Completeness Before Post: Bernays, Hilbert, and the Development of Propositional Logic

1999 ◽  
Vol 5 (3) ◽  
pp. 331-366 ◽  
Author(s):  
Richard Zach
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Historical Context ◽  
Subsequent Development ◽  
Symbolic Logic ◽  
David Hilbert ◽  
Semantic Completeness ◽  
Lecture Notes ◽  
Truth Value

AbstractSome of the most important developments of symbolic logic took place in the 1920s. Foremost among them are the distinction between syntax and semantics and the formulation of questions of completeness and decidability of logical systems. David Hilbert and his students played a very important part in these developments. Their contributions can be traced to unpublished lecture notes and other manuscripts by Hilbert and Bernays dating to the period 1917–1923. The aim of this paper is to describe these results, focussing primarily on propositional logic, and to put them in their historical context. It is argued that truth-value semantics, syntactic (“Post-”) and semantic completeness, decidability, and other results were first obtained by Hilbert and Bernays in 1918, and that Bernays's role in their discovery and the subsequent development of mathematical logic is much greater than has so far been acknowledged.

scholarly journals Relative Necessity and Propositional Quantification

2019 ◽  
Vol 49 (4) ◽  
pp. 703-726
Author(s):  
Alexander Roberts
Keyword(s):  
Mathematical Logic ◽  
Propositional Logic ◽  
Recent Article ◽  
Symbolic Logic ◽  
Philosophical Logic ◽  
Alternative Account ◽  
Standard Account ◽  
Current Article ◽  
Modal Propositional Logic ◽  
Propositional Quantification

AbstractFollowing Smiley’s (The Journal of Symbolic Logic, 28, 113–134 1963) influential proposal, it has become standard practice to characterise notions of relative necessity in terms of simple strict conditionals. However, Humberstone (Reports on Mathematical Logic, 13, 33–42 1981) and others have highlighted various flaws with Smiley’s now standard account of relative necessity. In their recent article, Hale and Leech (Journal of Philosophical Logic, 46, 1–26 2017) propose a novel account of relative necessity designed to overcome the problems facing the standard account. Nevertheless, the current article argues that Hale & Leech’s account suffers from its own defects, some of which Hale & Leech are aware of but underplay. To supplement this criticism, the article offers an alternative account of relative necessity which overcomes these defects. This alternative account is developed in a quantified modal propositional logic and is shown model-theoretically to meet several desiderata of an account of relative necessity.

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