Marco Grossi

Some philosophers argue that a theory of logical validity should not interpret its own language, because a Russellian argument shows that self-applicability is inconsistent with the ability to capture all the interpretations of its own language. First, I set up a formal system to examine the Russellian argument. I then defend the need for self-applicability. I argue that self-applicability seems to be implied by generality, and that the Russellian argument rests on a test for meaning that is biased against self-applicability. I propose an alternative test which is unproblematic for self-applicability. I conclude that self-applicability can be vindicated and the Russellian argument can be resisted.

According to the interpretational theory of logical validity (IR), logical validity is preservation of truth in all interpretations compatible with the intended meaning of logical expressions. IR suffers from a seemingly defeating objection, the so-called cardinality problem: any instance of the statement ‘There are n things’ is true under all interpretations, since it can be written down using only logical expressions that are not to be reinterpreted; yet ‘There are n things’ is not logically true. I argue that the cardinality problem is indeed a serious problem for IR, when understood in terms of ‘asymmetry of information’. I then argue that IR can be rehabilitated by making quantifiers context-sensitive: what we do not reinterpret is the Kaplanian character of a quantifier, rather than its content. ‘There are n things’ is false in a context where fewer than n things are relevant, so it is not logically true in IR. I finally discuss some objections and ramifications of my account: I discuss how to make space for the possibility of an explicitly absolutely general quantifier in my framework, how terms can be logical even though context-sensitive, and how to recapture classical logic within my framework.

What are logical notions? According to a very popular proposal, a logical notion is something invariant under some “transformation” of objects, usually permutations or isomorphisms. The first chapter is about extending “invariance” accounts of logicality to intensional notions, by asking for invariance under arbitrary permutations of both possible worlds and objects. I discuss the results one gets in this extended theory of invariance, and how to fix many technical issues. The second chapter is about setting out a better theory of logicality. I discuss the limits of invariance frameworks, and the need for a theory of logicality with a more solid philosophical ground. I believe that the concept of information can play a major role in defining what logic is and what logical notions are. I spell out this intuition, by designing a new test for logicality. A notion is logical iff it behaves in a certain way, by checking only “structural aspects of information”, and it does so under arbitrary transformations of its “informational inputs”. In the last chapter I explore some interesting features of my theory. I show how, contrary to standard invariance, in mine logical notions tend to stay persistent across different models of information. I also spell out an intermediate notion of quasi-logicality to make sense of the formality of “world-sensitive” notions: notions whose behaviour changes across worlds. I finally propose a case study: deontic modals. I discuss how one can argue for their quasi-logicality, in my framework. The dissertation is concluded with a technical appendix, in which I prove that my theory is a restriction of standard permutation invariance when we model the space of information in a certain way: as a set of complete powersets of some sets.