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)

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)

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)

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...

There are good reasons to want logics, including many-valued logics, to have usable conditionals, and we have explored this in certain logics. However, it turns out that we “accidentally” chose some favourable logics. In this paper, we look at some of the unfavourable logics and describe where usable conditionals can be added and where it is not possible.

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.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

A questionnaire on the attitudes towards the functions of research ethical committees was sent to members of selected research ethical committees in Wessex and some controls. Almost all respondents felt there was a need for ethical review of research projects; 42 per cent thought there was a need for some training before joining a committee; 67 per cent thought the system could be improved and 47 per cent thought that monitoring or follow-up procedures should be adopted. Ethical committees were thought (...) to be purely advisory, as opposed to mandatory, by 33 per cent, and 63 per cent thought they should restrict their review to ethical problems as opposed to scientific or design problems. Views about the function of non-medical members ranged from 'none at all' to 'very important'. Of the 10 controls who were asked whether they would become a member of an ethical committee if asked, seven said that on balance they would and the reasons stated varied from the view that it was a 'very important committee' to the feeling that it was 'a necessary but irksome job'. (shrink)

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

An edition of the letters of Erasmus, regarded as one of the greatest humanist writers. All 12 volumes of this work have been reissued, complete with their scholarly apparatus of commentary and notes, as well as plates.

In order to model any macroscopic system, it is necessary to aggregate both spatially and taxonomically. If average processes are assumed, then kinetic equations of “population dynamics” can be derived. Much effort has gone into showing the important effects introduced by non-average effects (fluctuations) in generating symmetry-breaking transitions and creating structure and form. However, the effects of microscopic diversity have been largely neglected. We show that evolution will select for populations which retain “variability,” even though this is, at any given (...) time, loss-making, predicting that we shall not observe populations with “optimal behavior,” but populations which can “learn.” This lesser short-term efficiency may be why natural diversity is so great. Evolution is seen to be “driven” by the noise to which it leads. (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)

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)