Logic and Language

Franz Berto: Telling Negations from In-Australia Operators

Speaker: Franz Berto
Title: Telling Negations from In-Australia Operators
Time: 16:00 - 17:30
Location: to be announced

“Don’t we say ‘In Australia, the winter is in the summer’, ‘In Australia, mammals lay eggs’, ‘In Australia, swans are black’? If ‘In Australia’ can thus behave like ‘not’ [...], perhaps the tilde means ‘In Australia’?”

This old joke by Timothy Smiley turns on the Quinean change-of-subject charge for the negations of non-classical logics: which among them (if any) are real negations, rather than in-Australia operators? I provide a principled answer supporting a moderate logical pluralism: not everything called “negation” in the literature deserves its name, but more than one item does. This is achieved by grounding negation in a (twofold) core notion: the concept of compatibility, together with its polar opposite, incompatibility. The features of (in)compatibility set precise constraints on what counts as a negation. The residual differences between the operators that pass the threshold depend on concepts different from the core notion. Some substantive claims are made here: Whatever does not satisfy (Minimal) Contraposition and Double Negation Introduction is a mere in-Australia operator; Some paraconsistent negations (unsurprisingly) are Australian, but others may qualify as real; Intuitionistic negation qualifies; Classical-Boolean negation does as well, for constructivist and paraconsistent doubts on it do not turn on the basic concept of (in)compatibility.