Abstract
R. Suszko’s Sentential Calculus with Identity \( SCI \) results from classical propositional calculus \( CPC \) by adding a new connective \(\equiv \) and axioms for identity \(\varphi \equiv \psi \) (which we interpret here as ‘propositional identity’). We reformulate the original semantics of \( SCI \) using Boolean prealgebras which, introduced in different ways, are known in the literature as structures for the modeling of (hyper-) intensional semantics. We regard intensionality here as a measure for the discernibility of propositions (and hyperintensionality as a high degree of intensionality). As concrete examples of \( SCI \)-based intensional modeling, we review and study algebraic semantics of some Lewis-style modal logics in the vicinity of \( S3 \) and present conditions under which those modal systems can be restored, in a precise sense, as certain axiomatic extensions of \( SCI \). This generalizes work of Suszko which is focused on the modal systems \( S4 \) and \( S5 \). Our approach is particularly intended as a proposal to consider and to further study \( SCI \) (and its extensions) as a general framework for the modeling of (hyper-) intensional semantics.
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Notes
\(\chi [x:=\varphi ]\) is the result of substituting \(\varphi \) for every occurrence of variable x in \(\chi \). The concept can be formally defined in the obvious way by induction on the construction of \(\chi \).
The intension (sense) of a formula is a mental content expressed by the formula’s syntax: e.g. \(x\wedge y\) and \(y\wedge x\) have different intensions. However, intension and syntax are not always in one-to-one correspondence. In a logic with propositional quantifiers, one may argue that the distinct formulas \(\forall x(x\rightarrow x)\) and \(\forall y(y\rightarrow y)\) have the same intension.
The epistemic notion of belief, represented by an operator B, is usually viewed as hyperintensional: even if \(\varphi \) and \(\psi \) are logically equivalent, \(B\varphi \) and \(B\psi \) may have different truth-values depending on the agents mental state. Such a context could be formalized adequately by a model where \(\varphi \) and \(\psi \) denote different propositions.
Actually, this notion is stronger than the notion of admissible defined in Bloom and Suszko (1972, Definition 1.6) where only a finite set of axiom schemes of \( CPC \) is considered (recall that we consider here the set of all theorems of \( CPC \) as axioms of \( SCI \)). However, if \( TRUE \) is also closed (see the next item), then both definitions coincide. In fact, if \( TRUE \) is closed and admissible in the sense of Bloom and Suszko (1972, ), and \(\varphi \) has the form of a theorem of \( CPC \), then by induction on the length of derivations it follows that \(\gamma (\varphi )\in TRUE \) (interpret the condition of being closed as modus ponens).
There is, in general, no unique preorder witnessing that the algebraic structure is a Boolean prealgebra. Actually, we could modify our definition of Boolean prealgebra by requiring the existence of an appropriate preorder instead of presenting a particular one. However, since such an alternative definition would involve further conceptual changes and technical difficulties, we decided to work with Definition 2.1 where a particular preorder is explicitly given.
\(\mathcal {M'}\) is not necessarily a Boolean algebra. For example, \([\varphi ]\vee [\psi ]=[\varphi \vee \psi ]\ne [\psi \vee \varphi ]=[\psi ]\vee [\varphi ]\) is possible. Even if \(\mathcal {M'}\) is a Boolean algebra, the preorder \(\preceq \) may be strictly coarser than the underlying lattice order (cf. Lemma 2.4). In fact, \(\preceq \) is the lattice order iff \(\varPsi \) contains all instances of the Fregean Axiom \((\varphi \equiv \psi )\leftrightarrow (\varphi \leftrightarrow \psi )\).
The model is called intensional because different intensions (different formulas) correspond to different denotations (propositions), so intensions and denotations are in one-to-one correspondence. Of course, such a model can be particularly viewed as hyperintensional since even logically equivalent formulas \(\varphi \ne \psi \) have different semantics.
The construction of an intensional model for such a first-order logic is not trivial because of the impredicativity of propositional quantifiers. Note that the bound variable x in the formula \(\forall x\varphi \) ranges over a propositional universe which contains the proposition denoted by \(\forall x\varphi \) itself.
Consider classical tautology \(\varphi \leftrightarrow (\varphi \leftrightarrow \top )\), rule AN, principle K and MP.
In the following, we will write such an expression also as \(\varphi \equiv _{\mathcal {L}} box ( id (\varphi ))\).
We will write such an expression also as \(\varphi \equiv _{{\mathcal {L}}_\equiv } id ( box (\varphi ))\).
Note that the same argument is applicable if we consider the axioms (3’), (4), (5). If \(\varphi \) is such an axiom, then \( box (\varphi )\) is the corresponding axiom of modal system \( S1SP \), \( S4 \), \( S5 \), respectively.
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Lewitzka, S. Some Remarks on Semantics and Expressiveness of the Sentential Calculus with Identity. J of Log Lang and Inf 32, 441–471 (2023). https://doi.org/10.1007/s10849-023-09396-z
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DOI: https://doi.org/10.1007/s10849-023-09396-z