Abstract
This paper attempts a comprehensive account of the comparative merits of paracomplete and dialetheic approaches to the semantic paradoxes. It argues that aside from issues about conditionals, there can be no strong case for paracomplete approaches over dialetheic, or dialetheic over paracomplete, and indeed that in absence of conditionals, the two approaches are plausibly seen as notational variants. Graham Priest disagrees: many of his arguments favoring dialetheic solutions over paracomplete do not turn on issues about conditionals. The paper discusses his arguments on these points in some detail. On the matter of conditionals, it argues that extant paracomplete approaches are far better than extant dialetheic approaches, a fact that traces in part to dialetheists having focused too heavily on “relevant logics”. It holds out hope for better dialetheic treatments of conditionals (including of how the distinct kinds of conditionals interact), but it also suggests that this is the one area where paracomplete approaches are inevitably better.
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Notes
- 1.
So if ‘consistent’ means Post-consistent, the paraconsistent view is that it is perfectly consistent to accept some contradictions. Priest prefers to use ‘consistent’ to mean negation-consistent (i.e., implies no contradictions); in that terminology, he thinks that the correct treatment of the paradoxes is inconsistent. This choice of terminology strikes me as unnecessarily provocative, but of course the issue is merely verbal.
- 2.
‘Dialetheic’, meaning that the view accepts truth-value gluts (sentences such that both they and their negations are true), is insufficient, mainly because of the aforementioned glut theories in classical logic.
‘Paraconsistent’ is also insufficient to describe the view, since some people advocate paraconsistent logic not because they want to accept contradictions (or contradictory pairs) or even think that it can be rationally acceptable to do so, but because they think that logic should respect certain kinds of relevance requirements in addition to the preservation of rational acceptability.
Indeed, even the conjunction ‘dialetheically paraconsistent’ is insufficient to describe the view: the classical glut theorist’s position that accepts the truth of the Liar sentence but not the Liar sentence itself could be combined with the idea that logical consequence involves relevance requirements over and above the preservation of rational acceptability. The correct description is “view that accepts contradictions in a paraconsistent logic”.
- 3.
Actually one can use Kleene semantics for other logics too, e.g. the non-transitive logic on which validity requires only that the drop in values from premises to conclusion not exceed \(\frac{1}{2}\).
- 4.
He requires only that \(|True(\langle A\rangle )|\) be at least |A|, and 0 when |A| is 0; so that \(|True(\langle A\rangle )|\) can be 1 when |A| is \(\frac{1}{2}\). In LP, this does not undermine the inference from \(True(\langle A\rangle )\) to A (since the value \(\frac{1}{2}\) is designated), but the inference from \(\lnot A\) to \(\lnot True(\langle A\rangle )\) does not come out valid. (Of course when A is a sentence asserting its own untruth, \(|True(\langle A\rangle )|\) as well as |A| must be \(\frac{1}{2}\).)
- 5.
Because of the restriction to \(\omega \)-models, there is no conflict with standard results dating to Gödel and Tarski on how adding even a restricted truth predicate to arithmetic yields new arithmetic consequences when the usual composition rules plus extended induction are assumed.
- 6.
That is, although naive truth can be added to these logics in an \(\omega \)-conservative fashion.
- 7.
The symmetry between ‘valid’ and ‘valid*’ would be neater in a multiple conclusion framework.
- 8.
Either take ‘accept’ to mean ‘accept with certainty’ or take ‘accept the premises’ to mean ‘accept their conjunction’. (There is a more general formulation of the connection of validity to acceptance and rejection that doesn’t require either course, but these will do for present purposes.)
- 9.
Once one adds a conditional to the language, one can explain the verbal nature of the dispute in the conditional-free fragment slightly differently: we could use the conditional to define a weakening operator W (WB is \(Q\rightarrow B\), where Q is a Liar sentence) and a strengthening operator S (SB is \(\lnot (Q\rightarrow \lnot B)\)); the advocate of \(K_{3}\) interprets the acceptances and rejections of the LP-theorist as acceptances and rejections of the result of prefixing with a W, and analogously in the reverse direction using an S.
- 10.
If we formalize “All A are B” as \(\forall x_{Ax}Bx\), and v is a variable not free in either A or B, define \(A\rightarrow B\) as \(\forall v_{v=v\wedge A}(v=v\wedge B)\).
- 11.
See, for instance, Priest (2008, p. 167). Paragraph 9.4.6 advocates totally anarchic worlds, at which, for instance, the valuation rules for \(\wedge \) fail. As a result, the inference from \(A\rightarrow B\wedge C\) to \(A\rightarrow B\) will come out invalid, by virtue of normal worlds from which such anarchic worlds are accessible. But in paragraph 9.4.7 and the subsequent development in the book, there is a shift to only moderately non-normal worlds in which the basic laws of FDE are held inviolate.
- 12.
Some of these would not be acceptable if we allowed for anarchic worlds, but as indicated above, I will ignore those. Priest does too in the logics of paradox he discusses.
- 13.
A precisification and proof of the \(\omega \)-conservativeness of naive truth in sentential Łukasiewicz continuum-valued logic can be found in Field (2008, Chap. 4). (I claim no originality: the result seems to have been known among the cognoscenti long before, as is indicated by the frequent allusions to the Brouwer theorem in related contexts.)
- 14.
- 15.
I won’t discuss here why I find it less promising to look for a theory that admits (?) but not the \(\exists \)-Exportation rule that is its infinitary generalization.
- 16.
I don’t think the Brady conditional is suitable as an “ordinary” conditional either, for the same reasons to be offered in the next section in connection with Priest’s conditionals. I’ll come back to Brady’s conditionals in Sect. 6.8, in connection with the naive theory of classes.
- 17.
The official system of Field (2008) didn’t include this last axiom. Nor did it include the full axiom \(\models \,(A\rightarrow B)\wedge (A\rightarrow C)\rightarrow [A\rightarrow (B\wedge C)]\), but only its rule weakening. However, the “first variation” described in 17.5 does yield the full system described in the text. I should have used that as the official system. (It also avoids a problem about truth-preservation pointed out in Standefer 2015.)
- 18.
In Field (2014) I introduced as an alternative to (or generalization of) the revision construction “higher order fixed point construction” whose main laws are similar, but which avoids certain odd laws noted in Standefer (2015). But the construction that follows also avoids those odd laws, but in a revision-theoretic context, thus reducing the motivation for the more complicated “higher order fixed point” approach. In any case, the revision-theoretic and higher order fixed point approaches have a great deal in common, and what I say here about the revision-theoretic holds with little change for the higher order fixed point.
- 19.
X can be viewed as a set of pairs of ordinals (one for \(\triangleright \)-conditionals and the other for \(\rightarrow \)-conditionals), though a slightly more abstract “fiber bundle” representation can be used to exhibit the value-dependencies more faithfully.
- 20.
In some places he also contemplates weakening B by giving up even the rule form of contraposition: see the discussion of “quasi-naivety” below.
Priest advocates treating negation with the Routley * operator, which “interchanges gaps and gluts”. Endorsing excluded middle involves ruling “gap worlds” out of the normal worlds, so evidently he takes the Routley * operator to yield a non-normal world whenever applied to a normal world.
- 21.
Interestingly, the joint paper seems to suggest that its relevant conditional should be contraposable, though somewhat equivocally: see p. 595 middle.
- 22.
Without relying on (#), there’s a way to modify the account of \(\rightarrow \) to get (2) by brute force: see Beall (2009, pp. 125–6). But it has the serious disadvantage of invalidating (4a), and indeed even the rule form of that.
- 23.
Restricted quantifier conditionals are slightly better behaved than ordinary ones at non-normal worlds, which is what allows for (2) and (4).
The use of the “Routley *” to handle negation in the Beall et al. paper creates further difficulties for achieving acceptable laws.
- 24.
Another law validated in \(\mathbf{B}\) and similar systems, but that fails for the ordinary conditional, is Transitivity: \(A\triangleright B,B\triangleright C\,\models \,A\triangleright C.\) That failure is almost an immediate consequence of the failure of (AS): Transitivity gives \((A\wedge B)\triangleright B,B\triangleright C\,\models \,(A\wedge B)\triangleright C\), and since \(\models \,(A\wedge B)\triangleright B\) we get (AS).
- 25.
I’m not necessarily claiming that relevant conditionals are of no interest in connection with work on the paradoxes, only that they are of no use either as “ordinary” conditionals or for restricted quantification. Weber (2018) grants that they are of no use for these purposes, but proposes that they may nonetheless be of use in a third role, that of providing identity conditions for properties. I have a mixed reaction to this (see Field 2018b), but need take no stand here.
- 26.
These broadly classical theories are forced to reject other classical metarules too, like reasoning by cases, the rule that if \(\Gamma ,A\,\models \,C\) and \(\Gamma ,B\,\models \,C\) then \(\Gamma ,A\vee B\,\models \,C\). This is forced on them because they accept excluded middle and explosion: given excluded middle, reasoning by cases leads to the rule
(*) If \(\Gamma ,A\,\models \,C\) and \(\Gamma ,\lnot A\,\models \,C\) then \(\Gamma \,\models \,C\);
and with even the minimal truth rules and Explosion, (*) cannot be accepted in face of the Liar Paradox. Indeed, independent of any conditional, there is an obvious conflict simply between (*), (T-Elim), and
(Alternative Weak T-Introd) \(A,\lnot True(\langle A\rangle )\,\models \,C\).
- 27.
Recently he has shown some sympathy with keeping both Conditional Introduction and a version of Modus Ponens, by going for a radically non-classical logic that restricts the structural contraction rule. (I say that he keeps a version of Modus Ponens because he treats \(A,A\rightarrow B\,\models \,B\) not as equivalent to \(A\wedge (A\rightarrow B)\,\models \,B\), which he rejects, but to the weaker \(A\circ (A\rightarrow B)\,\models \,B\) where \(\circ \) is a new connective (“fusion”) that is substantially stronger than \(\wedge \).) I will ignore this more recent turn in his thought.
- 28.
This assumes very minimal laws for disjunction, and that consequence is transitive.
- 29.
(a) The entailment is hard to explain unless the quantifier-restricting conditional is weak; indeed, (b) it can actually be shown to entail that it’s weak, using the definition of \(\rightarrow \) in terms of restricted quantification suggested in note 10.
- 30.
By contrast, if we restrict Modus Ponens for a conditional \(\ggg \) that is strong in the sense that \(A\negthinspace \ggg \negthinspace B\,\models \,\lnot A\vee B\), then we must restrict Explosion. [For Explosion gives \(A,\lnot A\,\models \,B\), which easily yields \(A,\lnot A\vee B\,\models \,B\), which by strongness assumption yields \(A,A\negthinspace \ggg \negthinspace B\,\models \,B\), i.e. Modus Ponens.] So resolving the \(\ggg \)-Curry for a strong conditional by restricting Modus Ponens would make the standard paraconsistent resolution of the Liar very natural. This however doesn’t provide much of an argument for the unity of the two paradoxes for a typical dialetheist: the only conditional that the typical dialetheist holds to be both strong and to fail Modus Ponens is the \(\supset \), and for it the resolution of the Curry Paradox just is the resolution of the Liar.
- 31.
The dual line for the paracomplete theorist would be to reject the rule
- (\(\lnot \)T-Elim):
-
\(\lnot True(\langle A\rangle )\,\models \,\lnot A\)
while accepting \(A\leftrightarrow True(\langle A\rangle )\) plus \(\lnot A\rightarrow \lnot True(\langle A\rangle )\). This would allow us to accept the untruth of the falsity-Liar \(Q*\), though we’d still have to reject both its falsity and its non-falsity (so it would still be inappropriate to call it a “truth-value gap”). And for the untruth Liar, we’d still need to take a paracomplete attitude to its truth as well as to its falsity. Again this seems like excess complication.
- 32.
A further conclusion may not be far behind: that such extremely weak theories are deductively inconsistent. The idea would be to use the numeralwise expressibility claimed in (iv) to run an analog of the informal proof claimed in (i) within the weak arithmetic. I don’t know whether Priest meant to endorse that further step, though pp. 238–9 of Priest (2006) are naturally read that way.
- 33.
One natural account of informal provability (which I think fits well with Priest’s discussion, e.g. in making the usual Gödel sentence \(G_{PA}\) informally provable in PA even though not formally so) is that for a sentence to be informally provable from PA is for it to be formally provable in an appropriate expansion PA\(^{T}\) of PA which includes a theory of truth/satisfaction or of properties. Then not only the argument that informal provability leads to paradox, but the details of what is informally provable, will depend on the truth/satisfaction/property theory. In a theory with an unstratified truth predicate one might proceed roughly as follows:
-
For each \(\alpha \), PA\(_{\alpha }\) is PA with an added predicate ‘True’ and composition laws, and induction extended to include it, plus some formalization of “for all \(\beta <\alpha \), every theorem of PA\(_{\beta }\) is true”.
The proper formalization of this requires a system of ordinal notations, so the PA\(_{\alpha }\) will be defined only for \(\alpha \) less than some countable ordinal \(\Gamma \) (presumably a limit ordinal). [One natural candidate is the Feferman-Schütte ordinal \(\Gamma _{0}\) (Feferman 1962), which will be the first one for which the ordering relation among notations for ordinals less than it can’t be shown to be a well ordering by reasoning within \(\cup \){PA\(_{\beta }\) : \(\beta <\alpha \)}. There may be arguments for an even earlier stopping point.] Then one can either
(i) take our informal theory to be \(\cup \){PA\(_{\alpha }\) : \(\alpha <\Gamma \)}, or
(ii) take “our informal theory” to be vague, and its precisifications stratified, via an increasing sequence \({\{\alpha _{\xi }:\xi <\Gamma \}}\) of increasingly complex limit ordinals less than \(\Gamma \).
(I’m inclined to think that (ii) is the better course.) In case (i), the informal theory we employ goes beyond the bounds of anything we can recognize as acceptable (though we may be able to recognize each of the fragments PA\(_{\alpha _{\xi }}\) as acceptable); so have no reason to believe a Gödel sentence for the entire informal theory. In case (ii), the Gödel sentence involving the vague predicate ‘informally provable’ is itself vague, and its precisification involving the notion of informal provability\(_{\xi _{1}}\) is only informally provable\(_{\xi _{2}}\) when \(\xi _{2}>\xi _{1}\). In both cases, obvious arguments to paradox are blocked. And we’ll be able to show that the usual Gödel sentence \(G_{PA}\) of PA is informally provable; as is the usual Gödel sentence of PA + \(G_{PA}\); and so on for a substantial transfinite sequence of iterated Gödel sentences.
-
- 34.
As remarked above, we must modify his presentation slightly to avoid the claim that the paradoxical argument is in PA. Rather, we must view the argument as done in a theory PA* that expands PA to include a truth or satisfaction predicate (or an ontology of properties and a property-instantiation predicate), and perhaps an informal provability predicate too though that might be definable using truth/satisfaction/instantiation. The new predicates are allowed in instances of the induction schema.
- 35.
Of course, the notion of property is loose enough that there is no one right answer to the question of their identity conditions.
- 36.
For instance, let U be \({\{x:x=x\}}\) and V be \({\{x:\lnot (x=x\ggg \lnot (x=x))\}}\). In the Brady theory, we can prove that everything is in U and everything is in V. (And no dialetheia are involved: we can reject the negation of these claims.) Nonetheless, we can define sets W for which we can prove that \(U\in W\) and \(V\notin W\) (and these claims aren’t dialetheic either); for instance, where \(W={\{y:\forall z(z=z\ggg z\in y)\}}\). This doesn’t violate “extensionality” as he defines it because U and V aren’t “coextensive” as he defines it, despite both being universal sets.
- 37.
By “the obvious formulation of a naive theory” I mean one that includes
- Abstraction Schema:
-
\(\forall u_{1} \ldots \forall u_{n}\forall z[z\in {\{x:A(x,u_{1} \ldots u_{n})\}}\lll \ggg A(z,u_{1} \ldots u_{n})]\)
(for the same \(\ggg \) as used in Extensionality), where we allow as instances of the schema formulas that themselves include the abstraction operator. It turns out that Øgaard’s proof depends on this formulation, rather than one which instead merely has
- Comprehension Schema:
-
\(\forall u_{1} \ldots \forall u_{n}\exists y\forall z[z\in y\lll \ggg A(z,u_{1} \ldots u_{n})]\),
and which takes that schema to apply only to formulas in its own language. (The Comprehension Schema allows us to introduce abstracts for formulas not containing the abstraction operator, but by the italicized clause, one can’t then assume that Comprehension applies to all formulas containing such abstracts.) Nothing in our paper rules out that there be a naive class theory with “genuine extensionality” that restricts itself to Comprehension rather than Abstraction. But nothing gives a whole lot of hope for one either, or suggests that the prospects for one are any better in a paraconsistent logic than in a paracomplete one. Moreover, I’m not sure what value Comprehension without Abstraction would have.
- 38.
It’s important that the truth theory be naive: without that, the semantics would not be intuitively correct. And it’s probably important that the account of truth and satisfaction be compositional, but as stressed earlier, one gets this automatically in the kinds of naive theory we’ve discussed.
- 39.
I’m taking ‘t denotes x’ to be existence-entailing: no term denotes Santa Claus.
- 40.
All we can expect is the inference from \(\exists n[Fn\wedge (\forall m<n)(Fm\vee \lnot Fm)]\) to \(\exists n[Fn\wedge (\forall m<n)\lnot Fm]\). Of course, if F is a predicate for which excluded middle is assumed to hold generally, this yields the principle discussed in the text, for that F. The use of the least number principle within mathematics is not affected.
- 41.
This could be extended to terms MOD in the non-classical part of the language, though the formulation would require more care since we can’t assume it determinate what such terms denote.
- 42.
On any naive dialetheic theory, sole truth is equivalent to truth, and sole falsity to falsity; so any dialetheia will also come out both solely true and solely false. With quasi-naivety this isn’t quite so, but they are equivalent as regards the Liar sentence Q. (And even for the \(Q*\) of Sect. 6.6, Priest’s view entails that it is solely false as well as dialetheic.)
- 43.
A preliminary argument that it is embarrassing: (C\(_{F}\)) would say of W that we should accept it if and only if we reject it. But the point of talk of rejection is to exclude acceptance: we can’t accept and reject the same sentence (barring shift in meaning), as Priest agrees. So we must neither accept it nor reject it. But such agnosticism wouldn’t bring much relief: if you have a view on which either accepting an additional claim or rejecting it leads immediately to trouble, it seems to me that the view is already in trouble. (An argument that doesn’t rely on this objection to agnosticism will be given near the end of this subsection.)
- 44.
Incidentally, it would really be somewhat more natural to define a sequence of weaker and weaker negation-like operators \(N^{\alpha }\), and to define \(NPF_{\alpha }(x)\) to mean \(N^{\alpha }True(x)\). The difference with what is in the text isn’t substantial. It’s worth stressing that either way, this is unlike stratified approaches to truth in that the \(N^{\alpha }\) and \(NPF_{\alpha }\) are all defined from the conditional; we do not need a hierarchy of primitive notions.
- 45.
Given a predicate \(NPF_{\infty }\), we define \(NPF_{\infty +1}(x)\) in the manner above: \(NPF_{\infty }(x)\vee (True(x)\rightarrow NPF_{\infty }(x))\). Letting the iterated Curry \(\kappa _{\infty }\) be equivalent to \(NPF_{\infty }(\langle \kappa _{\infty }\rangle )\), we’d have to reject \(NPF_{\infty }(\langle \kappa _{\infty }\rangle )\); but \(True(\langle \kappa _{\infty }\rangle )\rightarrow NPF_{\infty }(\langle \kappa _{\infty }\rangle )\) by construction and so \(NPF_{\infty +1}(\langle \kappa _{\infty }\rangle \) is provable.
- 46.
The claim holds far more generally, but the discussion in Sect. 6.7 was restricted in this way, and I don’t need any added generality here.
- 47.
We can avoid this situation only by having the converse situation, which is what a naive theorist gets if he or she evades the Curry Paradox by keeping Conditional Introduction. This route involves accepting the Curry sentence K and its equivalent \(K\rightarrow \bot \), and regarding the instance of Modus Ponens taking us from these to \(\bot \) as invalid in that reasoning in accordance with it is bad. At the same time, it involves regarding the inference as preserving truth, since given naivety, to say that it preserves truth is equivalent to K.
- 48.
No set-theoretically definable model can have this property, by Tarski’s Theorem on the undefinability in a classical language of truth in that language. Hamkins (2003) shows that the question of the existence of undefinable models with that property is independent of ZFC: you need a forcing construction to establish the consistency of the claim that such models exist.
- 49.
If there are “Hamkins models” of the sort mentioned in the previous footnote, then such models are inexpressible without expanding the classical part of the language. But this is so quite independent of the dialetheic or paracomplete truth theory. And in any case, such a model would approximate “intendedness” only for ‘True’-free sentences. For sentences with ‘True’, designatedness is distinct from “truthy” notions in the theories in question here: notions like truth, determinate truth, and so forth are non-classical notions; whereas, as discussed in Sect. 6.11.3, designatedness is a different sort of notion used for getting the logic right.
- 50.
I’m not totally certain that he’d be so concessive, since in many places he seems to assume that model theory has a role incompatible with what I’ve suggested:
(i) As discussed above near the end of Sect. 6.9, Priest’s argument that the paracomplete theorist has a problem with definite descriptions seems to turn on confusing classical model with reality. It is obviously true that in a given classical model, there will be a least number that satisfies the predicate ‘is not \(10^{6}\)-denoted in L’. But ‘denote’ (and hence ‘\(10^{6}\)-denoted’) is a non-classical notion, and so the classical model is an imperfect guide to the non-classical reality. (In a property-theoretic analog of model theory, done in a paracomplete logic, there would be no argument for a least number that satisfies ‘is not \(10^{6}\)-denoted in L’ in a given model.)
(ii) In his earlier discussions he tended to attribute to advocates of paracomplete theories the belief in “truth-value gaps”, which led to complete confusion between those theories and theories in classical logic that genuinely posit truth-value gaps. (Priest was far from alone in this attribution, I regret to say.) Paracomplete theories, of course, reject the existence of gaps: they reject the claim that there are sentences such that neither they nor their negation is true, since this is equivalent (in the logic common to all paracomplete and paraconsistent theories under discussion) to the claim that there are sentences such that both they and their negations are true. The way the confusion doubtless came about was in confusing truth with having designated value in some (unspecified) classical model. A parallel confusion would be one that attributed to the paraconsistent dialetheist the rejection of the claim that the Liar sentence Q isn’t true, on the basis of their rejection of the claim that their value in models is undesignated. That would ignore the distinction between the paraconsistent dialetheist and the classical truth-value glut theorist who accepts that both Q and \(\lnot Q\) are true but doesn’t accept \(\lnot Q\) (since he rejects the inference from \(True(\langle \lnot Q\rangle )\) to \(\lnot Q\)). Paraconsistent dialetheist theories are totally different from classical glut theories; paracomplete theories are equally different from truth-value gap theories.
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Field, H. (2019). Paraconsistent or Paracomplete?. In: Başkent, C., Ferguson, T. (eds) Graham Priest on Dialetheism and Paraconsistency. Outstanding Contributions to Logic, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-030-25365-3_6
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