Raymundo Morado's Talks at GTAL-CLE Seminars
In 1800, Kant famously declared exhausted our research into logic. According to him, “we do not require any new discoveries in Logic” (“wir brauchen auch zur Logik keine neuen Erfindungen”). Mankind had found practically all there was to find about inference and validity. Turned out the news of the end of logic were greatly exaggerated. So, are there any limits to logic? Certainly, we do very little syllogistic logic anymore, and we expect no big surprises from classical propositional calculus. Maybe logic ended in 1879, with the publication of the first complete system of first-order quantificational logic; maybe in 1932 with the normal systems of strict implication. Yet, logic keeps expanding both its depth and its breadth. We have discovered truths about the logical connectives that intrigued the stoics, and we have expanded the power of classical logic with amazing conservative extensions. Still, some developments seem to challenge our very notion of logicality. Kant was talking from a certain perspective of what logic is that excluded from the set go many recent developments. If logic is the science of necessary inference, non-deductive forms of reasoning must fall outside its realm; mathematical induction is in, induction in zoology is out. If logic is the science of abstract concepts, there can be a logical theory of classical quantifiers, but not of all fallacies. Each idea of logic sets its limits. Limits can be good, as Kant’s dove attests. But there can be also good reasons to evolve our concepts out of the old limits and to allow them to encompass new or unsuspected facets of reality.
I propose to see if we can find a principled extension of our ideas of logic when confronted with non-classical systems, especially with rival logics. I believe there are important lessons for the philosophy of logic to be gleaned from the examination of logics such as the intuitionistic, free, or quantum ones.
I will illustrate this with the case of the paraconsistent logics of relevance which are of great importance both theoretical and practical.
Then we shall examine some of the general problems of constraining excessively our notion of logicality and illustrate this with a discussion of the family of non-monotonic formalisms. This will lead us to consider some formal questions that can help us hone pertinent concepts.
And this in turn will be useful to tackle the ultimate limit: the general issue of what justifies logic itself. We shall finally mention some open problems in this area of the philosophy of logic(s). These are mostly fundamental topics, and we shall only require the minimal symbolic apparatus of a first semester in logic.