Abstract
It has been argued recently (Beall in Spandrels of truth, Oxford University Press, Oxford, 2009; Beall and Murzi J Philos 110:143–165, 2013) that dialetheist theories are unable to express the concept of naive validity. In this paper, we will show that \(\mathbf {LP}\) can be non-trivially expanded with a naive validity predicate. The resulting theory, \(\mathbf {LP}^{\mathbf {Val}}\) reaches this goal by adopting a weak self-referential procedure. We show that \(\mathbf {LP}^{\mathbf {Val}}\) is sound and complete with respect to the three-sided sequent calculus \(\mathbf {SLP}^{\mathbf {Val}}\). Moreover, \(\mathbf {LP}^{\mathbf {Val}}\) can be safely expanded with a transparent truth predicate. We will also present an alternative theory \(\mathbf {LP}^{\mathbf {Val}^{*}}\), which includes a non-deterministic validity predicate.
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Notes
Notice that, as for any formula B, \(\bot \vdash B\), an application of \(\textit{VP}\) will prove \(\vdash \textit{Val}(\langle \bot \rangle , \langle B \rangle )\). Then, by an application of \(\textit{MetaVD}\), \(\vdash B\).
Additionally, there are in fact many connections between our project and the one presented in Goodship (1996) by Laura Goodship, named by Beall (2011) as “the Goodship Project.” Those links will become explicit when we explain how we get self-referential sentences in our theory of naïve validity, in Sect. 3.
Though in principle it is possible to internalize validity either with a validity constant or a validity predicate, we will choose this last approach, since adding a validity operator has expressive limitations comparable to the ones of a truth operator.
Let \(\mathcal {L}\) be a first order language that includes the language of first-order Peano Arithmetic, and let Th be a theory that extends \(\mathbf {PA}\). Assume then that there is available a name-coding device \(\langle \cdot \rangle \) of formulas of \(\mathcal {L}\) over \(\omega \), and let \(\mathbf {\jmath }\) abbreviate the sequence of variables \(j_1, \ldots , j_n\) by \(\mathbf {\jmath }\). Then the Weak Diagonal Lemma will be proved in Th.
- Weak Diagonal Lemma::
For every formula \(A(j,\mathbf {\jmath })\in \mathcal {L}\), there is a formula \(B(\mathbf {\jmath }) \in \mathcal {L}\) such that
$$\begin{aligned} Th \vdash B(\mathbf {\jmath })\leftrightarrow A(\langle B(\mathbf {\jmath })\rangle , \mathbf {\jmath }) \end{aligned}$$
To prove the the Strong Diagonal Lemma, the theory must contain terms for each recursive function.
- Strong Diagonal Lemma::
For every formula \(A(j,\mathbf {\jmath })\in \mathcal {L}\), there is a term t such that
$$\begin{aligned} Th \vdash t = \langle A(t, \mathbf {\jmath })\rangle \end{aligned}$$
For more about the Diagonal Lemmas, see Burguess et al. (2007).
At least when we restrict our attention to formulas with one free variable.
We will like to thank an anonymous referee for helping us clarify this point. As she points out, although strong self-reference might be expressed in the meta-language, it is not expressible in the object language on pain on triviality. We think she is right, if identity is understood classically, e.g., if no identity assertion takes a non-classical value. We guess (though we do not have a proof to fully justify this claim) that a non-classical treatment of identities might help us recover the syntactic functions that we lack in the present framework. We think that such a project is worth exploring. But we will leave the accomplishment of this task for future work.
As there may be some doubts about whether a strong self-referential procedure has the same effect as the Definitional Equivalence Principle that takes part in the proof described in page 2, we will present a proof that does not use that principle.
This matrix was first introduced in the literature of paraconsistent logics by Antonio Sette in Sette (1973). It is also the one corresponding to the conditional defined in the system \(\mathbf {MPT}\), developed in Coniglio and Cruz (2014). The authors endorse that conditional precisely because it allegedly reflects the consequence relation of the theory, that is the same as in \({ LP}\): preservation of values \(1, \frac{1}{2}\).
Nevertheless, this does not mean that we straightforward reject validity “gaps” or “gluts”. Moreover, there are some approaches that support a non-classical view for validity. In particular, Meadows (2014) support a “gappy” theory of validity, while Pailos and Tajer (2017) defend a “glutty” version of it. Those approaches are interesting on there own, but we think they are not available for a supporter of \(\mathbf {LP}\). \(\mathbf {LP}\)’s notion of validity leaves no space neither for “gluts” nor for “gaps”, because it should be understand in a traditional way, as preservation of designated values—e.g., of “truth”—from premises to conclusions. Thus, the way a supporter of \(\mathbf {LP}\) deals with a validity predicate must be very different form the way she treats, for example, a truth predicate—e.g., a predicate that admits a “glutty” behaviour.
Though it might be strange to explain the behaviour of a predicate through a matrix, like a truth-functional connective, we hope to have make it clear enough how this predicate works. In fact, if we were working in a theory with a transparent truth predicate, every self-referential sentence that includes \({ Val}\) can be emulated with the help of the truth predicate and a suitable conditional that shares \({ Val}\)’s truth table. But without that kind of resource, it is not possible to made self-referential sentences just with truth functional constants.
This last move is not essential. But putting a mark on the distinguished propositional letters will make things easier to follow, as those propositional letters will play a key part in the self-referential procedure we are about to present.
Notice that neither \(x^{P}\) nor \(x^{C}\) belong to the language of the theory. Moreover, for every \(x^{*}\), there will be (infinitely) many formulas with the structure \({ Val}(\langle x^{P} \rangle , \langle x^{C} \rangle )\). For example, there will be one such that \(x^{P} = x^{*}\), but \(x^{C} = p\), one such that \(x^{C} = x^{*}\) and \(x^{P} = q \wedge r\), etc.
Notice that such valuations exist. Just consider the one that assigns the value \(\frac{1}{2}\) to every propositional letter.
We would like to highlight that is only a version of that principle holds in our system. As we already mentioned, we will have, for every formula-schema \(B_{y}\), one and just one sentence of the form \(y^{*} \leftrightarrow B_{y}^{*}\) in Z. An unrestricted version of the principle makes all instances of such biconditionals true. For our purposes, the restricted version will be good enough.
As we have already explained, those biconditionals can also be read as a way to mimic (some relevant) instances of the weak diagonal lemma, that are themselves traditionally treated as a way to achieve self-reference. Therefore, this allow us to have in the language sentences that represent part of the instances of the (weak) diagonal lemma. Nevertheless, we would not have all of them. For example, we will not have cycles –e.g. sentences that refer to other sentences that (eventually) refer to them. But it will not be difficult to expand this procedure to include them all. Still, as our primary interest is not cycles (nor, for example, a validity version of a Yablo’s chain) this seems to be good enough to achieve our goals. (For more about the Yablos Paradox, see e.g. Yablo 1993).
We would like to thank Dave Ripley for his help with this result.
Classical values, in this framework, will be associated with the left and the right sides of a three-sided sequent. But it can be prove that the Beall–Murzi can only receive the intermediate value, because the sequent that is empty on the left and on the right, and has only the Beall–Murzi sentence in the middle, will have a proof. That proof is very similar to the one that Ripley in Ripley (2012) gave for the sequent that has only the Liar sentence in the middle and is empty on the extremes. In fact, the Beall–Murzi sentence can be used to define a constant for the intermediate value, and, with its help, it is easy to define constants for the classical values.
VAL reflects the semantic behaviour of the validity predicate. What VAL express is that every sentence of the form \({ Val}(\langle A \rangle , \langle B \rangle )\) will either receive the value 0 or the value 1 (but never the value \(\frac{1}{2}\)) in every valuation.
For example, Meadows (2014) gives a theory of naive validity that cannot be expanded with a transparent truth predicate without becoming trivial.
Such valuations also exist in this case. We only need to consider, once again, the one that assigns \(\frac{1}{2}\) to every propositional letter.
Or will validate inferences that involve validity assertions that are not intuitively valid.
The awkwardness, in cases like these, seems to be not different from the one generated by a conditional that is true if its antecedent is false, or its consequent is true, or some other sufficient extensional condition.
Though there is some open debate about whether \(\textit{MetaVD}\), as we already mentioned, should be part of the list. An inferentialist like David Ripley, for example, rejects \(\textit{MetaVD}\) as an adequate rule for a validity predicate.
A similar strategy was adopted, for example, by Meadows (2014).
A similar solution is considered in Barrio et al. (2016). In that paper, the authors assess different ways to add a validity predicate to the non-transitive logic ST.
Other rules of this kind that Negri mentioned in Negri (2005) are the following:
The validity-version of T might be the following:
And here is a the validity-version of S4:
It cannot occur in the three places, because then there will be some finite stage n where the formula appears for the first time in the branch in the three sides. But then that sequent will be an axiom, and therefore the branch will be closed.
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Acknowledgements
This work was supported by The National Scientific and Technical Research Council (CONICET), and has benefited greatly from discussions with Graham Priest, David Ripley, Eduardo Barrio, Natalia Buacar, Lucas Rosenblatt, Damian Szmuc, Diego Tajer and Paula Teijeiro.
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This work was supported by The National Scientific and Technical Research Council (CONICET), and has benefited greatly from discussions with Graham Priest, David Ripley, Eduardo Barrio, Natalia Buacar, Lucas Rosenblatt, Damian Szmuc, Diego Tajer and Paula Teijeiro.
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Pailos, F.M. Validity, dialetheism and self-reference. Synthese 197, 773–792 (2020). https://doi.org/10.1007/s11229-018-1731-x
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DOI: https://doi.org/10.1007/s11229-018-1731-x