Classical C-systems: Logical achievements and logical problems (2015)
(Arnon Avron, Tel Aviv University, IL)
What is a classical LFI, and when are two such logics identical?
We discuss two fundamental questions concerning classical LFIs (i.e., LFIs which are based on positive classical logic). We start with the observation that according to the official definition, an axiomatic extension of classical logic is an LFI w.r.t some unary connective *iff it has a bottom element, and * satisfies three simple minimal conditions (that, for atomic P, neither of P and *P implies the other, and their conjunction does not implies the bottom element). This means that the class of classical LFIs is rather broad in comparison to the family of logics on which most of the investigations on LFIs have concentrated on. Accordingly, we provide a characterization of this family.
Another question that we address is when should we see two LFIs as identical. For example: is da Costa $C_1$ identical with the system $Cila$x, or $Cla$? Our reply is that with neither.
To provide adequate answers to both questions we describe and use appropriate semantic and proof-theoretical tools.
Click for slides.
We discuss two fundamental questions concerning classical LFIs (i.e., LFIs which are based on positive classical logic). We start with the observation that according to the official definition, an axiomatic extension of classical logic is an LFI w.r.t some unary connective *iff it has a bottom element, and * satisfies three simple minimal conditions (that, for atomic P, neither of P and *P implies the other, and their conjunction does not implies the bottom element). This means that the class of classical LFIs is rather broad in comparison to the family of logics on which most of the investigations on LFIs have concentrated on. Accordingly, we provide a characterization of this family.
Another question that we address is when should we see two LFIs as identical. For example: is da Costa $C_1$ identical with the system $Cila$x, or $Cla$? Our reply is that with neither.
To provide adequate answers to both questions we describe and use appropriate semantic and proof-theoretical tools.
Click for slides.