The partitions of a given set stand in a well known one-to-onecorrespondence with the equivalence relations on that set. We askwhether anything analogous to partitions can be found which correspondin a like manner to the similarity relations (reflexive, symmetricrelations) on a set, and show that (what we call) decompositions – of acertain kind – play this role. A key ingredient in the discussion is akind of closure relation (analogous to the consequence relationsconsidered in formal logic) having nothing especially to do (...) with thesimilarity issue, and we devote a final section to highlighting some ofits properties. (shrink)

After a summary of earlier work it is shown that elementary or Kalmar arithmetic can be interpreted within the system of Russell's Principia Mathematica with the axiom of infinity but without the axiom of reducibility.

The representation of quantification over relations in monadic third-order logic is discussed; it is shown to be possible in numerous special cases of foundational interest, but not in general unless something akin to the Axiom of Choice is assumed.

Gentzen’s and Jaśkowski’s formulations of natural deduction are logically equivalent in the normal sense of those words. However, Gentzen’s formulation more straightforwardly lends itself both to a normalization theorem and to a theory of “meaning” for connectives . The present paper investigates cases where Jaskowski’s formulation seems better suited. These cases range from the phenomenology and epistemology of proof construction to the ways to incorporate novel logical connectives into the language. We close with a demonstration of this latter aspect by (...) considering a Sheffer function for intuitionistic logic. (shrink)

The logic of paradox, LP, is a first-order, three-valued logic that has been advocated by Graham Priest as an appropriate way to represent the possibility of acceptable contradictory statements. Second-order LP is that logic augmented with quantification over predicates. As with classical second-order logic, there are different ways to give the semantic interpretation of sentences of the logic. The different ways give rise to different logical advantages and disadvantages, and we canvass several of these, concluding that it will be extremely (...) difficult to appeal to second-order LP for the purposes that its proponents advocate, until some deep, intricate, and hitherto unarticulated metaphysical advances are made. (shrink)

A 1971 paper by Roman Suszko, ‘Identity Connective and Modality’, claimed that a certain identity-free schema expressed the condition that there are at most two objects in the domain. Section 1 here gives that schema and enough of the background to this claim to explain Suszko’s own interest in it and related conditions—via non-Fregean logic, in which the objects in question are situations and the aim is to refrain from imposing this condition. Section 3 shows that the claim is false, (...) and suggests a diagnosis as to why it might have been made, in terms of quantifier scope distinctions as they bear on schematic formulations. In between, Section 2 discusses an issue brought up by this discussion but not involving the mistaken claim itself, and shows that Suszko was familiar with two different ways to set an upper or lower finite bound to the domain—a contrast which some contemporary readers may associate with David Lewis. (shrink)

The present study examined the role of father sensitivity and couple coparenting quality in the first 2 years of life in relation to the development of externalizing behavior problems in middle childhood, focusing on the unique role of fathers. In this study, 125 mothers, fathers, and their first-born children were followed from 8 months to age 7 years. Paternal sensitivity was rated when infants were 8 and 24 months old. Fathers were videotaped at home playing, feeding, and changing their 8-month-old (...) infants’ clothes. They also were videotaped in a lab playing with their 24-month-olds and solving a variety of challenging tasks. At 24 months, competitive coparenting was assessed via videotaped triadic family interactions at home in which families participated in a variety of tasks. Teachers rated externalizing behavior problems when the children were age 7. Continuity in paternal sensitivity was documented from 8 to 24 months, and paternal sensitivity at 8 months predicted externalizing behavior in middle childhood through father sensitivity at 24 months. Moreover, paternal sensitivity at 8 months predicted competitive coparenting which, in turn, forecast externalizing behavior problems in middle childhood, even after controlling for maternal sensitivity at 8 and 24 months. These findings highlight the unique role of paternal caregiving quality during the first year of life on couple coparenting and children’s subsequent development of externalizing problems and have implications for creating effective interventions to prevent children from developing externalizing disorders. (shrink)

One of the most dominant approaches to semantics for relevant (and many paraconsistent) logics is the Routley-Meyer semantics involving a ternary relation on points. To some (many?), this ternary relation has seemed like a technical trick devoid of an intuitively appealing philosophical story that connects it up with conditionality in general. In this paper, we respond to this worry by providing three different philosophical accounts of the ternary relation that correspond to three conceptions of conditionality. We close by briefly discussing (...) a general conception of conditionality that may unify the three given conceptions. (shrink)

The paper reviews a number of approaches for handling restricted quantification in relevant logic, and proposes a novel one. This proceeds by introducing a novel kind of enthymematic conditional.

Logics that have many truth values—more than just True and False—have been argued to be useful in the analysis of very many philosophical and linguistic puzzles. In this paper, which is a followup to, we will start with a particularly well-motivated four-valued logic that has been studied mainly in its propositional and first-order versions. And we will then investigate its second-order version. This four-valued logic has two natural three-valued extensions: what is called a “gap logic”, and what is called a (...) “glut logic”. We mention various results about the second-order version of these logics as well. And we then follow our earlier papers, where we had added a specific conditional connective to the three valued logics, and now add that connective to the four-valued logic under consideration. We then show that, although this addition is “conservative” in the sense that no new theorems are generated in the four-valued logic unless they employ this new conditional in their statement, nevertheless the resulting second-order versions of these logics with and without the conditional are quite different in important ways. We close with a moral for logical investigations in this realm. (shrink)

Nelson has presented a constructive arithmetic with a negation opera- tion () dierent from the ordinary intuitionistic one . In [5] he presents a variant of Kleene's realization semantics for intuitionistic arithmetic, and proves that relative to this interpretation the arithmetic language with { has the same expressive power as the usual intuitionistic one, and fact certain theories of arithmetic incorporating his negation are equivalent to corresponding systems of intuitionistic arithmetic. A Fitch style natural deduction formulation ) of the pure (...) rst order logic that may be extracted, from Nelson's work may be obtained from that for intuitionistic rst order logic by rst dropping the intuitionistic rule of negation intro- duction and then adding rules guaranteeing the De Morgan equivalences, their quantier analogues, and the equivalence of anegated implication with the conjunction of its antecedent and the negation of its consequent. Given the De Morgan equivalences, conjunction and the existential quantier may he treated as dened operators. (shrink)

The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as ‘disguised’ quantifiers, and I think (...) that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worldsConsidered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages.A final note. Quine has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.29 He has also dropped hints about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by the concept of identity. He has dubbed the resulting concept a ‘pre-individuative’ concept of existence, or a concept of ‘entity without identity.’ What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: “is that the same coffee that was in the cup last night?” I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit quantifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Leibniz or a David Lewis, conceives of possible worlds as entities without identity. (shrink)

We show that the actuality operator A is redundant in any propositional modal logic characterized by a class of Kripke models (respectively, neighborhood models). Specifically, we prove that for every formula ${\phi}$ in the propositional modal language with A, there is a formula ${\psi}$ not containing A such that ${\phi}$ and ${\psi}$ are materially equivalent at the actual world in every Kripke model (respectively, neighborhood model). Inspection of the proofs leads to corresponding proof-theoretic results concerning the eliminability of the actuality (...) operator in the actuality extension of any normal propositional modal logic and of any “classical” modal logic. As an application, we provide an alternative proof of a result of Williamson’s to the effect that the compound operator A□ behaves, in any normal logic between T and S5, like the simple necessity operator □ in S5. (shrink)

Not the least admirable of the late David Lewis’s attributes was his disdain for technical terminology and jargon. His writings are a model demonstrating that, with skill and care, it is possible to discuss even the most mathematical aspects of logic and semantics in clear English prose, and with only a minimum of symbolism. The main text of Parts of Classes [1, hereafter: PoC], a 120-page essay on the foundations of set theory, follows Aristotle in using letters as variables, and (...) has a few dashes and ellipses where it mentions schemata, but is otherwise free of logical notation. The essential advantages of what Church called “the logistic method” do not depend on its typographical accidents!) It is only natural that writers following him have taken one of his expressive words and turned it into a technical term: he allowed ‘atomless gunk’ as a possible complication in the mereological analysis of things, and ‘gunk’ is now a technical term in mereology. Gunk is stuff every part of which contains further parts, stuff which can be divided endlessly without ever arriving at an indivisible atom. (shrink)

The French philosopher Jean Buridan's work on the logical paradoxes is currently attracting more attention than it has for several centuries. In part this is due to a general resurgence of interest in the paradoxes, but the immediate occasion is the recent publication of G. E. Hughes's edition, translation, and commentary on the chapter of Buridan's Sophismata most immediately concerned with the paradoxes. It is worth noting, therefore, that Buridan's theory fails, and in a way that makes it seem unlikely (...) that it can be developed into a serious rival of more modern theories. (shrink)

Ian Hacking's [6] is a spirited romp though a broad field of metaphysics, touching on a variety of important questions, and appealing to deep results in mathematical logic while remaining free of logical pedantry. Philosophical journals might be more fun to read if others could write in his style. It is an essay in applying the theory of logic expounded in more detail in his very interesting [7]. Unfortunately, despite my sympathy for his project, I have a number of criticisms (...) to make of his argument. Some of them are picky logical points, but the major conclusions of [6] are affected. (shrink)

Il platonista sostiene che le verità della matematica e della logica siano letteralmente vere, ossia che descrivano (in qualche modo: non voglio legare la definizione a una particolare teoria semantica) realtà che non sono create o decise da noi. Il convenzionalista sostiene invece che le proposizioni che chiamiamo verità della matematica siano in qualche misura convenzionali: esse esprimerebbero convenzioni che abbiamo adottato noi, o certe loro conseguenze. Le due posizioni sono apparenteme...

This paper is an informal presentation of the ideas presented formally in ”Relative-Sameness Counterpart Theory”. Relative-sameness relations -- such as being the same person as -- are like David Lewis’s “counterpart” relations in the following respects: (i) they may hold over time or across worlds between objects that aren’t cross-time or cross-world identical (I propose), and (ii) there are a multiplicity of them, different ones of which may be variously invoked in different contexts. They differ from his counterpart relations, however, (...) in that they are weak equivalence relations (transitive, symmetric and weakly reflexive). The likenesses to counterpart relations make them suitable for an analysis of de-re temporal and modal predications. The difference renders the resulting counterpart theory immune to standard criticisms of Lewis’s Counterpart Theory (e.g., in Hazen 1979, and Fara and Williamson 2005). The use of sameness as opposed to similarity relations in the analysis of de-re temporal and modal predication renders the resulting truth conditions as statable in terms that proponents of Kripke’s identity-based analysis can accept. (shrink)

C I Lewis showed up Down Under in 2005, in e-mails initiated by Allen Hazen of Melbourne. Their topic was the system Hazen called FL (a Funny Logic), axiomatized in passing in Lewis 1921. I show that FL is the system MEN of material equivalence with negation. But negation plays no special role in MEN. Symbolizing equivalence with → and defining ∼A inferentially as A→f, the theorems of MEN are just those of the underlying theory ME of pure (...) material equivalence. This accords with the treatment of negation in the Abelian l-group logic A of Meyer and Slaney (Abelian logic. Abstract, Journal of Symbolic Logic 46, 425–426, 1981), which also defines ∼A inferentially with no special conditions on f. The paper then concentrates on the pure implicational part AI of A, the simple logic of Abelian groups. The integers Z were known to be characteristic for AI, with every non-theorem B refutable mod some Zn for finite n. Noted here is that AI is pre-tabular, having the Scroggs property that every proper extension SI of AI, closed under substitution and detachment, has some finite Zn as its characteristic matrix. In particular FL is the extension for which n = 2 (Lewis, The structure of logic and its relation to other systems. The Journal of Philosophy 18, 505–516, 1921; Meyer and Slaney, Abelian logic. Abstract. Journal of Symbolic Logic 46, 425–426, 1981; This is an abstract of the much longer paper finally published in 1989 in G. G. Priest, R. Routley and J. Norman, eds., Paraconsistent logic: essays on the inconsistent, Philosophica Verlag, Munich, pp. 245–288, 1989). (shrink)

In a recent article in Analysis, Graeme Hunter and William Seager (1981) attempt to rescue counterpart theory (CT) from some objections of Hazen 1979. They see these objections as arising from ‘uncritical use of the translation scheme originally proposed by Lewis’, and intend to meet them by refraining from use of that scheme. But they do not offer a new scheme; they say ‘…it is no more necessary to have one to capture the sense of modal idiom than it (...) is to capture the sense of quantificational idiom…Appeal to truth value is the single most important criterion of correct translation’ (Hunter and Seager 1981:72). Thus, where the scheme of Lewis (1968) translates a truth by a falsehood or conversely, Hunter and Seager simply produce a sentence which they claim to be a better translation, without articulating any structural principles they employ to arrive at their candidate. A friend of CT should not be happy with this procedure. Let Lm be first-order modal language and Lc be the language of counterpart theory. Sentences of Lc are intended to interpret or elucidate the meanings of Lm-sentences in the strong sense that they should be the outputs of an adequate theory of meaning, which we can formulate as a model theory, for Lm, written in Lc plus set theory. The interest of CT resides in its potential to provide an alternative approach to the model theory of modal logic, and from the clauses of any such alternative model theory we should be able to read off a translation scheme for Lm into Lc, just as we can in the standard case. If CT’s motivation is sound, which Hunter and Seager do not dispute, yet there is no translation scheme and so no counterpart-theoretic model theory, one might suspect some defect in the original language Lm. And in fact, Hunter and Seager find certain Lm-sentences multiply ambiguous with respect to Lc; but if a counterpart-theoretic model theory is possible, to say.. (shrink)