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.
References
Anderson, A., & Belnap, N. (1975). Entailment: the logic of relevance and necessity (Vol. 1). Princeton University Press.
Anderson, A., Belnap, N., & Michael Dunn, J. (1992). Entailment: The logic of relevance and necessity (Vol. 2). Princeton University Press.
Beall, J. C. (2009). Spandrels of truth, Oxford.
Beall, J. (2015). Free of detachment: logic, rationality, and gluts. Nous, 49(2), 410–423.
Beall, J. C. & Allen Logan, S. (2017). Logic: The basics (2nd ed.), Routledge.
Bimbó, K. (2014). Proof theory; Sequent calculi and related formalisms. Boca Ration, FL: CRC Press.
Brandom, R. B. (1994). Brandom, making it explicit. Havard University Press.
Fine, K. (1988). Semantics for quantified relevance logic. Journal of Philosophical Logic, 17(1), 27–59.
Field, H. (2008). Saving truth from paradox. Oxford University Press.
Fuhrmann, A. (1990). Models for relevant modal logics. Studia Logica, 49, 501–514.
Gödel, K. (1933). Eine Interpretation des intuitionistischen Aussagenkalküls. Ergebnisse eines mathematischen Kolloquiums, 4, 39–40.
Kleene, S. C. (19 45). On the interpretation of intuitionistic number theory. Journal of Symbolic Logic,10, 109–124.
Kremer, P. (1993). Quantifying over propositions in relevance logic: Nonaxiomatisability of primary interpretations of \(\forall p\) and \(\exists p\). Journal of Symbolic Logic, 58(1), 334–349.
Kripke, S. (1975). Outline of a theory of truth. The Journal of Philosophy, 72(19), 690–716.
Linnebo, Ø. (2013). The potential hierarchy of sets. Review of Symbolic Logic, 6(2), 205–228.
Linnebo, Ø., & Shapiro, S. (Forthcoming). Actual and Potential Infinity. Nous.
Mares, E. (1993). Classically modal complete relevant logics. Mathematical Logic Quarterly, 39(1), 165–177.
Mares, E., & Goldblatt, R. (2006). An alternative semantics for quantified relevant logic. Journal of Symbolic Logic, 71(1), 163–187.
Meyer, R. K. (1973). Intuitionism, entailment, negation. In H. Leblanc (Ed.), Truth, syntax, and modality (pp. 168–198), North Holland.
Meyer, R. K., & Martin, E. P. (1973). Classical relevant logics. Studia Logica, 32, 51–63.
Meyer, R. K., & Martin, E. P. (1982). Solution to the P-W problem. The Journal of Symbolic Logic, 47(4), 869–887.
Meyer, R. K., & Routley, R. (1974). Classical relevant logics II. Studia Logica, 33, 183–194.
Meyer, R. K., Routley, R., & Dunn, J. M. (1979). Curry’s paradox. Analysis, 39, 124–128.
Michael Dunn, J. (1993). Star and Perp: two treatments of negation. Philosophical Perspectives, 7, 331–357.
Paoli, F. (2002). Logics, substructural: A primer. Springer.
Priest, G. (2000). Inconsistent models of arithmetic part II: The general case. Journal of Symbolic Logic, 65, 1519–1529.
Priest, G. (2006). In contradiction: A study of the transconsistent (2nd ed.). Oxford University Press.
Priest, G. (2008). An introduction to non-classical logics (2nd ed.), Cambridge.
Priest, G., & Sylvan, R. (1992). Simplified semantics for basic relevant logics. Journal of Philosophical Logic, 21(2), 217–232.
Restall, G. (1992). A note on Naïve set theory in LP. Notre Dame Journal of Formal Logic, 33(3), 422–432.
Restall, G. (2000). An introduction to substructural logics, Routledge.
Restall, G. (2005). Multiple conclusions. In P. Hajek, L. Valdes-Villanueva, & D. Westerstahl (Eds.), Logic, Methodology and Philosophy of Science: Proceedings of the Twelfth International Congress (pp. 189–205). Kings College Publication.
Restall, G. (2013). Assertion, denial and non-classical theories. In K. Tanaka (Ed.), Paraconsistency: Logic and applications (pp. 81–99). Springer.
Ripley, D. (2012). Conservatively extending classical logic with transparent truth. Review of Symbolic Logic, 5(2), 354–378.
Ripley, D. (2013). Paradoxes and failure of cut. Australasian Journal of Philosophy, 91(1), 139–164.
Robles, G., & Mendez, J. M. (2018). Routley-Meyer ternary relational semantics for intuitionistic-type negations. Academic Press.
Routley, R., & Meyer, R. K. (1973). The semantics of entailment. In H. Leblanc (Ed.), Truth, Syntax, and Modality. Proceedings of the Temple University Conference on Alternative Semantics (pp. 199–243), North-Holland, Amsterdam, Netherlands.
Routley, R., & Routley, V. (1972). The semantics of first degree entailment. Nous, 6, 335–358.
Routley, R., Plumwood, V., Meyer, R. K., & Brady, R. T. (1982). Brady, relevant logics and their rivals: The basic philosophical and semantical theory, Ridgeview.
Schroeder-Heister, P., & Dosen, K. (1994). Substructural logics. Oxford University Press.
Tennant, N. (1987). Anti-realism and logic: Truth as eternal. Oxford University Press.
Tennant, N. (1997). The taming of the true. Oxford University Press.
Tennant, N. (2015). A new unified account of truth and paradox. Mind, 124, 571–605.
Tennant, N. (2017). Core logic. Oxford University Press.
Thomas, M. (2014). Expressive limitations of naive set theory in LP and minimally inconsistent LP. Review of Symbolic Logic, 7(2), 341–350.
Thomas, M. (2017). A kripke-style semantics for paradox-tolerant nontransitive intuitionistic logic. Retrieved November 6, 2017, from https://drive.google.com/file/d/0Bx_KuRX8hgkZenBSbXZ6YnZNRkk/edit.
van Dalen, D. (2001). Intuitionistic logic. In L. Goble (Ed.), The blackwell guide to philosophical logic (pp. 224–258). Blackwell.
Weber, Z. (2012). Transfinite cardinals in paraconsistent set theory. Review of Symbolic Logic, 5, 269–293.
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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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