Intuitionistic G\"odel-L\"ob logic, \`a la Simpson: labelled systems and birelational semantics

We derive an intuitionistic version of G\"odel-L\"ob modal logic ($\sf{GL}$) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, $\sf{\ell IGL}$, by restricting a non-wellfounded labelled system for $\sf{GL}$ to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that $\sf{GL}$'s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of $\sf{GL}$ are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that $\sf{\ell IGL}$ coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic $\sf{IGL}$. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of $\sf{IGL}$.Comment: 25 pages including 8 pages appendix, 4 figure

Strong Types for Direct Logic

This article follows on the introductory article “Direct Logic for Intelligent Applications” [Hewitt 2017a]. Strong Types enable new mathematical theorems to be proved including the Formal Consistency of Mathematics. Also, Strong Types are extremely important in Direct Logic because they block all known paradoxes[Cantini and Bruni 2017]. Blocking known paradoxes makes Direct Logic safer for use in Intelligent Applications by preventing security holes. Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions has been a progressive development and not “game stoppers.” Contradictions can be helpful instead of being something to be “swept under the rug” by denying their existence, which has been repeatedly attempted by authoritarian theoreticians (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations. Mathematics here means the common foundation of all classical mathematical theories from Euclid to the mathematics used to prove Fermat's Last [McLarty 2010]. Direct Logic provides categorical axiomatizations of the Natural Numbers, Real Numbers, Ordinal Numbers, Set Theory, and the Lambda Calculus meaning that up a unique isomorphism there is only one model that satisfies the respective axioms. Good evidence for the consistency Classical Direct Logic derives from how it blocks the known paradoxes of classical mathematics. Humans have spent millennia devising paradoxes for classical mathematics. Having a powerful system like Direct Logic is important in computer science because computers must be able to formalize all logical inferences (including inferences about their own inference processes) without requiring recourse to human intervention. Any inconsistency in Classical Direct Logic would be a potential security hole because it could be used to cause computer systems to adopt invalid conclusions. After [Church 1934], logicians faced the following dilemma: • 1st order theories cannot be powerful lest they fall into inconsistency because of Church’s Paradox. • 2nd order theories contravene the philosophical doctrine that theorems must be computationally enumerable. The above issues can be addressed by requiring Mathematics to be strongly typed using so that: • Mathematics self proves that it is “open” in the sense that theorems are not computationally enumerable. • Mathematics self proves that it is formally consistent. • Strong mathematical theories for Natural Numbers, Ordinals, Set Theory, the Lambda Calculus, Actors, etc. are inferentially decidable, meaning that every true proposition is provable and every proposition is either provable or disprovable. Furthermore, theorems of these theories are not enumerable by a provably total procedure

Strong Types for Direct Logic

This article follows on the introductory article “Direct Logic for Intelligent Applications” [Hewitt 2017a]. Strong Types enable new mathematical theorems to be proved including the Formal Consistency of Mathematics. Also, Strong Types are extremely important in Direct Logic because they block all known paradoxes[Cantini and Bruni 2017]. Blocking known paradoxes makes Direct Logic safer for use in Intelligent Applications by preventing security holes. Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions has been a progressive development and not “game stoppers.” Contradictions can be helpful instead of being something to be “swept under the rug” by denying their existence, which has been repeatedly attempted by authoritarian theoreticians (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations. Mathematics here means the common foundation of all classical mathematical theories from Euclid to the mathematics used to prove Fermat's Last [McLarty 2010]. Direct Logic provides categorical axiomatizations of the Natural Numbers, Real Numbers, Ordinal Numbers, Set Theory, and the Lambda Calculus meaning that up a unique isomorphism there is only one model that satisfies the respective axioms. Good evidence for the consistency Classical Direct Logic derives from how it blocks the known paradoxes of classical mathematics. Humans have spent millennia devising paradoxes for classical mathematics. Having a powerful system like Direct Logic is important in computer science because computers must be able to formalize all logical inferences (including inferences about their own inference processes) without requiring recourse to human intervention. Any inconsistency in Classical Direct Logic would be a potential security hole because it could be used to cause computer systems to adopt invalid conclusions. After [Church 1934], logicians faced the following dilemma: • 1st order theories cannot be powerful lest they fall into inconsistency because of Church’s Paradox. • 2nd order theories contravene the philosophical doctrine that theorems must be computationally enumerable. The above issues can be addressed by requiring Mathematics to be strongly typed using so that: • Mathematics self proves that it is “open” in the sense that theorems are not computationally enumerable. • Mathematics self proves that it is formally consistent. • Strong mathematical theories for Natural Numbers, Ordinals, Set Theory, the Lambda Calculus, Actors, etc. are inferentially decidable, meaning that every true proposition is provable and every proposition is either provable or disprovable. Furthermore, theorems of these theories are not enumerable by a provably total procedure

Independent Combinatoric Worm Principles for First Order Arithmetic and Beyond

Treballs Finals del Màster de Lògica Pura i Aplicada, Facultat de Filosofia, Universitat de Barcelona, Curs: 2019-2020, Tutor: Joost J. JoostenIn this thesis we study Beklemishev’s combinatorial principle Every Worm Dies, EWD which although true, it is unprovable in Peano Arithmetic (PA). The principle talks about sequences of modal formulas, the finiteness of all of them being equivalent to the one-consistency of PA. We present the elements of proof theory at play here and perform two attempts at generalizing this theorem. One is directed towards its relationship with some known fragments of PA while the other aims to see its connection with fragments of second order arithmetic

Reflection ranks and ordinal analysis

It is well-known that natural axiomatic theories are well-ordered by consistency strength. However, it is possible to construct descending chains of artificial theories with respect to consistency strength. We provide an explanation of this well-orderness phenomenon by studying a coarsening of the consistency strength order, namely, the $\Pi^1_1$ reflection strength order. We prove that there are no descending sequences of $\Pi^1_1$ sound extensions of $\mathsf{ACA}_0$ in this order. Accordingly, we can attach a rank in this order, which we call reflection rank, to any $\Pi^1_1$ sound extension of $\mathsf{ACA}_0$. We prove that for any $\Pi^1_1$ sound theory $T$ extending $\mathsf{ACA}_0^+$, the reflection rank of $T$ equals the proof-theoretic ordinal of $T$. We also prove that the proof-theoretic ordinal of $\alpha$ iterated $\Pi^1_1$ reflection is $\varepsilon_\alpha$. Finally, we use our results to provide straightforward well-foundedness proofs of ordinal notation systems based on reflection principles

On the Semantics of Intensionality and Intensional Recursion

Intensionality is a phenomenon that occurs in logic and computation. In the most general sense, a function is intensional if it operates at a level finer than (extensional) equality. This is a familiar setting for computer scientists, who often study different programs or processes that are interchangeable, i.e. extensionally equal, even though they are not implemented in the same way, so intensionally distinct. Concomitant with intensionality is the phenomenon of intensional recursion, which refers to the ability of a program to have access to its own code. In computability theory, intensional recursion is enabled by Kleene's Second Recursion Theorem. This thesis is concerned with the crafting of a logical toolkit through which these phenomena can be studied. Our main contribution is a framework in which mathematical and computational constructions can be considered either extensionally, i.e. as abstract values, or intensionally, i.e. as fine-grained descriptions of their construction. Once this is achieved, it may be used to analyse intensional recursion.Comment: DPhil thesis, Department of Computer Science & St John's College, University of Oxfor

Worms and Spiders: Reflection calculi and ordinal notation systems

We give a general overview of ordinal notation systems arising from reflection calculi, and extend the to represent impredicative ordinals up to those representable using Buchholz-style collapsing functions

Epistemic Modality, Mind, and Mathematics

This book concerns the foundations of epistemic modality. I examine the nature of epistemic modality, when the modal operator is interpreted as concerning both apriority and conceivability, as well as states of knowledge and belief. The book demonstrates how epistemic modality relates to the computational theory of mind; metaphysical modality; the types of mathematical modality; to the epistemic status of large cardinal axioms, undecidable propositions, and abstraction principles in the philosophy of mathematics; to the modal profile of rational intuition; and to the types of intention, when the latter is interpreted as a modal mental state. Chapter \textbf{2} argues for a novel type of expressivism based on the duality between the categories of coalgebras and algebras, and argues that the duality permits of the reconciliation between modal cognitivism and modal expressivism. Chapter \textbf{3} provides an abstraction principle for epistemic intensions. Chapter \textbf{4} advances a topic-sensitive two-dimensional truthmaker semantics, and provides three novel interpretations of the framework along with the epistemic and metasemantic. Chapter \textbf{5} applies the fixed points of the modal $\mu$-calculus in order to account for the iteration of epistemic states, by contrast to availing of modal axiom 4 (i.e. the KK principle). Chapter \textbf{6} advances a solution to the Julius Caesar problem based on Fine's "criterial" identity conditions which incorporate conditions on essentiality and grounding. Chapter \textbf{7} provides a ground-theoretic regimentation of the proposals in the metaphysics of consciousness and examines its bearing on the two-dimensional conceivability argument against physicalism. The topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{4} is availed of in order for epistemic states to be a guide to metaphysical states in the hyperintensional setting. Chapter \textbf{8} examines the modal commitments of abstractionism, in particular necessitism, and epistemic modality and the epistemology of abstraction. Chapter \textbf{9} examines the modal profile of $\Omega$-logic in set theory. Chapter \textbf{10} examines the interaction between epistemic two-dimensional truthmaker semantics, epistemic set theory, and absolute decidability. Chapter \textbf{11} avails of modal coalgebraic automata to interpret the defining properties of indefinite extensibility, and avails of epistemic two-dimensional semantics in order to account for the interaction of the interpretational and objective modalities thereof. The hyperintensional, topic-sensitive epistemic two-dimensional truthmaker semantics developed in chapter \textbf{2} is applied in chapters \textbf{7}, \textbf{8}, \textbf{10}, and \textbf{11}. Chapter \textbf{12} provides a modal logic for rational intuition and provides four models of hyperintensional semantics. Chapter \textbf{13} examines modal responses to the alethic paradoxes. Chapter \textbf{14} examines, finally, the modal semantics for the different types of intention and the relation of the latter to evidential decision theory