Melbourne Logic Day
Old Quad, Room 142A
University of Melbourne
Parkville 3010
Australia
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11am-12noon
Michał Stronkowski—Almost Structural Completeness
The notion of structural completeness for logics/deductive systems has received considerable attention for many years. A translation to algebra gives: a quasivariety is structurally complete if it is generated by its free algebras. It appears that many deductive systems (quasivarieties), like S5 or MVn fails structural completeness for a rather immaterial reason. Therefore the adjusted notion was introduced:almost structural completeness. We investigate almost structural completeness from an algebraic perspective and obtain a characterization of this notion for quasivarieties. We also provide examples of modal logics which are not structurally complete but are almost structurally complete.
12:10pm-1:10pm
Lloyd Humberstone— Prior's OIC Nonconservativity Example Revisited
In his 1964 note, “Two Additions to Positive Implication,” A. N. Prior showed that standard axioms governing conjunction yielded a nonconservative extension of the pure implicational intermediate logic, OIC (“order implicational calculus”), of R. A. Bull. Here we seek to illuminate this example by transposing it to the setting of modal logic, and also, in a final (‘postcript’) section, to relate it to the propositional logic of what have been called ‘Hilbert algebras with infimum’.
2pm-3pm
Greg Restall—From Defining Rules to Cut Elimination, and its consequences
I show how a collection of defining rules (invertible rules characterising connectives) can be converted uniformly and systematically (in the presence of the structural rules of Cut and Identity) converted into Left and Right Rules, in such a way that the conditions under which Cut can be seen as a natural and straightforward consequence. I explain the motivation, and the upshot for accounts of harmony, and the use of sequent rules to define logical concepts.
3:10pm-4:10pm
Rohan French—Predicate Logic with Scopes
We present a sequent calculus for a novel reformulation of classical predicate logic in which variables play not only an ‘object representing’ role, but also a ‘scope indicating’ role. Using this sequent calculus we then present some applications to problems in the philosophy of language concerning opaque contexts and propositional attitudes.
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#unimelb, #logic