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Capturing consequence. (English) Zbl 07063892
Summary: First-order formalisations are often preferred to propositional ones because they are thought to underwrite the validity of more arguments. We compare and contrast the ability of some well-known logics – these two in particular – to formally capture valid and invalid arguments. We show that there is a precise and important sense in which first-order logic does not improve on propositional logic in this respect. We also prove some generalisations and related results of philosophical interest. The rest of the article investigates the results’ philosophical significance. A first moral is that the correct way to state the oft-cited superiority of first-order logic vis-à-vis propositional logic is more nuanced than often thought. The second moral concerns semantic theory; the third logic’s use as a tool for discovery. A fourth and final moral is that second-order logic’s transcendence of first-order logic is greater than first-order logic’s transcendence of propositional logic.
MSC:
03A05 Philosophical and critical aspects of logic and foundations
03B05 Classical propositional logic
03B10 Classical first-order logic
03B65 Logic of natural languages
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