This paper studies proof systems for the logics of super-strict implication ST2–ST5, which correspond to C.I. Lewis’ systems S2–S5 freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating STn in Sn and backsimulating Sn in STn, respectively(for n=2,...,5). Next, G3-style labelled sequent calculi are investigated. It is shown that these calculi have the good structural properties that are distinctive of G3-style calculi, that they are (...) sound and complete, and it is shown that the proof search for G3. ST2 is terminating and therefore the logic is decidable. (shrink)

This paper introduces the logics of super-strict implications, where a super-strict implication is a strengthening of C.I. Lewis' strict implication that avoids not only the paradoxes of material implication but also those of strict implication. The semantics of super-strict implications is obtained by strengthening the (normal) relational semantics for strict implication. We consider all logics of super-strict implications that are based on relational frames for modal logics in the (...) modal cube. it is shown that all logics of super-strict implications are connexive logics in that they validate Aristotle's Theses and (weak) Boethius's Theses. A proof-theoretic characterisation of logics of super-strict implications is given by means of G3-style labelled calculi, and it is proved that the structural rules of inference are admissible in these calculi. It is also shown that validity in the S5-based logic of super-strict implications is equivalent to validity in G. Priest's negation-as-cancellation-based logic. Hence, we also give a cut-free calculus for Priest's logic. (shrink)

This paper studies proof systems for the logics of super-strict implication \(\textsf{ST2}\) – \(\textsf{ST5}\), which correspond to C.I. Lewis’ systems \(\textsf{S2}\) – \(\textsf{S5}\) freed of paradoxes of strict implication. First, Hilbert-style axiomatic systems are introduced and shown to be sound and complete by simulating \(\textsf{STn}\) in \(\textsf{Sn}\) and backsimulating \(\textsf{Sn}\) in \(\textsf{STn}\), respectively (for \({\textsf{n}} =2, \ldots, 5\) ). Next, \(\textsf{G3}\) -style labelled sequent calculi are investigated. It is shown that these calculi have the good structural (...) properties that are distinctive of \(\textsf{G3}\) -style calculi, that they are sound and complete, and it is shown that the proof search for \(\mathsf {G3.ST2}\) is terminating and therefore the logic is decidable. (shrink)

We introduce a system PSI for a strict implication operator called Priorean strict implication. The semantics for PSI is based on partial Kripke models without accessibility relations. PSI is proved sound and complete with respect to that semantics, and Prior's system Q and related systems are shown to be fragments of PSI or of a mild extension of it.

I have argued that a strong kind of physicalism based on the strict implication thesis can consistently reject both eliminativism and reductionism (in any nontrivial sense). This piece defends that position against objections from Andrew Melnyk, who claims that either my formulation doesn't entail physicalism, or it must be interpreted in such a way that the mental is after all reducible to the physical. His alternatives depend on two interesting assumptions. I argue that both are mistaken, thereby, making (...) this kind of nonreductive physicalism clearer and more clearly defensible. (shrink)

C. I. Lewis intended his systems S1–S5 as contributions to the study of “strict implication”, but in his formulation, strict implication is so thoroughly intertwined with other notions, such as possibility and negation, that it remains a problem, to separate out the properties of strict implication itself. I shall solve this problem for S2–5 and von Wright's M. The results for S3–5 are given below, while the implicative parts of S2 and M, which are (...) rather more complicated, are given in §5.In this presentation, ‘⊃’ stands for strict implication, and missing brackets for ‘⊃’ are restored by associationtothe right. Astrictformula is one of the form A ⊃ B. Axiom schemes are used throughout. (shrink)

The present paper introduces and studies the variety WH of weakly Heyting algebras. It corresponds to the strict implication fragment of the normal modal logic K which is also known as the subintuitionistic local consequence of the class of all Kripke models. The tools developed in the paper can be applied to the study of the subvarieties of WH; among them are the varieties determined by the strict implication fragments of normal modal logics as well as (...) varieties that do not arise in this way as the variety of Basic algebras or the variety of Heyting algebras. Apart from WH itself the paper studies the subvarieties of WH that naturally correspond to subintuitionistic logics, namely the variety of R-weakly Heyting algebras, the variety of T-weakly Heyting algebras and the varieties of Basic algebras and subresiduated lattices. (shrink)

Suppose P is the conjunction of all truths statable in the austere vocabulary of an ideal physics. Then phsicalists are likely to accept that any truths not included in P are different ways of talking about the reality specified by P. This ‘redescription thesis’ can be made clearer by means of the ‘strict implication thesis’, according to which inconsistency or incoherence are involved in denying the implication from P to interesting truths not included in it, such as (...) truths about phenomenal consciousness. Commitment to the strict implication thesis cannot be escaped by appeal to a posteriori necessary identities or entailments. A minimal physicalism formulated in terms of strict implication is preferable to one based on a priori entailment. (shrink)

The aim of this paper is to determine the plausibility of Robert Kirk’s strict implication thesis as an explication of physicalism and its relation to Jackson and Chalmer’s notion of application conditionals, to the notion of global supervenience and to a posteriori identities. It is argued that the strict implication thesis is subject to the same objection that affects the notion of global supervenience. Furthermore, reference to an idealised physics in the formulation of strict (...) class='Hi'>implication threatens to make the thesis vacuous. Third, Kirk’s claim that the strict implication thesis does not entail reduction of the mental to the physical (excluding phenomenal properties) is untenable if a functional model of reduction is preferred over Nagel’s classical model. Finally, Kirk’s claim that the physical facts entail in an a priori way the fact that certain brain states feel somehow seems to be unfounded. (shrink)

The aim of this paper is to determine the plausibility of Robert Kirk's strict implication thesis as an explication of physicalism and its relation to Jackson and Chalmer's notion of application conditionals, to the notion of global supervenience and to a posteriori identities. It is argued that the strict implication thesis is subject to the same objection that affects the notion of global supervenience. Furthermore, reference to an idealised physics in the formulation of strict (...) class='Hi'>implication threatens to make the thesis vacuous. Third, Kirk's claim that the strict implication thesis does not entail reduction of the mental to the physical (excluding phenomenal properties) is untenable if a functional model of reduction is preferred over Nagel's classical model. Finally, Kirk's claim that the physical facts entail in an a priori way the fact that certain brain states feel somehow seems to be unfounded. (shrink)

Review of the paper mentioned in the title.

It can be said that Formal Logic begun by studying an idealization of the statements ’if p, then q’, something coming from long ago in both Greek and Scholastic Philosophy. Nevertheless, only in the XX Century it arrived at a stage of formalization once in 1910 Russell introduced and identified the ’material conditional’ with the expresion ”not p or q”. In 1918, and from paradoxical conditionals like ”If the Moon is a cheese, it is a Lyon’s face”, Lewis critiziced the (...) material conditional and introduced the so-called ’strict conditional’ as the modal necessity of the material one. In 1934, Huntington proved that both material and strict implications are coincidental in the setting of Boolean algebras and, hence, that they only can be actually different in algebraic structures weaker than Boolean algebras. Boolean algebras have too much laws for supporting the difference of both conditionals. This paper is nothing else than an algebraic trial to find simple binary Boolean operations able to express Lewis’ strict implication, and contains a new and simpler proof than that of Huntington, made by identifying ’possible’ with ’non self-contradictory’. By the way, it is shown that this proof is only valid in Boolean algebras, but neither in proper De Morgan algebras, nor in proper Ortholattices, with which it still remains open where the two conditional relations are actually different. (shrink)

In his main work Summa Logicae written around 1323, William of Ockham developed a system of propositional modal logic which contains almost all theorems of a modern calculus of strict implication. This calculus is formally reconstructed here with the help of modern symbols for the operators of conjunction, disjunction, implication, negation, possibility, and necessity.

We present systems of Natural Deduction based on Strict Implication for the main normal modal logics between K and S5. In this work we consider Strict Implication as the main modal operator, and establish a natural correspondence between Strict Implication and strict subproofs.

Editor's Note: This paper was found in galley proof form from the journal Mind in the C.I. Lewis Archives in the Special Collections Department of the Stanford University Libraries, call number M174, Box 18, Folder 1. There are two copies of the proofs in this folder, one includes Lewis's corrections. The version that appears here incorporates all of Lewis's corrections. Where these corrections are substantive, the original wording is give in a footnote. The paperwas withdrawn from publication by Lewis early (...) in 1935. The proofs were found with Lewis's other papers in his house in Menlo Park after his death in 1964. (shrink)