Abstract
Term Functor Logic is a term logic that recovers some important features of the traditional, Aristotelian logic; however, it turns out that it does not preserve all of the Aristotelian properties a valid inference should have insofar as the class of theorems of Term Functor Logic includes some inferences that may be considered irrelevant (e.g. ex falso, verum ad, and petitio principii). By following an Aristotelian or syllogistic notion of relevance, in this contribution we adapt a tableaux method for Term Functor Logic in order to avoid irrelevance and we offer some sort of intuitive semantics in terms of traditional inferential situations (e.g. propter quid, quia, non causa ut causa, and non sequitur).
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Notes
Although we are exemplifying this logic with syllogistic inferences, this system is capable of representing relational, singular, and compound inferences with ease and clarity. Furthermore, TFL is arguably more expressive than classical first order logic [Englebretsen 1987, p.172].
That these inferences are valid in TFL can be swiftly shown as follows. First, ex falso is valid because TFL admits indirect proof methods (Sommers and Englebretsen, 2000, p.147), and so the conjunction of the premises and the rejection of the conclusion entails a contradiction; second, verum ad is valid because again, by way of indirect methods, the rejection of a tautology is a contradiction; and last, petitio is valid because the sum of the premises is equal to the conclusion and the number of particular premises is equal to the number of particular conclusions.
In TFL relations are understood as terms. Hence, in this example, the statement “every Boy Loves some Girl" has three terms, namely, Boy, Girl, and Loves. Consequently, the statement “every Boy Loves some Girl" would be represented by the expression B + (+ L + G).
For sake of brevity, but without loss of generality, here we omit the syllogisms that require existential import.
References
Alvarez, E., & Correia, M. (2012). Syllogistic with indefinite terms. History and Philosophy of Logic, 33(4), 297–306. https://doi.org/10.1080/01445340.2012.680704
Anderson, A., Dunn, J., & Belnap, N. (1975). Entailment: The Logic of Relevance and Necessity (Vol. 1). Princeton University Press.
Aristotle. (1989). Prior analytics. Hackett classics series. Hackett.
Bastit, M. (2011). Jan Lukasiewicz contre le dictum de omni et de nullo. Philosophia Scienti, 15, 55–68.
Carnap, R. (1930). Die alte und die neue Logik. Erkenntnis, 1, 12–26.
Castro-Manzano, J. M. (2020). Distribution tableaux, distribution models. Axioms, 9(2), 41. https://doi.org/10.3390/axioms9020041
Castro-Manzano, J. M. (2021). Traditional logic and computational thinking. Philosophies, 6(1), 12. https://doi.org/10.3390/philosophies6010012
Copi, I., Cohen, C., & McMahon, K. (2013). Introduction to logic: Pearson New International Edition. Pearson Education Limited.
Correia, M. (2017). La lógica aristotélica y sus perspectivas. Pensamiento. Revista de Investigación e Información Filosófica, 73(275), 5–19. https://doi.org/10.14422/pen.v73.i275
D’Agostino, M., Gabbay, D. M., Hahnle, R., & Posegga, J. (1999). Handbook of tableau methods. Springer.
de Morgan, A. (1864). On the Syllogism, No. IV., and on the Logic of Relations. Transactions of the Cambridge Philosophical Society, 10, 331.
Dosen, K. (1992). The first axiomatization of relevant logic. Journal of Philosophical Logic, 21(4), 339–356.
Englebretsen, G.: The New Syllogistic. 05. P. Lang (1987)
Englebretsen, G. (1996). Something to reckon with: The Logic of Terms: Books collection. University of Ottawa Press.
Englebretsen, G. (2017). Bare facts and naked truths: A new correspondence theory of truth. Taylor & Francis.
Englebretsen, G., & Sayward, C. (2011). Philosophical logic: An introduction to advanced topics. Bloomsbury Academic.
Frege, G., & Angelelli, I. (1973). Begrifsschrift und andere Aufsatze. Wissenschaftliche Buchge-sellschaft.
Geach, P. T. (1962). Reference and generality: An examination of some medieval and modern theories. Contemporary Philosophy / Cornell University.
Haack, S. (1978). Philosophy of Logics. Cambridge University Press.
Kreeft, P., & Dougherty, T. (2004). Socratic logic: A logic text using socratic method, platonic questions & Aristotelian Principles. St. Augustine’s Press.
Kuhn, S. T. (1983). An axiomatization of predicate functor logic. Notre Dame Journal of Formal Logic, 24(2), 233–241. https://doi.org/10.1305/ndj
Mares, E. (2004). Relevant Logic: A philosophical interpretation. Cambridge University Press.
Meyer, R., & Martin, E. (2019). S (for syllogism) revisited. The Australasian Journal of Logic, 16(3), 49–67. https://doi.org/10.26686/ajl.v16i3.5466
Morado, R. (1983). Deducibility implies relevance? A cautious answer (on professor Orayen’s criticisms of relevant logic). Crítica: Revista Hispanoamericana De Filosofía, 15(45), 105–108.
Moss, L. (2015). Natural logic. In S. Lappin & C. Fox (Eds.), The handbook of contemporary semantic theory. Wiley.
Noah, A. (1980). Predicate-functors and the limits of decidability in logic. Notre Dame Journal of Formal Logic, 21(4), 701–707. https://doi.org/10.1305/ndjfl/1093883255
Noah, A. (2005). Sommers’s cancellation technique and the method of resolution. In D. S. Oderberg (Ed.), The old new logic: Essays on the Philosophy of Fred Sommers (pp. 169–182). MIT Press.
Priest, G. (2008). An introduction to non-classical logic: From if to is. Cambridge University Press.
Quine, W.V.O. (1971). Predicate functor logic. In: J.E. Fenstad (ed.) Proceedings of the Second Scandinavian Logic Symposium. North-Holland (1971)
Russell, B. (1901). A critical exposition of the philosophy of Leibniz: With an appendix of leading passages. Cambridge University Press.
Sommers, F.: The Logic of Natural Language. Clarendon Library of Logic and Philosophy, Clarendon Press; Oxford: New York: Oxford University Press (1982)
Sommers, F. (2005). Intelectual autobiography. In D. S. Oderberg (Ed.), The old new logic: Essays on the philosophy of Fred Sommers (pp. 1–24). MIT Press.
Sommers, F., & Englebretsen, G. (2000). An invitation to formal reasoning: The logic of terms. Ashgate.
Thom, P. (2007). Logic and ontology in the syllogistic of Robert Kilwardby. Studien Und Texte Zur Geistesgeschichte Des Mittelalters. Brill.
Veatch, H. (1970). Intentional logic: A logic based on philosophical realism. Archon Books.
Walton, D. (2003). Relevance in argumentation. Taylor & Francis.
Woods, J. (2014). Aristotle’s Earlier Logic. Studies in logic. College Publications.
Acknowledgements
We would like to thank George Englebretsen for comments on a previous draft; and to the anonymous reviewers for their precise corrections.
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Appendix A: Syllogistic
Appendix A: Syllogistic
Syllogistic is a term logic that has its origins in Aristotle’s Prior Analytics (Aristotle, 1989) and deals with inference between categorical statements. A categorical statement is a statement composed by two terms, a quantity, and a quality. The subject and the predicate of a statement are called terms: the term-schema S denotes the subject term of the statement and the term-schema P denotes the predicate. The quantity may be either universal (All) or particular (Some) and the quality may be either affirmative (is) or negative (is not). These categorical statements have a type denoted by a label (either a (universal affirmative, SaP), e (universal negative, SeP), i (particular affrmative, SiP), or o (particular negative, SoP)) that allows us to determine a mood, that is, a sequence of three categorical statements ordered in such a way that two statements are premises (major and minor) and the last one is a conclusion. A categorical syllogism, then, is a mood with three terms one of which appears in both premises but not in the conclusion. This particular term, usually denoted with the term-schema M, works as a link between the remaining terms and is known as the middle term. According to the position of this middle term, four figures can be set up in order to encode the valid syllogistic moods. See Table
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Castro-Manzano, JM. Syllogistic Relevance and Term Logic. J of Log Lang and Inf (2024). https://doi.org/10.1007/s10849-024-09417-5
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DOI: https://doi.org/10.1007/s10849-024-09417-5