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Reports on Mathematical Logic

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No. 37

Sergei V. BABYONYSHEV:

**Fully Fregean logics**

A b s t r a c t. Frege's Principle asserts that the denotation of a propositional
sentence coincides with its truth value. In the context of algebraizable logics the principle can be interpreted as the
compositionality of interderivability relation $\Fr{S}$, defined formally by $\Fr{S}T=\{\langle
\phi, \psi\rangle\in\Fml^2\mid T,\phi \dashv\vdash_{\mathcal S}T,\psi \}$, for given deductive system
$\mathcal S$ and any $\mathcal S$-theory $T$. Of special interest are the deductive systems for which the property of being
Fregean is inherited by all full 2nd-order models, so called, \it{fully Fregean} deductive
systems. The main result of this paper is a characterization of fully Fregean deductive systems over countable languages
using properties of the strong Frege operator on the formula algebra. The example of a Fregean, but not fully Fregean deductive system
$\mathcal S$ is provided. $\mathcal S$ also turns out to be selfextensional, but not fully selfextensional, and, in
addition, the three principal algebraic semantics for $\mathcal S$ are different,
i.e., $\Alg^\ast\mathcal S\subsetneq\Alg\mathcal S\subsetneq\Var(\Alg\mathcal S)$.

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