Virtual modality. (English) Zbl 1049.03016
See also the author’s paper reviewed below [ibid. 136, No. 2, 281–304 (2003; Zbl 1049.03017)].
Summary: Model-theoretic 1-types over a given first-order theory \(T\) may be construed as natural metalogical miniatures of G. W. Leibniz’ “complete individual notions”, “substances” or “substantial forms”. This analogy prompts this essay’s modal semantics for an essentially undecidable first-order theory \(T\) in which one quantifies over such “substances” in a Boolean universe \(V(\mathbf C)\), where \(\mathbf C\) is the completion of the Lindenbaum algebra of \(T\).
More precisely, one can define recursively a set-theoretic translate of formulae \(\nu_{\mathbf N}(\varphi)\) of formulae \(\nu\) of a normal modal theory \(T_m\) based on \(T\) such that the counterpart ‘\(\xi_i\)’ of a the modal variable ‘\(x_i\)’ of \(L(T_m)\) in this translation-scheme ranges over elements of \(V(\mathbf C)\) that are 1-types of \(T\) with value 1 (sometimes called ‘definite’ \(\mathbf C\)-valued 1-types of \(T\)).
The article’s basic completeness result then establishes that \(\varphi\) is a theorem of \(T_m\) iff \([[\nu_{\mathbf N}(\varphi)\) is a consequence of \(\nu_{\mathbf N}(T_m)\) for each extension \(\mathbf N\) of \(T\) which is a subtheory of the canonical generic theory (ultrafilter) \(\mathbf u]] = 1 -\) or, equivalently, that \(T_m\) is consistent iff [[there is an extension \(\mathbf N\) of \(T\) such that \(\mathbf N\) is a subtheory of the canonical generic theory \(\mathbf u\), and \(\nu_N(\varphi)\) for all \(\varphi\) in \(T_m\)]] \(> 0\).
The proof of this completeness result also shows that an \(\mathbf N\) which provides a countermodel for a modally unprovable \(\varphi\) – or, equivalently, a closed set in the Stone space St(\(T\)) in the sense of \(\mathbf V(\mathbf C)\) – is intertranslatable with an ‘accessibility’ relation of a closely related Kripke semantics whose ‘worlds’ are generic extensions of an initial universe \(V\) via \(\mathbf C\).
This interrelation provides a fairly precise rationale for the semantics’ recourse to \(\mathbf C\)-valued structures, and exhibits a sense in which the Boolean-valued context is sharp.
Summary: Model-theoretic 1-types over a given first-order theory \(T\) may be construed as natural metalogical miniatures of G. W. Leibniz’ “complete individual notions”, “substances” or “substantial forms”. This analogy prompts this essay’s modal semantics for an essentially undecidable first-order theory \(T\) in which one quantifies over such “substances” in a Boolean universe \(V(\mathbf C)\), where \(\mathbf C\) is the completion of the Lindenbaum algebra of \(T\).
More precisely, one can define recursively a set-theoretic translate of formulae \(\nu_{\mathbf N}(\varphi)\) of formulae \(\nu\) of a normal modal theory \(T_m\) based on \(T\) such that the counterpart ‘\(\xi_i\)’ of a the modal variable ‘\(x_i\)’ of \(L(T_m)\) in this translation-scheme ranges over elements of \(V(\mathbf C)\) that are 1-types of \(T\) with value 1 (sometimes called ‘definite’ \(\mathbf C\)-valued 1-types of \(T\)).
The article’s basic completeness result then establishes that \(\varphi\) is a theorem of \(T_m\) iff \([[\nu_{\mathbf N}(\varphi)\) is a consequence of \(\nu_{\mathbf N}(T_m)\) for each extension \(\mathbf N\) of \(T\) which is a subtheory of the canonical generic theory (ultrafilter) \(\mathbf u]] = 1 -\) or, equivalently, that \(T_m\) is consistent iff [[there is an extension \(\mathbf N\) of \(T\) such that \(\mathbf N\) is a subtheory of the canonical generic theory \(\mathbf u\), and \(\nu_N(\varphi)\) for all \(\varphi\) in \(T_m\)]] \(> 0\).
The proof of this completeness result also shows that an \(\mathbf N\) which provides a countermodel for a modally unprovable \(\varphi\) – or, equivalently, a closed set in the Stone space St(\(T\)) in the sense of \(\mathbf V(\mathbf C)\) – is intertranslatable with an ‘accessibility’ relation of a closely related Kripke semantics whose ‘worlds’ are generic extensions of an initial universe \(V\) via \(\mathbf C\).
This interrelation provides a fairly precise rationale for the semantics’ recourse to \(\mathbf C\)-valued structures, and exhibits a sense in which the Boolean-valued context is sharp.
MSC:
| 03B45 | Modal logic (including the logic of norms) |
| 03A05 | Philosophical and critical aspects of logic and foundations |
| 03C90 | Nonclassical models (Boolean-valued, sheaf, etc.) |
