ∧ (♦◦ T ⊃ ♦◦ p) should entail ¬ ◻◦(¬p ⊃ T) ∨ ¬(♦◦ T ⊃ ♦◦ ¬p). But ◻◦ (p ⊃ T) ∧ (♦◦ T ⊃ ♦◦ p) equals ♦◦ p, while the consequent equals ⊥ ∨ (♦◦ T ∧ ¬♦◦ ¬p): so, given that ♦◦ T is a theorem, the consequent would reduce to ¬♦◦ ¬p, i.e. to ◻◦ p. Thus, if Secondary Boethius written in ⇒ were a theorem, ♦◦ p ⊃ ◻◦p would be a KDtheorem, which is obviously false. It turns out then that adding Secondary Boethius and adding Factor to CI⇒ yields the same result., i.e. the collapse of ♦◦ on ◻◦ . Must the absence of Secondary Boethius be considered a serious fault of the system? Montgomery and Routley in their powerful criticism of Angell’s connnexive logics (see [9]) held (p. 91) that if we read A ⇒ B and ¬A ⇒ B as subjunctive conditionals one cannot claim that they are conﬂicting in all contexts, since they can be asserted on the basis of diﬀerent background conditions. The remark however cannot be of use here since we are interpreting ⇒ as an operator for “analytic” implication. It is enough to remark that, according to the interpretation of A ⇒ B proposed at p. 4, both p and ¬p are superdeterminate of p ∨ ¬p, provided they are consistent. So it makes sense to assert that p ⇒ (p ∨ ¬p) and ¬p ⇒ (p ∨ ¬p) are both true provided ♦p and ♦¬p are true, which is a counterexample to Secondary Boethius. A second question concerns the collapse of the main arrow operator on other weaker or stronger operators. As we already saw with reference to →, in CIw we meet the equivalence between A → B, A ≡ B and A ↔ B, and in Pizzi and Williamson[18] it is proved that the same result holds for CI and CI.0 (see also Pizzi [15]). It is easy to prove that an analogous result may be extended to both CIw⇒ and CI⇒ thanks to the validity of the so-called Simple Cancellation Rule (see [20]): (SCR) ♦A ≡ ♦B iﬀ A ≡ B From SCR we have that A ≺ B, ♦A ≡ ♦B (so A ⇒ B) iﬀ A ≡ B. But, interestingly enough, this result does not hold for CI.0⇒ (=KT). Consider in fact the critical wﬀ (p ⊃ ◻ p) ⇒ T. It is a theorem of CI.0⇒ (=KT) in as much as ♦T ⊃ ♦(p ⊃ ◻p) is a well-known theorem of KT and such is obviously ◻ ((p ⊃ ◻ p) ⊃ T). But (p ⊃ ◻ p) ≡ T is not a KT-theorem, otherwise the collapse formula p ⊃◻ p would be such. So the rule A ⇒ B iﬀ A ≡ B does not hold in CI.0⇒.

C. E. A. Pizzi

So it is not in general true that in every consequential system as strong as KT A ⇒ B iﬀ A ≡ A is a valid rule. It also follows that in general

A ⇒ B does not imply A → B (otherwise the preceding equivalence would hold) and also that in general A ⇒ B does not imply B ⇒ A and

A ⇔ B (otherwise we would have A = B and A ≡ B). §4. Let us quickly summarize the reasons which allow saying that ⇒ may be defended a viable alternative to →. The more critical feature of ⇒ surely is the failure of contraposition. As already remarked, contraposition applies to ⇒ only with minor restriction on the consequent (which must be contingent) or on the antecedent (which must be necessary). These restrictions may seem unjustiﬁed if applied to analytic conditionals. Notice however that the justiﬁcation depends on how analytic implication is interpreted. Duncan Jones [3] and Parry [12] oﬀer two interpretations of analytic implication which do not to satisfy contraposition, and should be usefully compared with the one provided here in terms of superdetermination. Other properties of the truncated arrow are less controversial and could receive a justiﬁcation. (1) Simpliﬁcation holds at the condition that the two conjuncts are compossible: ♦◦ (A ∧ B) ⊃ ((A ∧ B) ⇒ B). This is more simple than the stronger restriction required for Simpliﬁcation in →-formulas, which needs also the restrictive clause ¬ ◻◦ B. Addition in the version ♦◦ p ⊃ (p ⇒ (p ∨ q)) is also more manageable. (2) When ⇒ is the main operator, it collapses on ≡ in CIw⇒, CI⇒ but not in CI.0⇒. So what may appear a trivialization holds only in weak consequential systems for the truncated arrow.10 (3) Secondary Boethius’, i.e. (A ⇒ B) ⊃ ¬(¬A ⇒ B), does not hold in CI⇒. This is not necessarily a fault, since one could ﬁnd that the law is quite counterintuitive under certain interpretation. Notice however that Aristotle’s Thesis holds for ⇒ in both the primary and the secondary variant. We want now to analyze a further reason to give some credit to the double arrow. We said at the beginning that the logic of context – dependent conditionals is not be coincident with the logic of analytic conditionals: this Notice however that Strong Boethius in the formulation (p ⇒ q) ⇒ ¬(p ⇒ ¬q), if added to CI.O⇒ (i.e. to KT) yields a system which contains a modal logic whose axioms are, beyond (T), ◻p ⊃ p, ◻♦T and ♦♦p ≡ ♦ ◻p, so ◻◻ p ≡ ◻ ♦p (see axioms 1D and 2F in Pizzi and Williamson [18], p. 579). Since one derives from this set of axioms ◻◻(♦p ≡ ◻p) (see [18], p. 583) axiom T leads directly to the equivalence ♦p ≡ ◻p and to the collapseformula p ⊃ ◻p. 10

Two Kinds of Consequential Implication

distinction has been the main achievement of classical conditional logic, in which however the basic “corner operator” > occurs both in counterfactual and tautological conditionals. A step toward a more discriminating language has been the introduction of a “circumstantial operator” * made by L.˚ Aqvist, which introduced a heavy sets of axioms for it in [2]. The minimal properties which one can assign to such operator are the following (see [14]). If we start from the modal systems, for our analysis it is enough to extend system KD (which translates CI⇒) with the following axioms for *: (Ax1*) ♦◦ p ⊃ ♦◦ *p (Ax2*) *p ⊃ p US, MP, Nec and Eq are the only rules along with the new rule: REq* A ≡ B / *A ≡ *B This new system will be called KD*. Notice that, given that ♦◦ T is a KD-theorem, such is ♦◦∗ T by Ax1*. Now there are two axioms of increasing strength which may be added to Ax1* and Ax2*: (Ax3*a) (*p ∧ *q) ⊃ *(p ∧ q) (Ax3*b) (*p ∧ q) ⊃ (*q ∧ p) Ax3*a follows from Ax3*b by substituting in it p ∧ q to q, so to reach (*p ∧ q) ⊃ *(p ∧ q) via *p ∧ p ≡ *p and Simpliﬁcation. Ax3*a then follows from a wﬀ which is a PC-consequence of *q ⊃ q, i.e. (*p ∧ *q) ⊃ (*p ∧ q), in conjunction with (*p ∧ q) ⊃ *(p ∧ q). Now we have to deﬁne a corner-operator for conditionals which are context-dependent and have also the properties of consequential conditionals. A ﬁrst intuitive proposal could be (Def > ) A > B =df *A → B. where the arrow → is deﬁned as at p.7. An obvious consequence of (Def > ) is that A > A and T > T turn out to be theorems. But by the modal deﬁnition of → we have ◻T ⊃ ◻*T, so by MP ◻*T, so by deﬁnition of the box T → *T, so by the cancellation rule mentioned at p.16 T ≡ *T, so *T. This result is however unwelcome, since it means that we cannot extend the calculus with axiom Ax3*b. In fact, an instance of Ax3*b would be (*T ∧ q) ⊃ (*q ∧ T); so, by the theorem *T, Exportation and Modus Ponens we reach soon the collapse q ≡ *q.

C. E. A. Pizzi

We observe, however, that this result cannot be obtained if we endorse the deﬁnition (Def >) A > B =df *A ⇒ B which yields the equivalence A > B = ◻(*A ⊃ B) ∧ (♦B ⊃ ♦*A). T > T turns out to be a theorem, but the translation ◻ (*T ⊃ T) ∧ (♦T ⊃ ♦*T) does not yield *T. In order to prove this fact we show that *T is underivable in the strongest system KD* + Ax3*b. To see this, we introduce in the language of KD a propositional constant w (but modifying the formation rules in such a way that the compounds of w are not well-formed formulas) and extend KD with the axiom ♦p ⊃ ♦(w ∧ p). ♦w is an obvious theorem of this system. We stipulate that, for every A,*A is translated into w ∧ A : then A > B, i.e. *A ⇒ B, is translated into (w ∧ A) ⇒ B. The new system KDw =KD+ ♦p ⊃ ♦(w ∧ p)+ Def ⇒ + Def > is easily proved to be consistent, decidable and non-trivial. The procedure consists simply in this: in order to test any wﬀ A in KDw simply test, for every p in A, ♦p ⊃ ♦(w∧p) ⊃ A in KD, treating in the tableau w as a variable not occurring in the other subformulas. It turns out that w ⊃ ♦w is a theorem even if ◻ w ⊃ w is not, due to the restriction on formation rules. There is no one-one translation between KD* + Ax3*b and KDw, even it may be easily proved by induction that all the translations of the theorems of the former systems are theorems of the latter. In particular such is Ax3*b, i.e. (*p ∧ q) ⊃ (*q ∧ p), which is translated into the tautology ((w ∧ p) ∧ q) ⊃ ((w ∧ q) ∧ p). The wﬀ *T is translated into w ∧ T, so into w, which is easily seen not to be a theorem of KDw. So *T is underivable in KD* + Ax3*b. A more general result which is easily reached by a parallel argument is that p ⊃ *p is not a theorem of KD* + Ax3*b. With the same tools we may also prove that, in KD*+ Ax3*b +Def ⇒ + Def >, A > B does not collapse on A ⇒ B. We conclude that ⇒ is more suitable than → to give a basis to a logic of subjunctive and counterfactual conditionals. In this connection two remarks are in order: (i) Some properties of ⇒, i.e. non-monotonicity and non-contraposivity, are in common with the properties of synthetic conditionals, so that the deﬁnition of the synthetic operator in terms of ⇒ turns out to be especially natural. (ii) In the strongest system of synthetic implication KD*+ Ax3*b +Def ⇒ + Def > the relation > turns out to be transitive: given that *A ∧ B implies *B ∧ A, it is easy to realize that *A ⇒ B ∧ *B ⇒ C entails *A ⇒ C. It turns out then that the logic of consequential implication gives rooms for

Two Kinds of Consequential Implication

non-trivial system of non-monotonic transitive subjunctive conditionals, a quality which marks an important diﬀerence with Stalnaker-Lewis systems of classical conditional logics. References [1] Angell, R. B., A Propositional Logic with Subjunctive Conditionals, The Journal of Symbolic Logic 27: 327–343, 1962. [2] ˚ Aqvist, L., Modal Logic with Subjunctive Conditional and Dispositional Predicates, Journal of Philosophical Logic 2: 1–76, 1973. [3] Duncan Jones, A. E., Is strict implication the same as entailment? Analysis 2: 70–78, 1935. [4] Johnson, W. E., Logic, Cambridge U.P. (reprint Dover 1964), 1921. [5] Kneale, W. C., and M. Kneale, The Development of Logic, Clarendon U.P., 1962. [6] Mccall, S., Connexive Implication, Journal of Symbolic Logic 31: 415–433, 1966. [7] Mccall, S., Connexive Class Logic, Journal of Symbolic Logic 32: 83–90, 1967. [8] Mccall, S., Connexive Implication, in A. R. Anderson and N. Belnap, Entailment, vol. I, Princeton U.P., 1975, pp. 434–452. [9] Montgomery, M. and R. Routley, On systems containing Aristotle’s Thesis, Journal of Symbolic Logic 33: 82–96, 1968. [10] Nasti De Vincentis, M., Connexive Implication in a Chrysippean Setting, in C. Cellucci, M. Di Maio, and G. Roncaglia (eds.), Logica e filosofia della scienza: problemi e prospettive, Pisa, ETS, 1994, pp. 595–603. [11] Nasti De Vincentis, M., Logiche della connessivit` a, Verlag Paul Haupt, Bern, 2002. [12] Parry, W. T., Ein Axiomsystem fur eine neue Art von Implikation, Ergebnisse eindes Mathematischen Kolloquims 4: 5–6, 1933. [13] Pizzi, C., Una procedura di decisione mediante tableaux per una logica connessiva debole, Boll. Un.Calabria 4: 131–142, 1982–83. [14] Pizzi, C., Decision Procedures for Logics of Consequential Implication, Notre Dame Journal of Formal Logic 32: 618–636, 1991. [15] Pizzi, C., Weak vs Strong Boethius’ Thesis: a Problem in the analysis of Consequential implication, in A. Ursini and P. Aglian` o (eds.), Logic and Algebra, M. Dekker Inc. NY, 1996, pp. 646–654. [16] Pizzi, C., A Modal Framework for Consequential Implicaton and the Factor Law, Contemporary Mathematics 235: 313–326, 1999. [17] Pizzi, C., Aristotle’s Thesis between Paraconsistency and Modalization, Journal of Applied Logic 3: 119–131, 2005. [18] Pizzi, C., and T. Williamson, Strong Boethius’ Thesis and Consequential Implication, Journal of Philosophical Logic 26: 569–588, 1997. [19] Wansing, H., Connexive Logic, in E. N. Zalta (ed.), The Stanford Encyclopedia of Philosophy (Fall 2014 Edition). http://plato.stanford.edu/archives/fall2014/entries/ logic--connexive/.

C. E. A. Pizzi [20] Williamson, T., Veriﬁcation, Falsiﬁcation and Cancellation in KT, Notre Dame Journal of Formal Logic 31: 286–290, 1990.

C. E. A. Pizzi Emeritus Siena University viale Abruzzi 36 20131 Milano Italy [email protected]