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Although logical research into meta-inferences has a fairly long history, the last decade or so has seen much increased logical and philosophical attention in this area.
One reason for the spike in interest is the recent articulation and defence of solutions to the sorites and liar paradoxes that work by giving up on certain “classically valid meta-inferences”. Assessment of these arguments has raised many questions of both a logical and conceptual nature about meta-inferences and meta-inferential validity, including: what is a meta-inference, and what does it mean for one to be valid? Does the notion of a meta-inference iterate to yield reasonable notions of meta-meta-inferences, and so on? Do proponents of logical theories need to specify standards for meta-validity (and meta-meta-validity, and so on) as well as standards for plain validity, in order to properly characterise the principles they accept? and others besides. This note will review some of the recent work addressing these questions.
David Ripley’s paper One step is enough ( Journal of Philosophical Logic , vol. 51 (2022), pp. 1233–1259) engages with many of these issues. Proponents of meta-inferential solutions to semantic paradoxes (like Ripley) argue that the liar and the sorites paradoxes can be solved compatibly with all the rules of classical inferential logic, if only one gives up on standardly accepted meta-inferences. One focus of attention in this regard is the rule of cut, the validity of which effectively amounts to the transitivity of the entailment relation. As proponents of meta-inferential solutions have shown, classical logic is compatible with liar and sorites premises if the rule of cut is not assumed. One can accept that there is a valid argument whose conclusion is the liar sentence, and accept there is a valid argument from the liar to any unpalatable proposition you like, but refuse to “paste these derivations together” in the way allowed by cut, hence avoiding the unpalatable conclusion.
Some critics have argued that the appeal to meta-inferences here threatens a regress problem. If it is a good idea to give up classically valid meta-inferences to save classically valid inferences, why not pull the same stunt at the level of meta-meta-inferences, saving the classically valid meta-inferences as well? For example, I might just accept the rule of cut and deny the meta-meta-rule allowing me to go from cut, and the above pattern of classical validities, to the validity of your arbitrary proposition. Perhaps I can even argue I’ve done better than the straight meta-inferential theorist, in that I have saved more “classical logic”. But if this is right, this is obviously only the first step in a vicious regress, at the limit of which lies Lewis Carroll’s tortoise. These ideas were made formally precise in Barrio Pailos and Szmuc, A hierarchy of classical and paraconsistent logics ( Journal of Philosophical Logic , vol. 49 (2020), pp. 93–120), where it was shown that there are indeed three-valued systems of logic that, in a natural sense, validate all classical validities to the nth order, and that are compatible with transparent truth, but that disagree at the nth order, and even that there is such a logic that agrees with classical logic to all orders. Consequently, as, for example, Pailos himself has argued (in A fully classical truth theory characterised by substructural means, Review of Symbolic Logic , vol. 13 (2019), no. 2, pp. 249-268), there is a prima facie regress worry here for meta-inferential solutions.
Ripley’s paper is directed at this regress worry. He argues that although his favoured meta-inferential logic is, from a formal point of view, “only the first step in a grand meta-inferential adventure,” nevertheless from a philosophical point of view the first step provides a stable stopping point: one step is enough.
The paper gives the best introduction to the meta-inferential logics at issue I know of, and proves the key results concerning them in an admirably clear, rigorous, and elegant fashion. The reply offered to the regress worry is also simple and compelling. He points out that the regress worry assumes the meta-inferential paradox-solver should be committed to the view that more classical logic is better, where classical logic here has quite a specific understanding: namely, as meaning that the (meta-
$\cdots $
-)meta-inferential sequents made valid by the logic should be, insofar as this is possible, the same as the ones validated by the classical two-valued semantics. But, he argues, neither the meta-inferential paradox-solver nor the classical logician more generally has (or should have) such a commitment.
By way of elaboration, Ripley argues first that existing appeals to notions of “classicality” tend to focus either on the two-valued semantics or on the simple notion of validity for inferences—with no further commitments at the level of meta-inferences. In neither case is it true that this motivation supports the non-classical meta-inferential logics involved in the regress worry, which are three-valued and make substantive commitments beyond the level of inferences.
Of course, it is true that typical proponents of classical logic will expect the Tarskian condition of transitivity for logical entailment to obtain, and Ripley and his fellow meta-inferentialists must deny this. But so too, Ripley points out, must all the logics involved in pressing the regress worry, since even those which validate the sequent expressing the rule of transitivity are not themselves transitive. Ripley uses these observations to build a case that in fact his favoured logic is distinguished from the others in its plausibility, and I found his case compelling.
Isabella McAllister’s paper Classical logic is not uniquely characterisable (
Journal of Philosophical Logic
, vol. 51 (2022), pp. 1345–1365) is another strong example of interesting new work in this area. The logical results from Barrio, Pailos, and Szmuc that went into the regress worry involve the production of model-theoretic logics that validate the same inferences as classical logic to all finite orders; BPS point out that this shows classical logic cannot be characterised by its valid inferences at any order. McAllister generalizes these methods into the transfinite. She defines the notion of a meta-inference for an arbitrary ordinal
$\alpha $
, as well as an honorary notion of a meta-inference that ‘unions together’ all the ordinal levels. She then uses elegant generalisations of the techniques of BPS into the transfinite to defend the claim that, even in the case of classical propositional logic, no level
$\alpha $
is such that the set of classical validities is unique to classical logic: in the case of levels represented by ordinals this is because of the Barrio-style results; and for levels represented by proper-class-sized well-orderings, this is because we don’t have enough sets. McAllister goes on to present an argument that, in a certain paraconsistent set theory, classical logic is uniquely characterisable; with the one caveat being that it is in addition not uniquely characterisable as well. It is, I think, questionable what the significance of this latter result is when it comes to the characterisability question; but McAllister’s work is logically inventive, and philosophically driven, and consequently well worth close attention.
Another issue that has arisen in the context of discussions about meta-validity concerns the proper formal definition of validity, even just for “first-level” meta-inferences (like the rule of cut). Validity for inferences, for example, is defined in terms of the satisfaction of formulas by models: an inference from premises
$\Gamma $
to conclusion
$\phi $
being valid if every model for every formula in
$\Gamma $
is also a model for
$\phi $
. But what should the corresponding definition for a meta-inference be?
There are at least two clear candidate definitions: the local and the global. On the local definition, one first extends the notion of satisfaction in a model to inferences, and then gives a definition of validity for meta-inferences in terms of satisfaction of inferences analogous to the one given for inferences in terms of satisfaction of formulas. On the other hand, according to the global notion, one defines validity of meta-inferences directly in terms of validity of inferences: a meta-inference
$$ \begin{align*} \frac{\Gamma \Rightarrow \phi \quad \Sigma \Rightarrow \pi}{\Delta \Rightarrow \theta} \end{align*} $$
being valid just in case either one of the premises is invalid or the conclusion is valid.
Rea Golan’s paper There is no tenable notion of global meta-inferential validity ( Analysis , vol. 81 (2021), no. 3, pp. 411–420) addresses the question of which of these definitions is superior. Despite acknowledging that the global definition is the more intuitive, firstly because it seems to correspond to the intuitive idea that a valid meta-inference asserts the closure of valid inferences under a law, and secondly, because the notion of satisfaction for inferences employed in defining the local notion seems to be one with no intuitive or pre-theoretic support, nevertheless Golan argues that there is no tenable notion of global validity for meta-inferences and that consequently only the local notion is viable.
Golan’s argument is pleasingly simple. First, he takes it as an assumption that (a) meta-inferential validity is in some significant sense ‘logical’ and (b) that any logical property should be closed under uniform substitution. But since for sentence letters p, q the meta-inference from
$p \Rightarrow q$
to
$\Sigma \Rightarrow \Pi $
is globally valid for any p, q, closure under substitution entails that the inference form
$p\Rightarrow p$
to
$\Sigma \Rightarrow \Pi $
should be valid for arbitrary p, which in turn entails triviality of the inferential logic. After considering some tweaks to the definition to avoid this, Golan concludes that any such tweak will either collapse into the local notion or else be untenable. Hence the general result: there is no tenable notion of global meta-inferential validity.
Golan’s paper is rather short and as such it does not consider either all the possible objections to his argument or its possible consequences. One point worthy of note is that in standard modal logic it seems the global definition of meta-inferential validity is needed to secure the validity of the rule of necessitation, which is, notoriously, not locally valid. So more could be said about the upshots for modal logic. On the other hand, as to possible objections, it is salient that Golan takes meta-inferential validity to be in a significant sense “logical.” Here he is, I think, at odds with people like Ripley who think that meta-inferential validity is, so to speak, derivative from the notion of inferential validity. For Ripley, for example, the question of whether “classical logic” is closed under the rule of cut is not somehow a deep logical problem but rather something to be resolved by just checking whether the classical language at issue (given non-logical constants, such as a truth predicate, perhaps) yields a transitive notion of inferential validity. This is a signature/axiom dependent matter, and so perhaps there is no pressure on such a view to think of meta-inferential validity as distinctively “logical.” But despite these reservations, Golan’s case is a simple and plausible one for a surprising conclusion, and again is certainly well worth a look.
