In previous work the author has introduced a lambda calculus SLR with modal and linear types which serves as an extension of Bellantoni–Cook's function algebra BC to higher types. It is a step towards a functional programming language in which all programs run in polynomial time. In this paper we develop a semantics of SLR using BCK -algebras consisting of certain polynomial-time algorithms. It will follow from this semantics that safe recursion with arbitrary result type built up from N and (...) ⊸ as well as recursion over trees and other data structures remains within polynomial time. In its original formulation SLR supported only natural numbers and recursion on notation with first-order functional result type. (shrink)

A classical quantified modal logic is used to define a "feasible" arithmetic A 1 2 whose provably total functions are exactly the polynomial-time computable functions. Informally, one understands $\Box\alpha$ as "α is feasibly demonstrable". A 1 2 differs from a system A 2 that is as powerful as Peano Arithmetic only by the restriction of induction to ontic (i.e., $\Box$ -free) formulas. Thus, A 1 2 is defined without any reference to bounding terms, and admitting induction over formulas having arbitrarily (...) many alternations of unbounded quantifiers. The system also uses only a very small set of initial functions. To obtain the characterization, one extends the Curry-Howard isomorphism to include modal operations. This leads to a realizability translation based on recent results in higher-type ramified recursion. The fact that induction formulas are not restricted in their logical complexity, allows one to use the Friedman A translation directly. The development also leads us to propose a new Frege rule, the "Modal Extension" rule: if $\vdash \alpha$ then $\vdash A \leftrightarrow \alpha$ a for new symbol A. (shrink)

We use the category of presheaves over PTIME-functions in order to show that Cook and Urquhart's higher-order function algebra PV ω defines exactly the PTIME-functions. As a byproduct we obtain a syntax-free generalisation of PTIME-computability to higher types. By restricting to sheaves for a suitable topology we obtain a model for intuitionistic predicate logic with ∑ 1 b -induction over PV ω and use this to re-establish that the provably total functions in this system are polynomial time computable. Finally, we (...) apply the category-theoretic approach to a new higher-order extension of Bellantoni-Cook's system BC of safe recursion. (shrink)

Habermas’s Public Sphere: A Critique systematically analyzes the strengths and weaknesses of Habermas’s classic public sphere concept to reinvigorate it for evaluating the liberal promises and realities of modern societies.

The essays in Powerful Arguments reconstruct the standards of validity underlying argumentative practices in a wide array of late imperial Chinese discourses, from the Song through the Qing dynasties. The fourteen case studies analyze concrete arguments defended or contested in areas ranging from historiography, philosophy, law, and religion to natural studies, literature, and the civil examination system. By examining uses of evidence, habits of inference, and the criteria by which some arguments were judged to be more persuasive than others, the (...) contributions recreate distinct cultures of reasoning. Together, they lay the foundations for a history of argumentative practice in one of the richest scholarly traditions outside of Europe and add a chapter to the as yet elusive global history of rationality. (shrink)

Reading Habermas: Structural Transformation of the Public Sphere provides comprehensive guidance to understanding the complex methodologies of Habermas's global academic best seller. This timely guide parallels Habermas's publications from 2021 and 2022 about a new structural transformation of political will-formation caused primarily by the digital dominance of social media platforms.

Between 1819 and 1832 Friedrich Schleiermacher was giving lectures on the life of Jesus at the University of Berlin. The following article includes two partial editions, which document the introductory parts of the lectures from 1819/20 and 1829/30. Both are based on manuscripts written by Schleiermacher’s listeners. Especially to explore the development of Schleiermacher’s conceptual considerations this two partial editions should be a useful addition to the new critical edition of Schleiermacher’s Vorlesungen über das Leben Jesu published in 2018 by (...) Walter Jaeschke (KGA II/15). (shrink)

Between 1819 and 1832 Friedrich Schleiermacher was giving lectures on the life of Jesus at the University of Berlin. The following article includes two partial editions, which document the introductory parts of the lectures from 1819/20 and 1829/30. Both are based on manuscripts written by Schleiermacher’s listeners. Especially to explore the development of Schleiermacher’s conceptual considerations this two partial editions should be a useful addition to the new critical edition of Schleiermacher’s Vorlesungen über das Leben Jesu published in 2018 by (...) Walter Jaeschke (KGA II/15). (shrink)

Was Schleiermacher the first who has given lectures on the life of Jesus? – The article discusses this question and provides a list of this type of lectures between 1778 and 1834 at German universities.