Expressing Validity by Revision: Formal and Philosophical Aspects

Philosophical logic as discipline is about clarifying what is logical for various concepts, and a logical theory is a collection of arguments that are supposed to be (logically) valid for one or more concepts. Since validity is itself a concept for which arguments may be valid, a logical theory can contain arguments about the logical theory itself; that the logical theory is thus about itself. Recent research in philosophical logic has argued that a logical theory about itself should express the fact that if an argument is valid, then if its premises are true then its conclusion is true. However, the same research shows that the most natural formalization of this and similar insights is, together with certain assumptions about logical theories commonly assumed in for example mathematical reasoning, sufficient for proving any statement, and logical theories that are about themselves will therefore be useless since every argument will be deemed valid. The most prominent conclusion has been that a logical theory can only be about itself if we give up one or more of those assumptions about logical theories. The aim of this project was to develop an alternative approach to expressing within a logical theory which arguments are valid according to that logical theory in accordance with the above insights, but which can be applied to most logical theories without giving up various central assumptions about what a logical theory is. The project successfully developed both proof-theoretic and model-theoretic tools to expand a range of logical theories with valid arguments about the logical theory itself in a way that is provably conservative over the logical theory being expanded and which results in a consistent logical theory if the logical theory being expanded itself is consistent. In general then, the project?s hypothesis about how one can obtain logical theories that are about themselves without paradoxical consequences has to some extent been confirmed. Moreover, the tools can be justified from a philosophical perspective in terms of how it ensures that the logical theory is expanded only with arguments about the logical theory and nothing else, thereby ensuring its conceptual neutrality with regard to the original logical theory.

Gitt at prosjektets målsetning og tema faller godt innenfor det som kan beskrives som grunnforskning, så er det særdeles vanskelig å si noe om hvilken betydning eller nytteverdi de publiserte resultatene kan ha, ikke bare fornæringslivet og samfunnet for øvrig, men også for forskningsfeltet. Dette er noe som bare kan sies noe om i ettertid, og vil avhenge av hvorvidt resultatene blir betraktet som viktige av andre forskere innenfor forskningsfeltet. Allikevel kan det sies at prosjektet's resultater har potensialet til å forme den videre filosofiske diskusjonen rundt prosjektets tema.

Can a theory of valid arguments represent which arguments are valid according to itself? New research in philosophical logic suggests that this can only be achieved if the theory is non-contractive, i.e. that multiple occurrences of the same premise cannot always be replaced with one occurrence of that premise in a valid argument. The aim of this project is to show that this conclusion is wrong by providing and justifying a new condition for when a theory represents which arguments are valid according to itself, and furthermore develop model-theoretic and proof-theoretic tools for augmenting theories with a predicate representing their valid arguments. To justify the new condition, the project will critically examine the reasons for adopting the paradoxical conditions, and show that the novel paradox-free condition satisfies the same reasons. It will focus on the intended inferential role of a predicate representing the valid arguments and the relationship between valid arguments and truth. The result will be an account of what is required to represent what is valid within a theory that has far-reaching consequences for research into semantic paradoxes and our understanding of theories of valid arguments themselves. The model-theoretic tool for theories definable on n-valued models will be developed by modifying transfinite revision sequences in ways that incorporate lessons from "possible world semantics" and the connection between conditions for being a premise and being a conclusion in a sound inference and the definition of a conditional. The proof-theoretic tool will be developed by modifying labelled sequent calculi for modal logics to simulate successor- and limit-stages of transfinite revision sequences. The extent to which the calculi can approximate such sequences is an open question, and the aim is that the tool can be applied on n-sided sequents to obtain a suitable tool mimicking the model-theoretic construction for each theory.