Grice e Pizzi

 Since 1952 the kind of implication known as connerive implication has been the focus of an original research program. The main formal contributions in this area are due to Robert Angell (1] and Storrs McCall (8), but the basic idea of connexive implication was clearly outlined by Everett Nelson in the Thirties (13). Nelson was critical of the so-called Law of Simplification, viz. the principle that, for every p and every 4, the conjunction of p with q implies each one of the conjuncts. Clearly inferences of this form are valid when p and q are jointly consistent. But what should we say when they are not, for instance when q is just -p or when q is

-(P→P), which states that p cannot imply itself'?

The idea of connection which Nelson was trying to capture is characterized by the property that, if we have the truth of A - B (where "-" relation of connexive implication) we cannot also have the truth of A 4 -B. If we accept the logical principle A → B- -(A → -B) - which we shall name Boethias' rule following Kielkopf?- along with unrestricted substitution, then this leads to a rejection of Simplification in the form (p^q) → q. If we had, in fact, (pAg) → g as a law of logic, we would have by Uniform Substitution both (pAp) - p (asan instance of A → B) and also (pA-p) - -p (as an instance A → -B), a result: which is incompatible with Boethius' Rule,

If we assume that p

→ pis a valid formula, and there seems no reason not

to do so, and we accept it as an instance of A - B, then by applying Boethius Rule we obtain what is known as Aristotle's Thesis: -(p - -p). Aristotle's Thesis is the cornerstone of connexive implication, since it states a new version of the Principle of Non-Contradiction. Indeed, in connexive logic p — -p is the paradigm contradiction. If L is a symbol for an arbitrary contradiction, then it follows from Aristotle's Thesis that L- p cannot be a connexive thesis since p could be exactly L, that is, an arbitrary tautology. (Henceforth we will symbolize an arbitrary tautology by T). It is thus clear that connexive logies are "non-Scotian" in the sense that in such logics contradictions can imply only contradictions while tautologies are implied only by tautologies.

What is the correct formulation of Boethius' Rule in the object language? In the first papers written by Angell and MeCall we find the law : (p → q) - -(p → -g)-Angell's original system PAI (see (1]) was axiomatized as follows.

1. （p→4）→（19→7→0p→7）
2. (р→ -(gA)) - ((дЛр) → пг))
3. (р - q) → ((рАг) → (г Л))
4. (рА (q Аг)) → (дА (рАг))
5. (р → q) → -(р → -q)

1. -（pA-（p/p））
2. (p → q) - -(p→ 7g)
3. （g→p）→（p→g
4. пр - р

Transformation Rules:

RL. IfF SandPS → S' then I S' R2. If S and F S' then FS NS'

R3. If S and v is a propositional variable occurring in S, and S' is obtained by Uniform Substitution of any t for u, t S'

RA If S, and S' is got by replacing any part, or all, of S by an erpression equivalent through rules of abbremation, then 5'

It should be noted that Modus Ponens is formulated in terms of "—" and the same holds for R4, which amounts to a Rule of Replacement for - -equivalents.

Axiom 3 is a strong version of the so-called Factor Law (Factor for short). If we

define S and = as usual in terms of A, - and V; we obtain the standard propositional

calculus PC as a sub-system. Notice that Axiom 5 is equivalent to (p → q) → (pD

9). Thus, thanks to R1, any theorem of the form A - B also holds in the weaker

form A B. We then have at our disposal the derived rule A ++ B/A = B, but we

do not have the converse rule, which would amount to having as a rule Replacement of Proved Material Equivalents. This restriction leads to some paradoxical results, for example that (pAp) cannot be replaced by p since (pAp) - p is not a theorem of PAL (Note that we cannot derive this wff by using (pAg) - q since the latter is not a theorem of connexive logic).McCall's system CC1 (see (9]) turns out to be equivalent. to a system obtained by extending PAI with the following axioms:

1. p - (pAp) Ap)
2. (pAp) - ((p- p) - (pAp))
3. ((p → q) → q) - g)
4. (q Aq) - (р - р)
5. pV (((p→ p) → p) V ((gq) → p)))

For a detailed criticism of PAI and CC1 the reader is referred to (11]. These criti-cisins were accepted by Angell (see [2]), but the attempt to overcome the difficulties pointed out by Montgomery and Routley involves extending the formal language of connexive logic as it was initially formulated, McCall's recent reformulation of connexive logic - named CFL in 9) - also requires a reformulation of the language of the original formal system since its formation rules prohibit wifs with iterated

2. Analytic and synthetic consequential implication

The logic of consequential implication (see [15]) differs from the logic of connex-ive implication in a number of respects, which can be outlined as follows: Firstly.

Boethius' rule is represented in the object language by (p → q) D -(p - -q) and not by (p → g) - -(p

→ -q). We will distinguish these two wits by calling the

first the Weak Boethius' Thesis (WBT) and the second the Strong Boethius' Thesis

(SBT). Secondly, Factor holds only in the following weakened form:

（WWE）→（T→ →PAT））→（（p→9）3（p^r）→（qAr））.

Thirdly, a distinction is drawn between the logic of "analytical" and "syntheti-cal" conditionals. The latter are conditionals whose truth depends on a set of true statements which are contextually understood but not explicitely stated. Counter-factual conditionals are paradigm examples of context-dependent statements, and so they should be formalized as synthetical consequential conditionals. However, intuitions concerning the logical properties of synthetical conditionals are not clear.

It appears that in ordinary language we have a whole family of different condition-als, whose logical properties we frequently confuse. To clarify the situation we can state two minimal properties of the so-called "circumstantial operator «** , which can be read as "ceteris paribus" ("other things being equal") or "rebus sie stantibus" ("things being thus and so")%.

The minimal requirements for the logic of this operator are axiomatized as fol-

lows:

（i）

（i） （*ート）3 （p→上）

The most natural definition of a synthetical conditional is A > B = D/ *A - B.

But many other definitions are possible which satisfy the properties required forconsequential implication. The weakest connective of this family is defined as fol-lows:

1 > B = D/ (T → (*A 5 В)) A (-(Т → -В) Л-(Т→ - * А))

3. Translations between logics of consequential implication and standard modal logics

If we want to stress the similarities between connexive implication and consequential implication, we should note that they are both compatible with Nelson's informal treatment of implication. Historically speaking they both have a com-mmon ancestor in Chrysippus conception of conditionals and so may be called Chrysippean conditionals'.

If, however, we want to stress the differences, apart

from the analytical/synthetical distinction which is mirrored by the proposed extension of the object language, the most important difference between the two formal theories is just their attitude toward Factor.

Intuitions about Factor are not clearly related to Aristotle's Thesis and Beethius' Rule and they should be subjected to a specific analysis. Indeed it may be claimed that Factor is implausible in the light of the underlying motivations for introducing the notion of connexivity. To see why consider the following argument. Suppose that

p→ q stands for "If Smith is a bachelor is a male" pAr stands for "Smith is a bachelor and married" gAr stands for "Smith is a male and is married".

Then p

→ q stands for a statement describing a necessary connection and

pAr stands for a contradiction, while q Ar stands for a contingent statement.

Since the conjunction of p and r in this particular example is consistent, deriving r by application of Simplification is connexively sound. So along with (p - q) D ((pA) → (gr)) (Factor), we have also (gAr) → r and so, by transitivity of *-*", (p +q) → ((р\т) - r). So assuming the necessary statement p - q we conclude that "Smith is bachelor and married" (pAr), connexively implies "Smith is married" (r). But this result is connexively unsound, since the conjunction symbolized by pAr is inconsistent while r is not.

This argument could of course be questioned since it relies on the presupposition that some instances of Simplification should be accepted. Now it does seem plausible that at least the following weakened version of simplification should be a theorem of connexive logic, since it states that Simplification holds provided the antecedent is not equivalent to a contradiction and the consequent is not equivalent to a tautology:

(WS) (-(IHPAS)AT(TAT))3(0Ar)-)

In fact, this law can be proved even in the weakest calculus of consequential implication in the class of systems which will be introduced in the next section".It should be pointed out that consequential implication has different origins from connexive implication since it originated in modal logic as a variant. of strict implication. Given that contradictions may imply and be implied only by contra-dictions, and tautologies imply and are implied only by tautologies, the key idea of consequential implication can be expressed by saying that it connects two propositions A and B when we have:

(i) A strictly implies B: 0(A B) (ii) A and B have the same modal status.

The sense of (ii) is that if A → B is to hold then A and B are both necessary, or both impossible, or both possible, or both not-necessary. Summing up, a relation of consequential implication holds between A and B when we have C(A > B) A(0A = 0В) A (0A = 0В) A (-DA = -OB) A (-0A = -OB), which is equivalent to

•(AD B) A (DA = OB) A (04 = 0B), a wif which in normal modal systems equals the simple D(AS B) A (OBS DA) A (OB O QA). The equivalence between A → B and the latter formula suggests that we look for a translation between the languages of modal logie and consequential implication.

At this point it is useful to set out some results about the interrelations between modal systems and systems of consequential implication. For sake of simplicity we will confine ourselves to the analytical fragment of logics of consequential implica-

Let Lo be the set of wifs resulting from standard combinations of propositional variables p, q.r, parentheses (.), the primitive functors {L, 5, ) and the standard definitions of -, A, v. 0.

Let L. be a language which is like Lo with the only difference that replaces Let us define two mappings: @ from L.., to Lo and a from Lu to L., by the

following conditions:

1a, pip）=p

28.中（ ）=上

3a. o（AD B） =・A）コo（B）

1a. 0(4-B)=0((A) (B)^ (0(B)>0())^(0(B)0(A)

1b. 4(p) = p

2b. （上） 上

36.2（A3B）=4（4）つ（B）

4b. 0(0A)=T= 0(A)

A normal system in L_ is a set X C L containing all the truth-functional tautologies and the wiis derived from the following axioms:

(PC). All the theorems of the classical propositional calculus PC

（a） （p→q4→r））（pir）

(b) (T → (рал -(Т → -р) Л -(Т → 9)) Р (р → q)

(с) - (Т → - (рАг)) > ((р→ g) Р ((рАт) → (дЛг))

（d） （Jp→g）2（9→

1. (p → 1) D (1→p)
2. (1→ p)D (p→L)
3. p. - p

The rules are Uniform Substitution (US), Modus Ponens (MP) and Replacement of Proved Material Equivalents (Eq).

We shall call the smallest normal system of consequential implication CIw. If we add the Weak Boethius Thesis (p - q) D -(p → -q) (WBT) to CIw then we obtain a system which we shall call CI, and if we add (p → q) (pS q) we obtain another system which we shall call CIO.

Let us now consider the weakest normal system of modal logic, i.e. the well known system K which is axiomatized by adding to the standard propositional calculus PC

K1. 0(p)q D (Op 3 0g)

with MP,US, Nec (F A → - DA) as the only rules of inference.

We now define a translation between the systems X C L. and between

Y C Lo as follows:

We say that X translates Y when, for every A € L... we have A € X iff ф(A) € Y. We will say that (A) is the modal counterpart of A.

We say that Y translates X when, for every A € Lo, we have A € Y iff 4(A) € X. 4(A) will be called the consequential counterpart of A.

Using these definitions we can prove the following metatheorems [19]):

1. If Y translates X and X is normal in L.., then Y is normal in Lo.
2. If Fk 4 then Fciw #(A)
3. If X translates Y and Y is normal in La then X is normal in L...
4. If FCiw A then Fk (A)
5. For all A € L, Fciw A = 4(ó(A))
6. For all A € La, Fa A =ф(@(A))
7. K translates CIw and CIw translates K
8. If X is normal in L., and Y is normal in L, then X translates Y iff Y translates X.
9. Suppose that X° C L.., Y" C Lo and X is the smallest normal system L_, such that X" § X; Y is the smallest normal system in Lo such that Y° CY; (a) € Y whenever a € X"; 4(a) € X whenever a € Y. Then X and Y translate each other.

Proposition 9 states that and induce a one-one embedding between the theses of any normal system of modal logic and the theses of the system of consequential implication which translates it. Hence we can show that there is a one-one translation between CI = CIw + (p → q) D -(p→ -g) and K + Op 3 Op (ie. the deontic system KD) and also a one-one translation between CIO = CIw + (p →

9) D(pOg) and K+Op 3 p, i.e.

KT. Since -(p → -p) is equivalent to

(p→q) D-p→ ng), CI is the weakest system containing Aristotle's Thesis®.These results about translations provide us with a decision procedure for all extensions of CIw whose modal translation is decidable. Tableaux methods which are appliable to normal modal logics turn out to be practical methods to test the validity of consequential wifs.

A remarkable by-product of this modal translation is that it provides us with a tool for analyzing typically connexive wifs, and for studying the properties of systems which are intermediate between systems of connexive implication and systems of consequential implication.

An example of the kind of investigation which can

be carried out in this way concerns what we labelled earlier the Strong Boethius' Thesis SBT (which is axiom 8 of Angell's PAI). The first question to ask is, of course, whether SBT is a theorem of the basic systems of consequential implication CIw, CI, and CI.O.

This question was anwered negatively in [17] using a result of [22]. In fact, the system KT has the so-called double cancellation property (DCP), which we can state as follows:

(DCP) If X is a normal modal system, -x CA = OB and -x 0A = B, then

-x A = B.

Let us suppose that (p → q) - -(p- -g) is a theorem of CI.O; then, by Reductio, in KT we should have @(p → q) 3 (-(p → -q)) as a theorem, hence also (T - p) = (-(T → -p)), which we know to be impossible, since the latter

wif is equivalent to the non-theorem Op = Op. The Strong Boethius' Thesis SBT

cannot then be a theorem of any system at least as strong as CIO. Let us call e-normal every normal modal system such that the "erasure transformation" yields valid PC-wffs (see [4], P. 23).

Then, since Op Op is consistent with every e-

normal modal system, SBT is also consistent with any consequential system which translates an e-normal modal system.

The next question is: since SBT is consistent with CIw, which is the modal system translating CIw + SBT? The answer is as follows. Let us call the required system CIw- and let us call the smallest fragment of La which contains the following Kdf:

(1D) OT

(2F) 00p 3 00p

The semantic properties of Kdf are obtained by standard correspondence theory and can be described as follows:

Quasi-seriality: Wwva(wRy 3y aRy)

ofunctionality Vutzty （wRy AaRya（ヨr（wRがへ♥ぱRつ2＝3））

(The latter wif is equivalent to the simpler VwVrVy(wRy AzRy AaRa 52 =By an application of the Henkin technique for completeness proofs, we obtain the following completeness result:

THEOREM 4.1. A is a theorem if and only A holds at all the frames which are

quasi-serial and O-functional.

This characterization result allows us to find a quasi-serial and (Q-functional frame which refutes the converse of SBT. We have thus:

THEOREM 4.2. T2. -(p → ~g) → (pq) is not a CIw→ theorem.

(See (18)).

This result is not a trivial one, since in the light of the application of (DCP) we have, for system CI.O.

(a) Fcio A - Biff Icio B = A

from which it follows by replacement of material equivalents that

(b) Fcto A → Biff Icio B → A.

We thus have the rather unwelcome result that if SBT were added to CI.O the system would contain its converse as well, and also the equivalence + (A → B) -

-(A → -B).

Even if not strictly trivial, Ciw→ has properties which throw a negative light on the Strong Boethius Thesis. For example, it can be proved that the Denecessitation Rule (- DA → A) is admissible in any modal system X iff Modus Ponens for + (If Fcro A → B and Fcio A then Fcro B) is an admissible rule of its

consequentialist translation. Now in Kdf we have a proof of the wff (Op = p), while

(Op = p) is refuted (see (18)). This proves that Kdf does not admit Denecessitation,

and hence that CIw- does not admit Modus Ponens for →. But it can be proved that every extension of CIw- which admits Modus Ponens for -, (such as CI.O) contains the undesirable equivalences (p → q) = (g - p) and (p → q) = -(p → -q).

Having Modus Ponens for "—"

means the possibility of interpreting

"—" as

an implication connective, but this destroys the very possibility of entertaining non-trivially the Strong Boethius Thesis. It can also be proved that adding the characteristic axiom of CI.O, namely (p → q) D (p D4), to CIw-, yields the

equivalence p = (T = p), whose modal counterpart is the collapse - formula

P= Op).

5. Factor and consequential implication -

Let us now consider the formula which distinguishes connexive logic from consequential logic, namely Factor. In systems of connexive logic we find two variants of this law, which we we will call "Strong Factor" (SF) and "Weak Factor" (WF).

(SP) (p → q) → ((р^т) → (gAr))

(For the latter see, for instance, (9]).

An equivalential variant of WE may also be found in the literature, viz.

which is of course equivalent to (p - q) ((pAr) - (g))(see for instance (2]). WFEq is unproblematic, since it can be shown that it is a theorem of even the minimal system CIw. Since K is the modal translation of CIw, it may be proved that the following wils are K-valid (where "_" is the symbol for strict equivalence).

（（pニタコロ（（p/r）→（9^z））

（E）（（ニタ）^（ロp=D4））2（0（g^7）2口（pAr））

（m）（（#=4）^（0p^04）） 2（0（g^r）3Q（p/r））

Thus by applying the so-called Theorema Praeclarum ((PS q)A(r 5 s)) 5 ((PAr) D (gAs)) it turns out that (p → q) 5Ф(рЛг) - (gAr)) is K-valid, and hence that (p +q) 3((рлг) → (gA)) is a CI-theorem. The problem of derivability then concerns the two wffs SF and WF

The first result to be noticed is that SF is inconsistent with any system of consequential implication which contains the Weak Boethius Thesis or, which amounts to the same thing, Aristotle's Thesis. If SF were a theorem of CI, in fact, we would have the following proof:

1. (р - -р) - ((рЛ-р) - (-рЛ тр))
2. (р- -р) - ((рАтр) - -р)

3） （p→ Jp） =1

1. 1- ((рА-р) → тр)
2. 1→ (p - T)

SF(-P/g)

1), PC

+ -(p--p) = T, Eq

1. , 2), Eq
2. , Az. (d)

The modal counterpart of line 5) is the wif -OOp, which is inconsistent with every normal system containing OT, namely with the modal counterpart of Aristotle Thesis. In fact an instance of it is -QOT, while in KD from T we have

However, it is to be noted that WF is consistent with every extension of CIw translating some e-normal system. This can be easily proved by replacing every occurrence of "—" with "=" in the axioms and checking that (i) the resulting wifs are PC-valid and (ii) the rules preserve the PC-validity of the transformed wffs.

If we now apply the transformation to WF we obtain (P=q) > ((pAr) = (gA)).

which is a PC-thesis.

Thus, by a standard argument, we can prove that WE is

consistent with CIw and with every extension of CIw whose axioms have PC-valid

The problem with WF is indeed not inconsisteney but the fact that adding WF to Cl yields counterintuitive results, which may be compared to the result of adding Strong Boethius Thesis to system admitting Denecessitation. It is remarkable, in fact, that by adding WE to CI we lose the asymmetry of the arrow, since we may prove the equivalence between (p → q) and (q p). This may be seen looking at the following proof, in which A and A are introduced by the two definitions:

(Def) 0A =DJ -(T→-A).

Thanks to such definitions (one of which is of course redundant) and to the mentioned embedding results, we know that every theorem of K belongs to CI + DefO.It is useful to recall that in CI + DefO (we have the equivalence

（→）（口（pコg）^（コ（p） ^ （0g 3ロp）） =n→q

We may then exhibit the following proof:

1） （p→9）3（（pAr）→（9^r））

1. (р → q) 3 ((р\-р) → (gA-р))
2. (р → q) D (1→ (g-р))

WF

1. , тр/г
2. , 1= (р.Л-р)
3. , (d), (e), (f)
   （→）
4. , 5), Defu
   K
   7), (-)
   6), 8).

6）（p→g）コロ（p=q）

7）ロ（p=4つ（（ロp3ロ/）^（Op3^4） ^ロ（92p））

8) 0(p=q) > (9-p)

9） （p→9） 3（91p）

A simple consequence of 9) is the theorem

（1）（p→g）=（g→p）

which asserts the equivalence between → and -.

On the other hand, suppose we add

(S) (p - g) 3 (g -p)

as an axiom to CI, so to obtain a system CI+S. Obviously we have (-) as a theorem of CI+S. But since we already know that (p - q) > ((pAr) → (qA)) is a theorem of CI, we have by replacement (p → 4) - ((р\г) - (gA)), i.e. WF, as a theorem of CI+S. So, if X is any system containing CI, CIW is equivalent to CI+S.

6. Factor and a non-contrapositive variant of consequential implication

An interesting property of systems of consequential implications is that by introducing the definitions of the modal operators in terms of the arrow we may define different arrow-operators which are variants of the standard arrow operator which have the minimal properties originally required for connexive implication.

For example, we may define a new arrow in terms of O as follows

（→）4→B=Dロ（43B）^（QB3>4）

and also define a second couple of modal operators as

（ロロ4=D/T=4

（ペ）4=Dr→ロ4、

Of course we have that A - B imples A → B but not vice-versa, while it is straightforward to prove that D°A is equivalent to CA and 0°A is equivalent to

•A'

The logie of = can be proved to be slightly different from the one of →, even if it is clearly a logic of a connective endowed with the properties of consequential implication. Among its theorems we have in fact

（WB→）（p=9） 3ー（p= -9）(AT →) -(p→ p)

（1→）（（p34）^（p） コ（p=g）

（2=）（1＝4＝（4→1）

We lose Contraposition for → in its standard form but we have the advantage that Simplification holds in the manageable variant

(S →0(pAq) D((pAq) → q).

It may be proved (but we will not do so now) that the fragment of CI containing only truth-functional wffs, and →-wfis can be axiomatized in a system which we will name CI→, and that the truth-functional and →-fragment of CIO, CI.O=, is definitionally equivalent to CI.O itself*.

What we want to do now is to extend CI not with WF but with its →-variant which is

(WF →)(p → q) 3 ((рАг) → (9Л г)).

Since (Og A Op) implies (gAr) @(pAr), a straightforward result of this new axiomatization is that

(3 →) ((р » q) A(0q> D)) О ((рАт) → (дЛг)) ЛО(дАт) рО(рЛг))

is a theorem (by Theorema Praeclarum). But since (3 →) is indeed equivalent to (WF) thanks to (-), we have that every theorem of CI+WF is also a theorem of CI+WF=.

What we may now prove is that there is a one-one embedding between CI= +WF and a modal system which in the literature is known as KD!, where KD! is KD +045 DA (see [4), p. 83). An established result concerning KD! is that KD! is characterized by the class of the frames < W,R> whose accessibility relation

is both functional: Vryz(rRy AaRz Sy = 2) and serial: VaZycRy. Now we can

prove the following two theorems:

MTI: If -KD: A then Fci»+ WE WA MT2: If -cI»+WP A then F-KD: ' A

MT1 The proof is by induction on the length of the proofs. We already know that the consequential counterparts of axioms of KD are theorems of CI→+WF and that the rules of KD preserve such a property. What we have to add to what is already known is the proof that Op D Opie.-(T → -p) (T → p) is a theorem of CI+WF→. The proof is as follows:

1） （p→4）3（（p^r）→（g^r））

1. (p → 4) Р ((рА тр) → (g Л -р))
2. (р → q) D (1→ (фЛ -р))
3. (p→q) 00(93 p))

5） 0（pハリ→う（T→（9つ（P^q））

6） 0（pAg） → （pAq） →9））

7） 0（（p^4）つ（T→（92p^g）））

WF

1. , пр/т
2. , 1= (pA-p)
   3.Dejo'

4)p Ag/p
(S →)
6), 5)0p 2 0p

7）T/9,DefD%，F D°p=Op

MT2 (Sketch of the proof) We simply have to show that the modal counterparts of the axioms of CI+ WF→ are valid in all serial and functional frames, that is in all serial and functional models. We already know that the modal counterpart of the axioms of CI hold in all serial models, so a fortiori in all serial and functional models. We have simply to show that the modal counterpart of WF→ is valid in every serial and functional model.

This fact is established by the following closed tableaux, where the first world w sees one and only one world w10,

w'

The above wif is then KD!- valid and, by the completeness of KD!, a KD!-theorem.

Thus, since the wff D(p 5 q) 5 0((pAr) 5 (gAr)) is a theorem of all normal systems

of modal logic， （Op 3g）^（ogコ 0p）） 3 （口（（pAr） コ（gAr））^（>gAr）コ•（pAr））

is a KD! theorem. But this formula is the modal counterpart of WF→.

This completes the proof of the definitional equivalence of the two systems.

The partial collapse of modal distinctions which occurrs in KD! is mirrored by a counterintuitive theorem of CI+WF→: as we can easily check by using the KD!-tableaux, a theorem of CI+ WF → is the converse of Boethius Thesis, namely (CB) -(p→ ng) > (p → q)

which can be proved also in a -version.

7. A recent connexive logic containing Factor

The preceding negative result about weak and strong Factor Law casts a shadow over all systems of consequential implication containing WE. The analytic fragment of the system named CA*1 in [14) contains WF and, being closed under the replacement of material equivalents, it can be proved to contain also the undesirable equivalence (p → q) = (q → p).

This system then has an interest only as a limit

case of a connexive-consequential system. Another example is given by McCall's system CFL (see [8]), whose language does not allow the iteration of arrows, CFL is axiomatized as follows:

1. （p-42（（*→p）2（→g2（p34）つ（19コt）2（par））

3. （p→9）コ（（pAr）→（rAg））

1. （pA（g^r））→（（p^q） ^r））
2. (pA-p) - (qA-q)
3. p - (pAp)
4. (рАр) - р

9, -p → P

1. （（p/9）→（P^→P））^（pV→4）） 3（p→g））
2. (р - 9) 3 -(р- -q)
3. (9 → -p) 5 (p--g)
4. pコ（p→ （pap））
5. (p → (pp)) Рр

The only primitive rules are Uniform Substitution and MP for 3.

In CFL p → (pOp) is assigned the meaning of "p is true" (not. "p is necessary") and p - q turns out to be equivalent to (T → (p q)) A (q p). In [9) R.K. Meyer showed that if we define the arrow in this way:

(*)A → B =Dj (A - 3B) ^ (A = B)

then the first degree fragment of the systems S1-S5 is exactly CFL. The result is unwelcome, since the arrow seems to identify a particular subclass of material equivalences. On this subject, note also that we have (A - B) > (B 5 A) and

((A - B) A B) D A. So, if we want to interpret "—" as an implication connective, we have to face something which recalls the fallacia consequentis.

McCall sees two possible ways to solve this problem: 1) dropping the restriction to first degree wils, and 2) introducing axioms which are not equivalential.

It is worth noticing that the minimal system of consequential implication CIw satisfies both McCall's conditions. Its formation rules are here unrestricted, while axiom (f), ie. (L-p) > (p -L), is a simple example of a wff which does not admit Meyer's interpretation: the wff ((1 -3p) Ap =1) 3 ((p- 3 1) Ap al) is in fact underivable even in S5, so that (f) is not a theorem of CFL. However, a more direct move would be to remove the factor law WE and replace it with some of the weakened variants of it which have been examined in the present paper.

If we introduce this modification it is no longer true that the resulting system is coincident with the first degree fragment of S1-S5.

We conclude by noting that (p - q) D ((pAr) - q) is neither a law of connex-ive logies, nor of consequential implication logics. If it were, by substituting p for q we would have (pAr) - p, which is not a theorem of consequential implication logics. If we call (p → q) > (pAr) - q) the "principle of monotonicity", we can then say that → symbolizes a particular kind of monotonie implication. add that also Weak Factor (WF) may justifiably be said to express a monotonicity principle of implication". Thus the representation of the arrow as a symbol for aparticular kind of non-monotonic implication receives a support from the fact that we have to exclude Factor Law from logics of consequential implications and to work only with suitable modifications of it 12.

ANGELL, R.B. A Propositional Logic with Subjunctive Conditionals, Journal of Symbolie Logic, XXVII, 1962, pp. 327-343.

ANGELL, R.B. Tre logiche dei condizionali congiuntivi in Pizzi C. (a cura di) Leggi di Natura, Modalità, Ipotesi, Feltrinelli, Milano, 1978, pp. 156-180.

AQVIST, L. Modal Logic with Subjunctive Conditionals and Dispositional Properties, Journal of Philosophical Logic, 11, 1973, pp. 1-76.

CHELLAS, B. Modal Logic, Cambridge U.P., Cambridge 1980.

KIELKOPE, C. Formal Sentential Entailment, Univ, Press of America, Washington, 1977.

LEWIS, D.K. Counterfactuals. Oxford, Blackwell, 1973.

LOWE, J. If 4 and B then A, Analysis, XIV, 1985, pp. 93-98.

McCALL, S. Connexive Implication and the Syllogism, Mind, LXXVI, 1967, pp. 346-356.

MeCALL, S. Connexive implication in A.R. Anderson and N. Beinap, Entailment. The Logic of Relevance and Necessity, vol. I, Princeton U.P., 1975, pp. 434- 452.

1. MEYER, R.K. The Poorman's Connexivo Implication, Relevant Logic Newsletter, II, 1977.
2. MONTGOMERY, H. e ROUTLEY, R. On systems Containing Aristotle's Thesis, Journal of Symbolic Logic, XXXIII, 1968, pp. 82-96.
3. NASTI DE VINCENTIS M., Stoic implication and Stoic Modalities, in G. Corsi, C. Man-gione, M. Mugnai (eds.), Le teorie delle modaliti, Bologns, CLUBB, 1989, pp. 259-263.
4. NELSON, E. Intensional Relations, Mind, XXXI, 1930, pp. 440-453.
5. PIZZI, C. Boethius' Thesis and Conditional Logic, Journal of Philosophical Logic, VI, 1977,
6. PIZZI, C. Decision Procedures for Logics of Consequential Implication, Notre Dame Journal of Formal Logic, XXXII, 1991a, PP. 618-636.
7. PIZZI, C. Varieties of Non-Monotonie Conditionals, in A. Carsetti, M. Mondadori, G. Sandri (eds.), Semantica, Complessitá e Linguaggio Naturale, CLUEB, Bologna, 1991b, pp. 59-76.
8. PIZZI, C. Weak vs. Strong, Boethius' Thesis: a Problem in the analysis of Consequential Implication, in A. Ursini and P. Agliano (eds.)., Logi and Algebra, pp. 646-654, M. Dekker
9. PIZZI, C. Implicazione crisippen e dipenden contestuale, Diancia 3, 1998, pp. 25-44.
10. PIZZI, C. and WILLIAMSON T., Strong Boethius' Thesis and Consequential Implication, Journal of Philosophical Logic, XXVI, 1997, pp. 569-588.
11. THOMPSON, B.R. Why is Conjunctive Simplification Invalid?, Notre Dame journal of For mal Logic, XXXII, 1991, pp. 248-54.
12. VAN BENTHEM, J.F.A.K. Essays in Logical Semantica, Reidel, Dordrecht 1986.

122] WILLIAMSON T. Verification, Filsification and Cancellation in KT, Notre Dame Journal of Formal Logic, 1990, vol. 31, pp. 286-290.