Faculty of Arts
Department of Philosophy
Phil 422/Phil 522: Relevance and Substructural Logics
— Fall term (2017)
Relevance logics constitute a group of logics that were invented to formally capture inferences in which the conclusion shares some content with the premises. Informally speaking, relevance logics succeeded where C. Lewis's attempt failed: relevance logics solve the problem for which Lewis invented strict implication and his modal logics (S1, …, S5), namely, the problem of the “paradoxes of material implication.” Formally speaking, relevance logics typically do not permit the insertion of arbitrary premises into an inference (while logical correctness is retained), and they abandon the rule that yields an arbitrary conclusion from an inconsistent set of premises. Since their inception in the 1950s, relevance logics developed into an area comprising many attractive intensional logics, which are well-motivated philosophical logics with mathematically exciting properties.
Substructural logics — understood in a literal sense — are logics that result from the sequent calculus for two-valued logic by omission (of some structural rules or properties of sequents). Historically, intuitionistic logic was the first substructural logic, but the Lambek calculi may be the most widely known substructural logics. It turns out that more variations in a sequent calculus, such as extensions and additions, allow us to formalize further logics, including some relevance logics, modal logics and linear logic. Substructural logics have their origins in various disciplines: intuitionistic logic comes from mathematics, the Lambek calculi were introduced in linguistics to characterize grammaticality, certain modal logics are of great interest to philosophers and linear logic reflects the resource conscious attitude in computer science. Subsequently, these logics found applications in some of the same areas.
The course will deal with some relevance and substructural logics — without attempting to cover everything within this immense field. We will look at proof systems, at various semantics and at various properties of these logics (such as the question of decidability). The selection of the logics and properties will be guided by the importance of the logics and the significance of the results about them (including classical and new theorems from the last decade or so).
T, R 11:00 am – 12:20 pm
Texts: Bimbó, K., “Relevance logics,” D. Jacquette, (ed.), Philosophy of Logic, (Vol. 5 of the Handbook of the Philosophy of Science, D. M. Gabbay, P. Thaggard, J. Woods, (eds.)), Elsevier, Amsterdam, 2007, pp. 723–789. (required)
Restall, G., An Introduction to Substructural Logics, Routledge, London, (UK), 2000. (required)
Some journal articles (about the latest research results) may be recommended or linked in the e-classroom.
For further information, please contact the instructor at
The (official) course outline is available in the e-classroom during the course.
[Last updated on March 3rd, 2017.]