Logic in the 1930s: type theory and model theory

Abstract

In historical discussions of twentieth-century logic, it is typically assumed that model theory emerged within the tradition that adopted first-order logic as the standard framework. Work within the type-theoretic tradition, in the style of Principia Mathematica, tends to be downplayed or ignored in this connection. Indeed, the shift from type theory to first-order logic is sometimes seen as involving a radical break that first made possible the rise of modern model theory. While comparing several early attempts to develop the semantics of axiomatic theories in the 1930s, by two proponents of the type-theoretic tradition (Carnap and Tarski) and two proponents of the first-order tradition (Gödel and Hilbert), we argue that, instead, the move from type theory to first-order logic is better understood as a gradual transformation, and further, that the contributions to semantics made in the type-theoretic tradition should be seen as central to the evolution of model theory.

Projective Duality and the Rise of Modern Logic

“…44 But while Chasles clearly saw that Poncelet was wrong in believing that the justi ication of duality requires polar theory, Chasles still agrees 42 See e.g. (Badesa et al 2009), (Awodey & Reck 2002), (Schiemer et al 2017), and (Schiemer & Reck 2013) for more on the development of various metatheoretical concepts during that period. 43 The signi icance of transfer principles for the formation of metatheoretical conceptions has been emphasized in the context of a discussion of Frege's views on metatheory in (Tappenden 2005), an article that was a major inspiration for the current paper.…”

“… 42 See, e.g., [5, 6, 66, 67] for more on the development of various metatheoretical concepts during that period. …”

Hilbert, Duality, and the Geometrical Roots of Model Theory

“…He starts by first considering the consistency ('compatibility') of the axioms (without the axiom of completeness) and provides the first 'model' for the axioms. He says: 31 Hilbert's decision to create a separate 'axiom group' for the axiom of parallels is due to its special status and is a concession to his time. Throughout the history of mathematics, dozens of mathematicians have tried to prove or refute the axiom.…”

“…The next group contains only one axiom, the axiom of parallels, which says that for each straight line a lying in a certain plane α and each point A of this plane which does not lie on a, there is exactly one straight line b which is parallel to a, i.e., a line which has no point in common with a. 31 The axioms of the fourth group, the axioms of congruence, determine the notion of congruence of line segments and angles and it is here that the idea of measurement first enters the stage. Using the axioms stated thus far already allows one to develop a considerable portion of elementary geometry.…”

Carnap’s early metatheory: scope and limits

“…In them, not only the limitations of his approach were acknowledged, but also its innovative aspects and its significant influence on later developments in metalogic (Awodey and Reck 2002a;Reck 2004;Goldfarb 2005;Reck 2007). The third phase consists in fairly recent scholarship in which attention is drawn to previously neglected details of Carnap's early model theory and in which its role in the development of metalogic is spelled out more (Reck 2011;Schiemer 2012bSchiemer ,a, 2013Schiemer and Reck 2013;Loeb 2014a,b).…”