Abstract
We outline the main stages of Maksimova’s investigation and present in details her results published between 1972 and 1979 and concerning the study of pretabularity and interpolation properties in superintuitionisitc logics and in normal extensions of the logic S4.
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Notes
- 1.
See Chap. 3.
- 2.
Recall that the trivial logic coinciding with the set of all formulas is the only superintuitionistic logic, which is not intermediate.
- 3.
Recall that \( \langle A, \& , \vee , \supset , \lnot , 1\rangle \) is a pseudo-Boolean algebra if \( \langle A, \& , \vee \rangle \) is a bounded lattice with the greatest element 1 with restpect to its lattice ordering \(\le \) and the operations \(\supset \) and \(\lnot \) are such that:
$$ \begin{aligned} c\le a\supset b\ \ \text{ iff }\ \ a \& c\le b; \ \ b\le \lnot a\ \ \text{ iff }\ \ a \& b=0, \end{aligned}$$where 0 is the least element with respect to \(\le \). Every pseudo-Boolean algebra is a distributive lattice.
- 4.
Rasiowa and Sikorski (1963) was translated into Russian by V.A. Yankov and published in 1972.
- 5.
The logic \(L\mathsf{A}\) of a p.-B. algebra is defined in the same way as a logic of the regular model \(\langle \mathsf{A},\{ 1_\mathsf{A}\}\rangle \).
- 6.
Recall that \(P\supseteq A\) is a prime filter on a p.-B. algebra A if it satifies the conditions: a) \(P\ne A\); b) \( (x \& y)\in P\) whenever \(x,y\in P\); c) \((x\vee y)\in P\) iff \(x\in P\) or \(y\in P\).
- 7.
\(A\equiv B\) is an abbreviation for \( (A\supset B) \& (B\supset A)\).
- 8.
\(\alpha \leftrightarrow \beta \) is an abbreviation for \( (\alpha \rightarrow \beta ) \& (\beta \rightarrow \alpha )\) and \(\Diamond \alpha \) for \(\sim \Box \sim \alpha \).
- 9.
Maksimova and Rybakov (1974) use the generally accepted version of Gödel translation such that \(T(p)=\Box p\) for a propositional variable p and
$$ \begin{aligned} \begin{array}{lll} T(A \& B) &{}=&{} T(A) \& T(B), \quad T(A\supset B) = \Box (T(A)\rightarrow T(B)), \\ T(A\vee B) &{}=&{} T(A)\vee T(B), \quad T(\lnot A) = \Box (\mathord {\sim }T(A)), \end{array} \end{aligned}$$where \(A,B\in For_\mathscr {I}\).
- 10.
By non-logical terms we mean elements of the signature in case of first order logic and propositional variables in case of propositional logics.
- 11.
Recall that a p.-B.algebra A is well-connected if \(a\vee b=1\) implies \(a=1\) or \(b=1\) for \(a,b\in \mathsf{A}\).
- 12.
See the previous section for the definition of t.-B. algebra \(\mathscr {S}(\mathsf{A})\) assigned to a p.-B. algebra A and for the definition of characteristics \(\mu _1(\mathsf{Q}_\mathsf{A})\) and \(\mu _2(\mathsf{Q}_\mathsf{A})\) of the representing quasi-ordering for \(\mathsf{A}\).
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Acknowledgements
As a customer of the seminar “Non-standard logics” in Novosibirsk State University I am gratefull to Prof. Maksimova for creating and supporting this form of intellectual communication, which caused, in particular, the change of my field of scientific interests from the classical computability theory to non-classical logics.
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Odintsov, S. (2018). Maksimova, Relevance and the Study of Lattices of Non-classical Logics. In: Odintsov, S. (eds) Larisa Maksimova on Implication, Interpolation, and Definability. Outstanding Contributions to Logic, vol 15. Springer, Cham. https://doi.org/10.1007/978-3-319-69917-2_1
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