# Implicit Definitions, Second-Order Quantifiers, and the Robustness of the Logical Operators

Chapter
Part of the Synthese Library book series (SYLI, volume 373)

## Abstract

We use a modified version of E.Beth’s concept of implicit definitions to show that all the usual logical operators as well as the first and second order quantifiers are implicitly defined—and for essentially the same reason that involves an account of the logical operators using a concept of filter conditions. An “inferential” proposal is then suggested for a Gentzen-like account as a necessary condition for the familiar logical operators. We then explore the question of whether our proposal can also be taken as a sufficient condition. To this end, we discuss whether other operators, like a truth operator, the counterfactual conditional, the identity, and the modal operators are also logical operators. The paper closes with a brief discussion of what is called the robustness of the logical operators: What happens to the logical operators when there is a shift from one logical structure to another which extends it, and what happens when there is a shift from one structure to one in which it is homomorphically embedded.

## Keywords

Logical Operator Universal Quantification Filter Condition Negation Operator Introduction Condition
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