7,362 research outputs found
On the Syntax of Logic and Set Theory
We introduce an extension of the propositional calculus to include abstracts
of predicates and quantifiers, employing a single rule along with a novel
comprehension schema and a principle of extensionality, which are substituted
for the Bernays postulates for quantifiers and the comprehension schemata of ZF
and other set theories. We prove that it is consistent in any finite Boolean
subset lattice. We investigate the antinomies of Russell, Cantor, Burali-Forti,
and others, and discuss the relationship of the system to other set theoretic
systems ZF, NBG, and NF. We discuss two methods of axiomatizing higher order
quantification and abstraction, and then very briefly discuss the application
of one of these methods to areas of mathematics outside of logic.Comment: 34 pages, accepted, to appear in the Review of Symbolic Logi
A Revenge Problem Without the Concept of Truth
The vast majority of putative solutions to the liar paradox face the infamous revenge problem. In recent work, however, Kevin Scharp has extensively developed an exciting and highly novel âinconsistency approachâ to the paradox that, he claims, does not face revenge. If Scharp is right, then this represents a significant step forward in our attempts to solve the liar paradox. However, in this paper, I raise a revenge problem that faces Scharp's inconsistency approach
Operator arguments revisited
Certain passages in Kaplanâs âDemonstrativesâ are often taken to show that non-vacuous sentential operators associated with a certain parameter of sentential truth require a corresponding relativism concerning assertoric contents: namely, their truth values also must vary with that parameter. Thus, for example, the non-vacuity of a temporal sentential operator âalwaysâ would require some of its operands to have contents that have different truth values at different times. While making no claims about Kaplanâs intentions, we provide several reconstructions of how such an argument might go, focusing on the case of time and temporal operators as an illustration. What we regard as the most plausible reconstruction of the argument establishes a conclusion similar enough to that attributed to Kaplan. However, the argument overgenerates, leading to absurd consequences. We conclude that we must distinguish assertoric contents from compositional semantic values, and argue that once they are distinguished, the argument fails to establish any substantial conclusions. We also briefly discuss a related argument commonly attributed to Lewis, and a recent variant due to Weber
- âŠ