Σ21\Sigma^1_2 and Π11\Pi^1_1 mad families

We answer in the affirmative the following question of J\"org Brendle: If there is a $\Sigma^1_2$ mad family, is there then a $\Pi^1_1$ mad family?Comment: This will appear in the Journal of Symbolic Logic (March 2014

Reduced Implication-bias Logic Loss for Neuro-Symbolic Learning

Integrating logical reasoning and machine learning by approximating logical inference with differentiable operators is a widely used technique in Neuro-Symbolic systems. However, some differentiable operators could bring a significant bias during backpropagation and degrade the performance of Neuro-Symbolic learning. In this paper, we reveal that this bias, named \textit{Implication Bias} is common in loss functions derived from fuzzy logic operators. Furthermore, we propose a simple yet effective method to transform the biased loss functions into \textit{Reduced Implication-bias Logic Loss (RILL)} to address the above problem. Empirical study shows that RILL can achieve significant improvements compared with the biased logic loss functions, especially when the knowledge base is incomplete, and keeps more robust than the compared methods when labelled data is insufficient.Comment: ACML'2023 Journal Track(Accepted by Machine Learning Journal

Remark on Relevant Arithmetic

This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984)

Remark on Relevant Arithmetic

This is a brief note about the history of the analysis of the collection of theories, RM3modn, in Meyer and Mortensen "Inconsistent Models for Relevant Arithmetics" Journal of Symbolic Logic 49 (1984)

The automorphism group of the Fra\"iss\'e limit of finite Heyting algebras

Roelcke non-precompactness, non-simplicity, and non-amenability of the automorphism group of the Fra\"iss\'e limit of finite Heyting algebras are examined among others.Comment: Submitted to Journal of Symbolic Logic; 15 page

On lengths of proofs in non-classical logics

AbstractWe give proofs of the effective monotone interpolation property for the system of modal logic K, and others, and the system IL of intuitionistic propositional logic. Hence we obtain exponential lower bounds on the number of proof-lines in those systems. The main results have been given in [P. Hrubeš, Lower bounds for modal logics, Journal of Symbolic Logic 72 (3) (2007) 941–958; P. Hrubeš, A lower bound for intuitionistic logic, Annals of Pure and Applied Logic 146 (2007) 72–90]; here, we give considerably simplified proofs, as well as some generalisations

Jerzy Łoś Positional Calculus and the Origin of Temporal Logic

Most accounts, including leading textbooks, credit Arthur Norman Prior with the invention of temporal (tense logic). However, (i) Jerzy Łoś delivered his version of temporal logic in 1947, several years before Prior; (ii) Henrk Hiż’s review of Łoś’s system in Journal of Symbolic Logic was published as early as 1951; (iii) there is evidence to the effect that, when constructing his tense calculi, Prior was aware of Łoś’s system. Therefore, although Prior is certainly a key figure in the history tense logic, as well as modal logic in general, it should be accepted both in the literature that temporal logic was invented by Jerzy Łoś