Lavinia Picollo

My current research is mainly in metasemantics—the study of the mechanisms by which the expressions of a language come to have the meanings they do. I am particularly interested in logical and mathematical vocabulary. For ordinary terms like “cat”, one can tell a plausible story: we point at cats while uttering the word, thereby fixing its intended meaning. But for expressions such as “not” and “plus”, it is far less clear how their meaning is fixed.

On a relatively popular metasemantic view, the meaning of our words is determined entirely by the rules governing their use. These may be inference rules—intralinguistic rules such as axioms, ordinary inferences, and meta-inferences—or language-entry and language-exit rules, which connect perceptual states with statements, and statements with non-linguistic actions, respectively. Within the use-theoretic tradition, logical and mathematical inferentialists maintain that the meaning of logical and mathematical vocabulary is fixed exclusively by inference rules. I’m primarily concerned with these inferentialist approaches and their implications. Moderate versions cash out meanings in terms of language interpretations – typically as truth-conditions over models or classes of models. I am especially interested in what the meanings of our logical and mathematical terms could be on these views if they were fully determined by inference rules humans are capable of following. The underlying thread of my work is the worry that such rules are not powerful enough to confer on our words their unique intended meanings as moderates understand them. A related worry is that, although the rules deliver many truths, they are not powerful enough to fix the truth-value of every logical or mathematical sentence.

These worries are grounded in salient twentieth-century metatheoretic results due to Carnap, Löwenheim and Skolem, and Gödel. Arguably, under plausible assumptions about human cognitive capabilities, any inference rules we are in a position to follow can be captured by a recursively axiomatizable formal theory. If correct, this assumption is seemingly bad news for the inferentialist, as:

• Carnap showed that the intended (objectual) truth-conditions for our universal and existential first-order quantifiers are not fixed by the inference rules of any known recursive calculus (i.e. whose notion of proof is recursive). This is known as “Carnap’s Categoricity Problem” for the quantifiers.
• The Löwenheim-Skolem theorems establish that any first-order theory which can be interpreted as talking about infinitely many things—e.g. arithmetic and set theory—also has an interpretation of each cardinality. So no such theory pins down a unique structure up to isomorphism (a unique class of isomorphic models)—e.g. the natural number structure. No such theory is categorical.
• Gödel’s first incompleteness theorem shows that any recursively axiomatizable theory that is consistent and contains a modicum of arithmetic—e.g. arithmetic and set theory—is negation-incomplete: some sentences of the language of the theory will be neither provable nor refutable. So no such theory suffices to account for the determinacy of the relevant fragment of mathematical discourse—e.g. the determinacy of arithmetic. (Gödel’s result also entails that none of the relevant theories describes a unique structure up to isomorphism. But, combined with the Löwenheim-Skolem theorems, it has the further implication that no unique structure can be pinned down even if we restrict attention to structures of a particular cardinality—e.g. denumerable structures.)

These results appear to undermine the inferentialist’s attempt to secure intended logical and mathematical content. The central question is how far the inferentialist can withstand these pressures. To this end, advocates of logical and mathematical inferentialism have developed several ingenious strategies, including revisions to the notion of categoricity, alternative inferential frameworks, and appeals to open-endedness. In my work, I examine these proposals in detail and search for promising alternatives.