INTERLEAVING LOGIC AND COUNTING, A NEW VIEW OF COMBINED LOGICAL-ARITHMETICAL SYSTEMS

Johan van Benthem

University of Amsterdam and Stanford University

Abstract

Reasoning patterns in natural language combine logical and arithmetical features, transcending divides between qualitative and quantitative. This natural practice blends into ‘grassroots mathematics’ such as using pigeon-hole principles. And from these basic abilities everyone has, the trail leads to the pinnacles of abstract mathematics.

In the foundations of science and philosophy, one often finds a hierarchy: ‘logic first, arithmetic later”, or the other way around. However, our topic is the cooperation of logic and counting on a par, studied with small formal systems and gradually extending these. We show what can be defined, and where complexity barriers arise, through connections with classical results in mathematical logic.

After this, we return to natural language, confronting our formal systems with linguistic quantifier vocabulary, monotonicity reasoning, and procedural semantics. We conclude with some challenges coming from the cognitive psychology of reasoning and the possible need to remodel what we mean by ‘counting’.

Supported by:

• Department of Mathematics of the Universitat Politècnica de Catalunya
• Department of Mathematics and Computer Science of the University of Barcelona
• Department of Philosophy of the University of Barcelona
• Institut de Matemàtica de la Universitat de Barcelona (IMUB)