## Abstract

We consider a handful of solutions to the liar paradox which admit a naive truth predicate and employ a non-classical logic, and which include a proposal for *classical recapture*. Classical recapture is essentially the property that the paradox solvent (in this case, the non-classical interpretation of the connectives) only affects the portion of the language including the truth predicate—so that the connectives can be interpreted classically in sentences in which the truth predicate does not occur. We consider a variation on this theme where the logic to be recaptured is not classical but rather intuitionist logic, and consider the extent to which these handful of solutions to the liar admit of intuitionist recapture by sketching potential ways of altering their various methods for classical recapture to suit an intuitionist framework.

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## Notes

- 1.
To be sure, one can restrict both the truth rules and propose a different logic, but we do not consider such options here.

- 2.
Notable exceptions are the extensive work by Michael Dunn, Neil Tennant’s work on

*core logic*, and a forthcoming book by Robles and Mendez [36]. - 3.
- 4.
One can insert one’s preferred rules for \(\vee \) to the same effect (perhaps needing some additional applications of structural rules).

- 5.
Thomas develops a Kripke-style model-theoretic semantics for this framework, but those details do not matter here. We might add that it seems to us that the Ripley account of logical consequence, inspired by Restall and Brandom, is not appropriate for intuitionism. The intuitionist

*does*take it to be “out of bounds”, i.e., incoherent, to assert a sentence in the form \(\lnot \lnot A\) while denying*A*. But, of course \(\lnot \lnot A\vdash A\) is not intuitionistically valid. Equivalently, it is incoherent to deny an instance of excluded middle, but \(\vdash (A \vee \lnot A)\) is not valid. To put the point another way, suppose that one starts with an intuitionistic deductive system, and then defines a new technical term as follows: An argument \([\Gamma ,\phi ]\) is*Ripley-valid*just in case it is incoherent to assert every member of \(\Gamma \) and to deny \(\phi \). Then Ripley-validity is the same as classical validity. - 6.
This sketch isn’t new—propositional modal and many-valued logics have been studied extensively, and details are available in [28, Chap. 11].

- 7.
\(\Rightarrow \) and \(\Leftrightarrow \) stand in for the metalanguage conditional and biconditional, respectively.

- 8.
Arguably, mathematical predicates are

*stable*, in the sense that if an atomic sentence is true (respectively false) in a given world, then it is true (false) in all accessible worlds (see Linnebo [15] or [16]). So if our attention restricted to mathematical languages, we could just think of our extensions and anti-extensions as functions from predicates to*n*-tuples of the whole domain*D*. The translation from the present language into that of intuitionism, in the next sub-section, will have this effect. - 9.
*T*(*u*,*x*,*z*) is read as “*z*is the code of a complete computation of the Turing machine with code*u*, given input*x*. And if*z*is the code of a complete computation, then*U*(*z*) is the output. - 10.
Further details and references on the topic of

**HA**can be found many places, including [47]. - 11.
Beall himself was engaged in this endeavour in [3].

- 12.
These comments address

*propositional***LP**, but for first order**LP**the recapture of classical logic is obtained by demanding the extensions and anti-extensions of predicates are*exclusive*, which amounts to the same restriction. Also worth noting is that this approach has a predecessor in the discussion in [27, pp. 117–119]. - 13.
Beall may not want to hold this combination of views at present.

- 14.
Using Gentzen’s original formulation, the data type of the premise and conclusion are something like multisets or sequences.

- 15.
- 16.
The sans-serif titles refer to a correspondence between these arguments and

*combinators*, which are central in proof and model theoretic presentations of substructural logics. Roughly, with the structural rule (along with appropriate rules for \(\rightarrow \) and \(\wedge \)), one can prove each of the following. Conversely, these axioms allow one to prove that the structural rule is admissible in a Hilbert system (when the premise and conclusion sequents of the rule are appropriately translated as sentences). The details here are complex, and [6] provides a good overview of the area. For present purposes it is enough to note the correspondence. - 17.
There is another system, occasionally called

**BB**which is slightly weaker than**B**which still has**FDE**as it’s conditional-free fragment, but our focus is on**B**as this is most natural in the standard ternary relation semantics. - 18.
Strictly speaking not all of these tonicity conditions on

*R*are required for present purposes. All are required to interpret the*fusion*or*multiplicative conjunction*(as it’s called in the tradition of Linear Logic) and an additional conditional along with the usual conditional. Details on these connectives in the ternary relation semantics are available in [31] and elsewhere. We keep the full set of tonicity conditions here for the sake of broadest applications to a variety of logics including these additional connectives, but our attention is restricted to \(\rightarrow \). - 19.
Unlike the valuation defined in Sect. 4.1, we can get away with a binary valuation rather than a four-valued valuation since the negation here is interpreted by \(*\), rather than in terms of additional values.

- 20.
As an example of the minimality of

**B**’s conditional, one can only prove \((A\rightarrow B)\wedge (B\rightarrow A)\) in**B**when*A*,*B*are*syntactically*identical. This is closely related to classic work in relevant logic on the**T**-W problem (or**P**-W problem) as investigated in [21]. - 21.
The frame condition corresponding to this formula is \(Rxyz\Rightarrow Rxz^*y^*\).

- 22.
Those familiar with combinatory logic or lambda calculus may recognise in these principles the combinators called K and S. In combinatory terms, these have the following reduction behaviours: \(\textsf {K}xy\) reduces to

*x*, and \(\textsf {S}xyz\) reduces to (*xz*)(*yz*). Note the resemblance between these reduction schemata and the frame constraints corresponding to the axioms in question. - 23.
The difference here is largely one of temperament—if one is of a positive mindset, then compatibility seems right, if of a negative mindset, then perp or

*incompatibility*seems best. - 24.
For instance, additions to the model structure are required to obtain

**E**[2, pp. 171–172]. - 25.
They call the resulting system

**CR**. - 26.
Recall that, in Field’s system (Sect. 8), the conditional just became the material conditional upon recovery—upon assuming instances of excluded middle in that case. And that facilitated Plan B. Here we have no such luck, and, it seems, have to simply ignore the conditional that was introduced in order to do the mathematics internally.

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Tedder, A., Shapiro, S. (2019). Making Truth Safe for Intuitionists. In: Rieger, A., Young, G. (eds) Dialetheism and its Applications. Trends in Logic, vol 52. Springer, Cham. https://doi.org/10.1007/978-3-030-30221-4_8

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