Periodic Reporting for period 1 - Truth in PTS (Truth within Proof-Theoretic Semantics)
Berichtszeitraum: 2022-09-01 bis 2024-08-31
Arithmetic, syntax and truth all play an important role in our everyday discourse. However, the philosophical study of these notions has proven challenging due to their connection to paradox. Proof-theoretic semantics is a formal programme based on the fundamental assumption that the meaning of certain terms is entirely determined by the inference or proof rules that they obey. By taking a certain set of proof rules to constitute the definition of a logical term it is argued that the resulting logical laws are justified by the definition.
The study of logics in this setting has been advancing rapidly in recent years. But extensions to other formal systems such as arithmetic, syntax and truth, have been limited. This project produced foundational work necessary to move to formal systems and then investigated arithmetical, syntactical, and truth-based vocabulary in a proof-theoretic semantics.
The study of logics in this setting has been advancing rapidly in recent years. But extensions to other formal systems such as arithmetic, syntax and truth, have been limited. This project produced foundational work necessary to move to formal systems and then investigated arithmetical, syntactical, and truth-based vocabulary in a proof-theoretic semantics.
This project applied proof-theoretic semantics to formal theories of arithmetic, syntax and truth. It was separated into two phases the first of which looked at proof theoretic semantics on its own. And the second of which looked at the application.
In the first phase, the connection between proof theoretic valid systems and their logics was explored. This was in part motivated by recent discoveries of an earlier proof not having the implications it was previously believed to. The result of this work was twofold. First there were two formal papers. The first showing that we could modify proof theoretic validity to make it a semantics for intuitionistic logic. And the second showing which logic was valid for a traditional notion of proof-theoretic validity. Second there were two philosophical papers. The first of which looked at the consequences for traditional notions not being intuitionistic and defending the formalism provided in Stafford and Nascimento 2023. The second of which looked at whether or not proof theoretic validity justifies our belief in the correctness of the logics it is developed for.
The second phase started by looking at arithmetic. It attempted to justify arithmetical rules in two separate ways. First by considering the induction rule as a new rule for universal introduction special to the natural numbers. And second by taking an idea from type theory; treating ‘the natural numbers’ as a predicate to be defined by proof rules. It was shown that neither of these approaches would work and in discussing this it became clear that arithmetic will not be capable of being captured by proof rules that meet the conditions of proof-theoretic semantics. This then led to the question of whether or not this result also held for the theory of syntax. Quine famously said that the theories of arithmetic and syntax are the same. However approaches to sameness of theory are normally model theoretic and better suited for the work of set theorists and theoretical philosophers of physics and less so for proof theoretic semantics. This led to an investigation of the complexity of proofs. Developed off of this was a notion of theory equivalence which is sensitive to complexity. Thereby making it sufficiently proof theoretic. Finally there was work on the formal theories of truth and whether or not we can consistently add a truth predicate to proof theoretic validity. The work on this has so far been negative suggesting that you cannot but the project is ongoing.
In the first phase, the connection between proof theoretic valid systems and their logics was explored. This was in part motivated by recent discoveries of an earlier proof not having the implications it was previously believed to. The result of this work was twofold. First there were two formal papers. The first showing that we could modify proof theoretic validity to make it a semantics for intuitionistic logic. And the second showing which logic was valid for a traditional notion of proof-theoretic validity. Second there were two philosophical papers. The first of which looked at the consequences for traditional notions not being intuitionistic and defending the formalism provided in Stafford and Nascimento 2023. The second of which looked at whether or not proof theoretic validity justifies our belief in the correctness of the logics it is developed for.
The second phase started by looking at arithmetic. It attempted to justify arithmetical rules in two separate ways. First by considering the induction rule as a new rule for universal introduction special to the natural numbers. And second by taking an idea from type theory; treating ‘the natural numbers’ as a predicate to be defined by proof rules. It was shown that neither of these approaches would work and in discussing this it became clear that arithmetic will not be capable of being captured by proof rules that meet the conditions of proof-theoretic semantics. This then led to the question of whether or not this result also held for the theory of syntax. Quine famously said that the theories of arithmetic and syntax are the same. However approaches to sameness of theory are normally model theoretic and better suited for the work of set theorists and theoretical philosophers of physics and less so for proof theoretic semantics. This led to an investigation of the complexity of proofs. Developed off of this was a notion of theory equivalence which is sensitive to complexity. Thereby making it sufficiently proof theoretic. Finally there was work on the formal theories of truth and whether or not we can consistently add a truth predicate to proof theoretic validity. The work on this has so far been negative suggesting that you cannot but the project is ongoing.
The project lead to the identification of the logic of an old system of proof-theoretic validity (one of the two main approaches within proof-theoretic semantics). A question that has been open since the 70s has been answered. The project also lead to the development of a new system which has the desired property of being a proof-theoretic validity for intuitionistic logic. This paper published in 2023 has already been cited 16 times. The project also produced one of the first explorations of arithmetic and truth in proof-theoretic semantics and to the authors knowledge the first exploration of theories of syntax in this setting. The projects two workshops both helped bridge divides both within disciplines (and even subdisciplines) and interdisciplinary divides, with presenters from mathematics, philosophy and computer science.