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.

Carnap’s early metatheory: scope and limits

“…Thus, while a clear-cut object/metalanguage is still missing in Untersuchungen, Carnap's approach can be characterized as metatheoretic. Compare Schiemer and Reck (2013) for a further discussion of this point. In addition, it should be stressed that Carnap's convention to formulate both axiom systems and their metatheoretic properties within a single and interpreted Grunddisziplin was not uncommon at the time.…”

“…4-8. 22 See Schiemer and Reck (2013) for further details, as well as Andrews (2002) for a general discussion of the semantics of type theoretic languages. 23 It is important to emphasize that this reconstruction is most likely not how Carnap understood the phrase "statement ϕ holds in the basic system" in 1928.…”

“…This view has recently been corrected in work by Schiemer, Reck, and Loeb. In that work, it is shown that Carnap was well aware of the importance of capturing the notion of domain variability for his metatheoretic work and was not without resources to do so. See Loeb (2014a,b), Schiemer (2012bSchiemer ( , 2013, and Schiemer and Reck (2013).…”

“…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).…”