We have earlier seen that the extensional notions of suppression are eclipsed by even weak variable sharing (Theorems 1 and 2), and that the Ackermann Property is independent from ASP (Theorems 3and 4). The fol-lowing theorem shows that IJFP is in fact independent from both VSP and AP:
Corollary 5. IJFP, VSP and AP are all independent.
Proof. • The logic of Theorem6satisfies IJFP, but since it has the Mingle axiom it fails to satisfy both AP and since it validates the Kleene axiom it fails to satisfy VSP.
• TM,TwithA→(A→A) added, satisfies VSP ([16, Cor. 4.6]). Because of Mingle, however, it does not satisfy AP and because of Reductio it does not satisfy IJFP (Theorem8).
• Since E satisfies AP, that property does not entail IJFP since ((A → A)→B)→B is a theorem of E. Nor does it entail VSP (Theorem4)
We have so far seen that the anti-suppression properties fail to rule out the implicational paradoxes—even the first degree such. Furthermore, they classify principles claimed not to be clearly suppressive—Reductio—as suppressive, yet fail to capture what is claimed to be suppressive principles of E, namely the ∧-axiom and the pre- and suffixing axioms. We have already looked at the reasons given for thinking that the E-axiom and Reductio are suppressive.
What, then, are the reasons given for thinking that the three other mentioned E-axioms involve suppression? I will now show that the reasons given are also in this case rather wanting.
The ∧-axiom is said to yield to “similar counterexamples” as the E-axiom ([27, p. 268]), although none are given. The story with the E-axiom is that it suppresses the theoremA→A in the antecedent of the consequent of ((A→A)→B)→((A→A)→B). Thus if the∧-axiom is to be similar, it ought to be the case that for some theoremC,A∧B→(C→(A∧B)) is a theorem, but that A∧B → (A∧B) fails to be a logical theorem without it being forcibly added as a primitive axiom. This story, however, is hard to substantiate as logics with the variable sharing property tend not to have theorems on the formA∧B→(C→(A∧B)) since the weakening
24Conjunctive syllogism (Ax12) gives one (∼C → A)∧(A → C) → (∼C → C) from which Reductio, some simple double negation fiddling, and the transitivity of → yields (∼C →A)∧(A→C)→C. From this the contraposition axiom and some more double negation fiddling suffices for (∼A→C)∧(A→C)→C, from which the rules ofBBsuffices for yielding (A∨ ∼A→C)→C.
Figure 6. The crystal lattice
axiom A → (C →A) can’t be a theorem of such logics. We can strengthen this observation as follows:
Theorem 9. If A∧B →(C→(A∧B))is a theorem of any sublogic of R, thenC shares propositional variables with bothAandB.
Proof. Assume thatA∧B →(C→(A∧B)) is a theorem and for con-tradiction thatCdoes not share propositional variables with, say,B. Consider theCrystal latticedisplayed in Fig.6. That model is a known model of Rand will therefore evaluate A∧B →(C→(A∧B)) to some value different from 0. Assign every propositional variable inAto 2 and every variable in B to 3. Since both{2} and {3} are subalgebra it follows that both A and A will be evaluated to 2, that both B andB will be evaluated to 3, and that bothA∧B and (A∧B) will be evaluated to 1. Since C does not share propositional variables withB we can also assign to every remaining proposi-tional variable inC the value 2 so thatC will be evaluated to 2 as well. We
now get a contradiction since 1→(2→1) = 0.
Thus if the ∧-axiom is suppressive, it only suppresses certain specific theorems. Maybe there are such cases, but I haven’t been able to find any. I therefore conclude that if indeed the∧-axiom is a suppressive principle, it is so in a sense not adequately specified by Routley et al.
Let’s move on to the transitivity axioms. (A → B) → ((C → A) → (C → B)) and (A → B) → ((B → C) → (A → C)) are collectively called ESyll, short for Exported Syllogism, by Routley et al., and are both deemed to be suppressive. It is called exported in reference to the axiom they call Exportation, namely (A∧B→C)→(A→(B →C)), their view being that ESyll is an exported version ofConjunctive Syllogism(CSyll), (A→B)∧(B→ C) → (A → C). Now Exportation yields irrelevance since A∧B → B is a logical axiom and so Exportation therefore yields A → (B → B), and since they view irrelevance as an form of suppression, Exportation is regarded as a suppressive principle. Routley et al. know of course that ESyll does not result in irrelevance, but argue that the fact that it “is an exported form of CSyll should, if anything, increase our suspicions of it.” [27, p. 269]. Their main argument against ESyll, however, is as follows:
The contribution ESyll makes over and above CSyll would appear to be that of allowing the suppression of certaintrue implications
in certain positions. This is a very limited form of suppression and it is not one which produces spectacular effects, like more wholesale suppression, in the form of irrelevance. But if we are attempting to explicate the concept of sufficiency, it must be rejected for just the same reasons as the more obvious and general suppression prin-ciples and resulting irrelevant forms. ESyll prinprin-ciples provide good examples of those sorts of principles we alluded to earlier, namely those that allow a certain degree of suppression but not so much as to produce irrelevance. Remember that relevance is only a nec-essary condition for an adequate system; so the fact that ESyll can be added to certain systems without inducing irrelevance provides on its ownno justification for ESyll. [27, p. 271]
The thought here seems to be that the fact that A → B is true does not suffice for allowing it to be dropped from the logically true claim (A → B)∧(B →C)→(A→C). This, however, according to Routley et al., is what in effect is afforded by ESyll, which therefore is regarded as a suppressive principle.
Now Routley et al. would object to suppression of merely true formulas as well the suppression of theorems. Note, however, that in order for ESyll to be suppressive in the way specified in the above quote, Routley et al. need to use the concept of a rule in two different senses: for what is wrong with (A→ B)→((C→A)→(C→B)), they say, is that it forces the consequent upon us when the antecedent is merely true. However, the ruleA→B (C→A)→ (C→B) is a rule of most relevant logics—indeed all such logics considered in this paper and therefore Routley’s two favorite logicsDKandDLas well. They do state that rules are to be interpreted as theorem preserving25, but if that is the case, then the consequent of (A→B)→((C →A)→(C→B)) need not be forced upon us just because the antecedent is true. Furthermore, if one looks at how rules are in fact used, the claim that rules are to be interpreted as only theorem-preserving seems not to be adhered to by Routley et al. For instance, the already mentioned triviality proof for na¨ıve set theory in [27, p. 367] happily uses the primitive ruleA→(B →C)A◦B →C, where◦ is the intensional conjunction calledfusion to get w ∈w◦w∈ w →pfrom w∈w→(w∈w→p), where the latter formula is deduced from the formula
∃w∀x(x ∈ w ↔ (w ∈ w◦w ∈ w → p)). Thus they use rules as applying to formulas deducible from assumptions which are not logical theorems, and so use the rules as truth-preserving, not merely as theorem-preserving. But then if ESyll detaches upon having a mere true antecedent, that is because modus ponens is regarded as truth-preserving, not only astheorem-preserving.
25The following is a clear example of this:
For example, the rule of Modus Ponens, symbolised:A, A→B−− B, is read:
whereAandA→Bare theorems so isB. Generally the rules are not, in what follows, intended to reflect entailment relations, or inferential connections such as that expressed by ‘As ...; therefore... ’; they are simply schemes for generating theorems from theorems. [27, p. 287]
It is, of course, possible to restrict the pre- and suffixing rules, R3 & R4—
jointly refered to by Routley et al. as the affixing rules—of DK/DL to be only theorem-preserving without also restricting the other two rules—modus ponens and adjunction. That would then beget different logics thanDK/DLas defined both here as well as all other presentations ofDK/DLby the authors of [27]. Generally, however, it seems that it is more plausible to interpret rules stated in the same syntactical way as the same type of rules: either all rules are to be interpreted as theorem preserving, or all rules are to be interpreted as truth preserving. But then if R3 & R4 are regarded as suppressive when taken as truth-preserving rules, why think that beefing up the premise from merely true to logically true would make for an unsuppressive rule? Why is it permissible to suppressA→B in (A→B)∧(B →C)→(A→C) when A→B is a theorem, but not when it is merely true?
The only sensible answer, it seems, would be to conclude that it is not only ESyll which is suppressive—in a way not covered by any principle discussed by Routley et al. I should note—but also that the affixing rules are suppressive.
Routley et al. did worry that these rules could be suppressive as well.26In order to shed light on this matter, Routley et al. [27, §4.2] reaxiomatized the logic Bso as to includeSubstitutivity of Coimplicants, A↔BD(A)→D(B), as a primitive rule, whereD(B) is obtained fromD(A) by replacing one or more instance ofAbyB. In addition they added the axioms (C→A∧B)→(C→ B) and (A∨B→C)→(A→C). These are easily verified to be deductively equivalent axiomatizations. The possible suppressiveness of the affixing rules is then formulated with CSyll as well as this reaxiomatization ofBas a backdrop:
These reaxiomatisations of B and its extensions focus the issue as to the correctness of the affixing rules on some special cases concerning
& and∨. Thus if
(C→A&B)→(C→A)
is a correct principle for sufficiency or conditionality then Prefixing is admissible. If, however, this principle really hasA&B→Aas a further premiss and is obtained by suppression of this premiss from the correct
(C→A&B) & (A&B→A)→(C→A),
then the principle is not a genuine sufficiency one and should be rejected as an entailment principle. [27, p. 293]
26Their basic logic is in fact the logicAbeing simply Routley’s logicDKminus the affixing rules [27, pp. 221f]. This, then, sets them apart from the view of Lewis who, just like Routley et al., included Conjunctive Syllogism as an axiom of his logicsS2andS3, but renouncedS3 in favor ofS2when it turned out thatS3had the suffixing axiom as a theorem. His reaction was also that “it gives the inference (qr)(pr) wheneverpqis a premise. Except Footnote 26b continued
as an elliptical statement for ‘(pq)∧(qr)(pr) andpqis true,’ this inference seems dubious.” [12, p. 496]. Despite this, Lewis seems to have been untroubled by Parry’s proof that the suffixing rule holds forS2(cf. [12, p. 507]).
Thus Routley et al. regard it as an open issue whether not only ESyll, but also the affixing rules are suppressive. They seem to model this on the same idea as the rules (Ic)&(IIc) above. As with these rules, however, the sober judgement here is surely that there simply is no available evidence which points to either ESyll or the affixing rules being suppressive.
Routley et al. did not discuss the suppressiveness of the affixing princi-ples further in volume one ofRelevant Logics and their Rivals, and when, after the untimely passing of Sylvan, the second volume, [7], eventually came out 21 years later, Brady, the editor of the second volume, wrote in the preface that the focus on the issue of suppression would not be continued in that vol-ume. The reason given, however, was that Brady thought that the suppression exhibited by ESyll was benign and so focusing on suppression alone would not suffice for reaching the so-called depth-relevant logics—DR and beneath [7, p. 4]. Since no further account of suppression was ever given, the idea of the suppressiveness of the affixing principles in particular, but also suppres-sion in general, was left in a rather unsettled state. I conclude, rather, that the suppressiveness of the affixing principles seems simply to be ill argued for and badly motivated; there just seems to be no clear case made for the suppressiveness of these principles.
I have in this section tried to make sense of some of Routley et al.’s use of the notion of suppression which does not fit into the extensional notion precisified by ASP and JFP. It was suggested that this could be viewed as a slightly different intensional variant of suppression and the principles IASP and IJFP were suggested for capturing it. It was shown, however, that neither this notion, not even in combination with extensional suppression, sufficed for ruling out the implicational paradoxes. Such paradoxes are not, according to Routley et al., features of a good sufficiency relation, and thus it seems that analyzing such a relation in terms of the principles ASP, JFP, IASP and IJFP will not be satisfactory. Furthermore, we have seen that although intensional suppression did rule out theE-axiom ((A→A)→B)→B, it also ruled out Reductio which Routley et al. did not judge to be suppressive. It furthermore did not rule the other principles differentiating DK and E as suppressive, contra what was in fact claimed. The arguments for why such principles are suppressive and therefore unacceptable for entailment taken as a sufficiency relation were shown to be, to put it lightly, less than persuasive. I have, of course, not demonstrated that there is no notion of suppression which will do the job Routley et al. wanted suppression to do. However, the burden of proof is now on the suppressivist to specify the rules of the game; if one is to uphold the verdict thatT,EandRare suppressive, then one needs to specify what suppression here is and why such suppressive features can’t be features of entailment taken as a sufficiency relation.
7. Summary
This essay has looked closer at Val Plumwood and Richard Sylvan’s, with collaborators, notion of suppression. Plumwood and Sylvan developed it from
that of an enthymeme, a notion which had already been used to argue for the supremacy of the relevant logicEover classical, intuitionistic and modal logics by Anderson and Belnap. Plumwood and Sylvan, however, argued that it was freedom from suppression which was the fundamental feature of entailment.
Entailment, they claimed, is a sufficiency relation, and so does not admit any form of premise suppression. Anderson and Belnap’s idea of analysing relevance as variable sharing and premise use, they claimed, did not guarantee freedom from suppression; although variable sharing was regarded as an important property, it was viewed as no more than a mere negative test of adequacy and eclipsed by the richer notion of suppression.
I have shown in this paper that the extensional so-calledAnti-Suppression Principle and the Joint Force Principle are both weaker than the Variable Sharing Property, in fact properly weaker than even the Weak Variable Shar-ing Property. As such, the principles do not suffice for rulShar-ing out the implica-tional paradoxes, not even the ones of the first degree, contra what is claimed in both [27] and [28]. A notion of intensional suppression was specified to give content to the claim that theE-axiom ((A→A)→B)→B—an axiom even named Suppression in [27]—is suppressive. It was shown, however, that nei-ther this notion of suppression is strong enough, even when augmented with the extensional notion of suppression, to rule out the implicational paradoxes.
We saw that the argument given for rejecting the archetypal intensional sup-pressive principle, i.e. theE-axiom, was less than clear, and even though the intensional notion of suppression did rule two of the axioms ofEandRto be suppressive, only one of these axioms, the E-axiom, was regarded as clearly suppressive by Routley et al., whereas the other one, (A→ ∼A)→ ∼A, was in fact taken as an axiom of one of Sylvan’s favorite logics,DL. We also saw that the other principles which differentiateDL from the Anderson-Belnap logics TandEwere in fact neither extensionally nor intensionally suppressive. Thus neither this notion of suppression could do the job intended for it, namely to provide reasons for thinking that the Anderson-Belnap logics T, E, and R cannot be viewed as capturing notions of entailment viewed as sufficiency relations. Their arguments against the alleged suppressive principles of these logics were shown to be rather wanting, and their proof that there are logics which satisfies the variable sharing property, yet areobviouslysuppressive, was shown to be rather obviously fallacious.
Their failure to make any serious effort at specifying what suppression is in such a way as to make it clear that the logical principles charged of being sup-pressive can indeed be recognized as such, as well as their failure to compare the notion of suppression to the notions of relevance already available—specifically to variable sharing, since they claim this to be a test and a derivative feature of a good sufficiency relation—seems in hindsight rather peculiar. To think of entailment as a sufficiency relation may yield some philosophical insight.
If this is to be analyzed as freedom from suppression, however, honest toil in making the notion precise has to be made. Doubting that such toil will bear any fruit, I bid suppression-freedom farewell!
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