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Formal Frameworks for Modal Notions Conceived as Predicates

Periodic Reporting for period 1 - FOREMOTIONS (Formal Frameworks for Modal Notions Conceived as Predicates)

Reporting period: 2016-01-01 to 2017-12-31

"The project addressed a fundamental gap in contemporary approaches to formal philosophy. Philosophy aims to provide universal theses about reality, such as ""all truths can be known"", but the way in which these claims are analyzed in contemporary philosophy is far from universal. To properly formulate them, one usually has to distinguish between a language or conceptual framework in which the thesis is formulated, the object-language, and a language in which the thesis is analyzed, the meta-language. The naive way of combining the two languages leads to paradox. The project analyzed ways to avoid paradox while restoring the intended universality. This was accomplished by mainly: (i) a careful and original analysis of the deductive properties of theories of semantic and modal notions and their different logical assumptions; (ii) a new approach to the notion of necessity that, unlike standard modal logic, is compatible with the presence of self-referential constructions in the theories; (iii) a new analysis of the notion of implicit commitment for basic mathematical theories and its relationships with semantic and modal notions.

The project contributes to establish the European academic environment as the landmark for mathematically informed studies of modalities. This, combined with the use of powerful formal tools, contributes to the reconstruction of a vision of philosophy in continuity to and not in opposition with the exact sciences that represents one of the pillars of the European identity, as witnessed by great European thinkers such as Descartes and Leibniz.

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"The activities carried out in the project can be divided in three main clusters. Their description also explains in what way the work carried out in the project differs from the original work packages.

1) The first part of the project, months 1-6, focused on two main issues. The first was the development of new theories of truth and necessity (in two forms, as metaphysical necessity and as logical necessity), compatible with self-reference, and overcoming the Liar and Montague's paradox. The second was an abstract approach to the possibility of comparing axioms for semantic and modal notions.

2) The second part of the project, months 6-12, focused on the criteria for choosing a specific solution to modal and semantic paradoxes, e.g. whether one should be prepared to revise the principles of standard logic as a reaction to paradox. The theory-choice criterion proposed and defended in the project was based on the presence of a satisfactory amount of non-semantic, non-modal consequences. This proposal has been corroborated by new results in the proof theory of semantic and modal theories.

3) The third part of the project, months 12-19, focused on the notion of implicit commitment for basic formal theories: what principles is one bound to accept if she accepts a starting theory? The analysis put forward in the process identified a coherent path from basic combinatorial assumtions (e.g. basic arithmetical theories), and much stronger logical systems, based on the combination of semantic or modal axioms and the so-called ""reflection principles"", i.e. explicit logical assertions of the correctness of a starting theory."
"Each of the updated work packages isolated above displayed a tangible advancement of the state of the art:

1) Two theories of self-referential necessity were introduced. The first notion was based on logical principles (in particular, the notion of supervaluation) that harmonize very well with the notion of necessity but that were never applied to it. The second notion of necessity studied was the notion of ""necessarily following from"", that applies to arguments or pieces of reasoning that we regard as valid. New principles for ""follows from"" we presented, and a new, non-paradoxical model for its use introduced. Moreover, the new tools for comparing semantic and modal axioms proposed in the project combined ideas from traditional proof theory and scientific reductions in the philosophy of science, providing a wide array of strong notions of equivalence between different axioms for intensional notions.

2) The second work package provided the literature with a new approach to the comparison between different solutions to paradox. Traditionally, rival solutions to semantic and modal paradoxes were proposed where standard logical principles are restricted or not. The project analyzed and measured the costs of restricting logical principles in a mathematically precise way: this analysis was based on the amount of uncontroversial non-modal principles that could be captured by different solutions to paradox. The analysis showed that non-classical logical choices lead to a severe loss of non-semantic, non-modal mathematical principles.

3) Finally, the project activities provided both negative and positive results concerning the traditional notion of implicit commitment, according to which when we accept a theory we also implicitly ought to accept principles that outstrip the resources of this theory (i.e. the claim that the theory in question is non-contradictory). The project findings showed that some strategies proposed in the literature to combine ""reflection principles"" (what is provable in the theory is true) and modal and semantic notions are bound to be incoherent. They also indicated and developed a new path that overcome this incoherence and that is based on a specific choice of the underlying logical principles."
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