Consequential Implication and the Implicative Conditional
DOI:
https://doi.org/10.12775/LLP.2025.001Keywords
consequential implication, implicative conditional, Aristotle’s thesis, Boethius’ thesis, contraposition, connexivity, definable conditionals, strong super-strict implication, square of oppositionsAbstract
This paper compares two logical conditionals which are strengthenings of the strict conditional and avoid the paradoxes of strict implication. The logics of both may be viewed as extensions of KT, and the two conditionals are interdefinable in KT. The implicative conditional requires that its antecedent and consequent be both contingent. The consequential conditional may be viewed as a weakening of the implicative conditional, insofar as it also admits the case in which the antecedent and the consequent are strictly equivalent (either both necessary or both impossible). The two conditionals share a number of properties, among them Transitivity, Contraposition, Aristotle’s Thesis, Weak Boethius’ Thesis and Aristotle’s Second Thesis. They also share some restricted principles such as Possibilistic Monotonicity, Possibilistic Simplification and Possibilistic Right Weakening. They differ in relation to Identity, which is validated by consequential implication, while the implicative conditional only validates the restricted principle of Possibilistic Identity. The relations between the two conditionals are represented by two Aristotelian cubes of opposition, one involving the contrariety between If A, then B and If A, then ¬B, according to Weak Boethius’ Thesis, and the other the contrariety between If A, then B and If ¬A, then B, according to Aristotle’s Second Thesis. We also explore the relations between the two logical conditionals and natural language conditionals, emphasizing the dependence of the latter on the context, and the need to distinguish natural language conditionals which may be viewed as consequential or implicative, on one side, and concessive and some other types of conditionals, on the other.
References
Anderson, A. R., and N. D. Belnap, 1975, Entailment: The Logic of Relevance and Necessity, volume 1, Princeton University Press, Princeton.
Angell, R. B., 1962, “A propositional logic with subjunctive conditionals”, The Journal of Symbolic Logic, 27(3): 327–343. DOI: https://doi.org/10.2307/2964651
Åqvist, L., 1973, “Modal logic with subjunctive conditionals and dispositional predicates”, Journal of Philosophical Logic, 2: 1–76. DOI: https://doi.org/10.1007/bf02115609
Bennett, J., 2003, A Philosophical Guide to Conditionals, Oxford University Press, New York. DOI: https://doi.org/10.1093/0199258872.001.0001
Blanshard, B., 1969, The Nature of Thought, G. Allen & Unwin, London.
Bolzano, B., 1978, Grundlegung der Logik. Wissenschaftslehre I/II, Felix Meiner Verlag, Hamburg. French translation: Théorie de la Science Paris: Gallimard, 2011. DOI: https://doi.org/10.28937/978-3-7873-2626-6
Burgess, J. P., 1981, “Quick completeness proofs for some logics of conditionals”, Notre Dame Journal of Formal Logic, 22(1): 76–84. DOI: https://doi.org/10.1305/ndjfl/1093883341
Burks, A. W., 1955, “Dispositional statements”, Philosophy of Science, 22(3): 175–193. DOI: https://doi.org/10.1086/287422
Chellas, B. F., 1975, “Basic conditional logic”, Journal of Philosophical Logic, 4(2): 133–153. DOI: https://doi.org/10.1007/BF00693270
Crupi, V., and A. Iacona, 2022, “On the logical form of concessive conditionals”, Journal of Philosophical Logic, 51: 633–651. DOI: https://doi.org/10.1007/s10992-021-09645-1
Davis, W. A., 1983, “Weak and strong conditionals”, Pacific Philosophical Quarterly, 64(1): 57–71. DOI: https://doi.org/10.1111/j.1468-0114.1983.tb00184.x
Douven., I., 2015, The Epistemology of Indicative Conditionals: Formal and Empirical Approaches, Cambridge University Press, Cambridge.
Douven, I., 2017, “How to account for the oddness of missing-link conditionals”, Synthese, 194: 1541–1554. DOI: https://doi.org/10.1007/s11229-015-0756-7
Ducrot, O., 1972/1991, Dire et ne pas dire Hermann, Paris.
Edgington, D., 1995, “On conditionals”, Mind, 104(414): 235–329. DOI: https://doi.org/10.1093/mind/104.414.235
Gherardi, G., and E. Orlandelli, 2021, “Super-strict implications”, Bulletin of the Section of Logic, 50(1): 1–34. DOI: https://doi.org/10.18778/0138-0680.2021.02
Gherardi, G., E. Orlandelli, and E. Raidl, 2024, “Proof systems for super-strict implication”, Studia Logica, 112: 249–294. DOI: https://doi.org/10.1007/s11225-023-10048-3
Gibbard, A., 1981, “Two recent theories of conditionals”, pages 211–247 in W. L. Harper, R. Stalnaker and G. Pearce (eds.), Ifs: Conditionals, belief, decision, chance and time, The University of Western Ontario Series in Philosophy of Science, vol. 15, Springer, Dordrecht. DOI: https://doi.org/10.1007/978-94-009-9117-0_10
Gomes, G., 2005, “Ordinary language conditionals”, Manuscript. https://www.academia.edu/93865496/Ordinary_Language_Conditionals_2005
Gomes, G., 2009, “Are necessary and sufficient conditions converse relations?”, Australasian Journal of Philosophy, 87(3): 375–387. DOI: https://doi.org/10.1080/00048400802587325
Gomes, G., 2013, “Pensamento e linguagem nas afirmações condicionais”, DELTA: Documentação de Estudos em Linguística Teórica e Aplicada, 29(1): 121–134. DOI: https://doi.org/10.1590/s0102-44502013000100006
Gomes, G., 2019, “Meaning-preserving contraposition of natural language conditionals”, Journal of Pragmatics, 152: 46–60. DOI: https://doi.org/10.1016/j.pragma.2019.08.003
Gomes, G., 2020, “Concessive conditionals without ‘even if’ and nonconcessive conditionals with ‘even if’ ”, Acta Analytica, 35(1): 1–21. DOI: https://doi.org/10.1007/s12136-019-00396-y
Gomes, G., 2024, “Necessary and sufficient conditions, counterfactuals and causal explanations”, Erkenntnis, 89(8): 3085–3108. DOI: https://doi.org/10.1007/s10670-023-00668-5
Humberstone, I., 1978, “Two merits of the circumstantial operator language for conditional logics”, Australasian Journal of Philosophy, 56(1): 21–24. DOI: https://doi.org/10.1080/00048407812341011
Jackson, F., 1979, “On assertion and conditionals”, Philosophical Review, 88: 565–589. DOI: https://doi.org/10.2307/2184845
Kielkopf, C. F., 1977, Formal Sentential Entailment, University Press of America, Washington.
Lenzen, W., 2021, “The third and fourth stoic accounts of conditionals”, pages 127–146 in M. Blicha and I. Sedlár (eds.), The Logica Yearbook 2020, College Publications, Rickmansworth.
Lewis, D., 1973, Counterfactuals, Blackwell, Oxford. DOI: https://doi.org/10.1007/978-94-009-9117-0_3
Lycan, W. G., 2001, Real Conditionals, Oxford University Press, Oxford. DOI: https://doi.org/10.1093/oso/9780199242078.001.0001
Martin, C., 2004, “Logic”, pages 158–199 in J. Brower and K. Guilfoy (eds.), The Cambridge Companion to Abelard, Handbook of the History of Logic, Cambridge University Press, Cambridge.
McCall, S., 1966, “Connexive implication”, The Journal of Symbolic Logic, 31(3): 415–433. DOI: https://doi.org/10.2307/2270458
McCall, S., 2012, “A history of connexivity”, pages 415–449 in D. M. Gabbay, F. J. Pelletier, and J. Woods (eds.), Logic: A History of its Central Concepts, volume 11 of Handbook of the History of Logic, North-Holland, Amsterdam. DOI: https://doi.org/10.1016/b978-0-444-52937-4.50008-3
Meyer, R. K., 1977, “S5–The poor man’s connexive implication”, The Relevance Logic Newsletter, 2: 117–123.
Mortensen, C., 1984, “Aristotle’s thesis in consistent and inconsistent logics”, Studia Logica, 43(1/2): 107–116. DOI: https://doi.org/10.1007/BF00935744
Nastide Vincentis, M., 2006, “Conflict and connectedness: Between modern logic and the history of ancient logic”, pages 229–251 in E. Ballo and M. Franchella (eds.), Logic and Philosophy in Italy, Polimetrica International Scientific Publisher, Monza.
Pizzi, C., 1980, “‘Since’, ‘even if’, ‘as if’ ”, pages 73–87 in M. L. Dalla Chiara (ed.), Italian Studies in the Philosophy of Science, Springer, Dordrecht. DOI: https://doi.org/ 10.1007/978-94-009-8937-5_6
Pizzi, C., 1991, “Decision procedures for logics of consequential implication”, Notre Dame Journal of Formal Logic, 32(4): 618–636. DOI: https://doi.org/10.1305/ndjfl/1093635934
Pizzi, C., 1993, “Causality and the transitivity of counterfactuals”, O que nos faz pensar, 7: 89–103.
Pizzi, C., 2018, “Two kinds of consequential implication”, Studia Logica, 106: 453–480. DOI: https://doi.org/10.1007/s11225-017-9749-5
Pizzi, C., 2022, “Axioms for a logic of consequential counterfactuals”, Logic Journal of the IGPL, 31(5): 907–925. DOI: https://doi.org/10.1093/jigpal/jzac052
Pizzi, C., 2024, “An introduction to Boethian logics”, pages 79–110 in H. Omori and H. Wansing (eds.), 60 Years of Connexive Logics, Springer.
Pizzi, C., and T. Williamson, 1997, “Strong Boethius’ thesis and consequential implication”, Journal of Philosophical Logic, 26(5): 569–588. DOI: https://doi.org/10.1023/a:1004230028063
Pollock, J. L., 1976, Subjunctive Reasoning, D. Reidel, Dordrecht.
Priest, G., 1999, “Negation as cancellation, and connexive logic”, Topoi, 18: 14–148. DOI: https://doi.org/10.1023/A:1006294205280
Raidl, E., 2019, “Completeness for counter-doxa conditionals – using ranking semantics”, The Review of Symbolic Logic, 12(4): 861–891. DOI: https://doi.org/10.1017/S1755020318000199
Raidl, E., 2021a, “Definable conditionals”, Topoi, 40(1): 87–105. DOI: https://doi.org/10.1007/s11245-020-09704-3
Raidl, E., 2021b, “Strengthened conditionals”, pages 139–155 in B. Liao and Y. N. Wáng (eds.), Context, Conflict and Reasoning. Logic in Asia Series, Springer, Singapore. DOI: https://doi.org/10.1007/978-981-15-7134-3_11
Raidl, E., 2021c, “Three conditionals: Contraposition, difference-making and dependency”, pages 201–217 in M. Blicha and I. Sedlár (eds.), The Logica Yearbook 2020, College Publications.
Raidl, E., 2023, “Neutralization, Lewis’ doctored conditional, or another note on ‘a connexive conditional’”, Logos & Episteme, 14(1): 101–118. DOI: https://doi.org/10.5840/logos-episteme20231415
Raidl, E., 2025a (forthcoming), “Between Lewis and Burgess I: Logics without rational monotonicity”, Journal of Philosophical Logic, pages 1–38.
Raidl, E., 2025b (forthcoming), “Between Lewis and Burgess II : Semantics without rational monotonicity”, Journal of Philosophical Logic, pages 1–52.
Raidl, E., and G. Gomes, 2024, “The implicative conditional”, Journal of Philosophical Logic, 53(1): 1–47. DOI: https://doi.org/10.1007/s10992-023-09715-6
Raidl, E., A. Iacona, and V. Crupi, 2021, “The logic of the evidential conditional”, Review of Symbolic Logic, 15(3): 758–770. DOI: https://doi.org/10.1017/S1755020321000071
Raidl, E., A. Iacona, and V. Crupi, 2023, “An axiomatic system for concessive conditionals,” Studia Logica, 112: 343–363. DOI: https://doi.org/10.1007/s11225-022-10034-1
Raidl, E., and H. Rott, 2023, “Threshold-based belief change”, Australasian Journal of Logic, 20(3): 429–477. DOI: https://doi.org/10.26686/ajl.v20i3.7408
Raidl, E., and H. Rott, 2024, “Towards a logic for ‘because’ ”, Philosophical Studies, 181: 2247–2277. DOI: https://doi.org/10.1007/s11098-023-01998-4
Ramsey, F. P., 1929, “General propositions and causality”, pages 133–151 in H. A. Mellor (ed.), Philosophical Papers, Cambridge University Press, Cambridge. DOI: https://doi.org/10.1017/cbo9780511527494.006
Rott, H., 1986, “Ifs, though and because”, Erkenntnis, 25(3): 345–370. DOI: https://doi.org/10.1007/BF00175348
Rott, H., 2020, “Notes on contraposing conditionals”. https://philsciarchive.pitt.edu/17092/
Rott, H., 2022, “Difference-making conditionals and the relevant Ramsey test”, Review of Symbolic Logic, 15(1): 133–164. DOI: https://doi.org/10.1017/S1755020319000674
Rott, H., 2024a, “Evidential support and contraposition”, Erkenntnis, 89: 2253–2271. DOI: https://doi.org/10.1007/s10670-022-00628-5
Rott, H., 2024b, “Difference-making conditionals and connexivity”, Studia Logica, 112: 405–458. DOI: https://doi.org/10.1007/s11225-023-10071-4
Routley, R., 1978, “Semantics for connexive logics. I”, Studia Logica, 37(4): 393–412. DOI: https://doi.org/10.1007/BF02176171
Routley, R., and V. Routley, 1985, “Negation and contradiction”, Revista Colombiana de Matematicas, 19(1–2): 201–230. https://repositorio.unal.edu.co/handle/unal/48791
Spohn, W., 2013, “A ranking-theoretic approach to conditionals”, Cognitive Science, 37(6): 1074–1106. DOI: https://doi.org/10.1111/cogs.12057
Spohn, W., 2015, “Conditionals: A unifying ranking-theoretic perspective”, Philosophers’ Imprint, 15(1): 1–30. http://hdl.handle.net/2027/spo.3521354.0015.001
Stalnaker, R. C., 1968, “A theory of conditionals”, pages 98–112 in N. Rescher (ed.), Studies in Logical Theory, American Philosophical Quarterly Monographs 2, Blackwell, Oxford. DOI: https://doi.org/10.1007/978-94-009-9117-0_2
Strawson, P. F., 1948, “Necessary propositions and entailment-statements”, Mind, 57(226): 184–200. DOI: https://doi.org/10.1093/mind/lvii.226.184
Thompson, B. E., 1991, “Why is conjunctive simplification invalid?”, Notre Dame Journal of Formal Logic, 32(2): 248–254. DOI: https://doi.org/10.1305/ndjfl/1093635749
Tichý, P., 1984, “Subjunctive conditionals: Two parameters vs. three”, Philosophical Studies, 45(2): 147–179. DOI: https://doi.org/10.1007/bf00372476
Wansing, H., and H. Omori, 2024, “Connexive logic, connexivity, and connexivism: Remarks on terminology”, Studia Logica, 112: 1–35. DOI: https://doi.org/10.1007/s11225-023-10082-1
Wansing, H., and D. Skurt, 2018, “Negation as cancellation, connexive logic, and qLPm”, The Australasian Journal of Logic, 15(2): 476–488. DOI: https://doi.org/10.26686/ajl.v15i2.4869
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