A calculus for the common rules of ∧ and ∨

We provide a finite axiomatization of the consequence , i.e. of the set of common sequential rules for and . Moreover, we show that has no proper non-trivial strengthenings other than and . A similar result is true for , but not, e.g., for ⁺.To the memory of Jerzy Supecki     

Some problems concerning the logic of grammatical modifiers

This paper consists principally of selections from a much longer work on the semantics of English. It discusses some problems concerning how to represent grammatical modifiers (e.g. slowly in x drives slowly) in a logically perspicuous notation. A proposal of Reichenbach's is given and criticized; then a new theory (apparently discovered independently by myself, Romain Clark, and Richard Montague and Hans Kamp) is given, in which grammatical modifiers are represented by operators added to a first-order predicate calculus. Finally some problems concerning applications of adjectives to that-clauses and gerundive-clauses are discussed.     

Kenneth Strike on Liberalism,Citizenship and the Private Interest in Schooling

After indicating a number of points of agreement with the argument 0eveloped by Kenneth Strike in his article Liberalism, Citizenship and the Private Interest in Schooling, this article identifies and explores a number of queries and criticisms which arise in relation to that argument. These queries and criticisms relate especially to the nature and extent of the expansiveness involved in Strike's conception of public or common educational influence, and to the implications and justification of the claim that private educational interests enjoy a greater salience and recognition on Strike's view of public or common educational influence than on some alternative views.     

Israel Scheffler on Religion,Reason and Education

Israel Scheffler has only recently written directly and about religion and education in religion, although these are matters in which he has a strong personal interest. Scheffler's views on these issues are outlined and critically appraised, with some reference to the views of R.S. Peters on similar questions. It is suggested that one of the major difficulties which arise in relation to Schelffer's position concern its account of the balance between acceptance and critical search for clarity needed on the part of students in their engagement with Jewish ritual. This difficulty brings into focus a numer of central questions which arise concerning the reinterpretive account of Jewish tradition which Scheffler offers.     

To and from philosophy — Discussions with Gödel and Wittgenstein

I propose to sketch my views on several aspects of the philosophy of mathematics that I take to be especially relevant to philosophy as a whole. The relevance of my discussion would, I think, become more evident, if the reader keeps in mind the function of (the philosophy of) mathematics in philosophy in providing us with more transparent aspects of general issues. I shall consider: (1) three familiar examples; (2) logic and our conceptual frame; (3) communal agreement and objective certainty; (4) the transcommunal universality of mathematics; (5) the big jump to the potential infinite; (6) the reconciliation of local creation with the hypothesis of discovery; (7) Platonism as realism plus conceptualism; (8) foundational studies and mathematical practice; and (9) the decomposition of philosophical disagreements. The views of Gödel and Wittgenstein are emphasized in order to add specificity to the discussions.     

Can Parts of Space Move? On Paragraph Six of Newton’s Scholium

Paragraph 6 of Newtons Scholium argues that the parts of space cannot move. A premise of the argument – that parts have individuality only through an order of position – has drawn distinguished modern support yet little agreement among interpretations of the paragraph. I argue that the paragraph offers an a priori, metaphysical argument for absolute motion, an argument which is invalid. That order of position is powerless to distinguish one part of Euclidean space from any other has gone virtually unremarked. It remains uncertain what the import of the paragraph is but it is not close to apparently similar arguments of Leibniz.     

On a statistic arising in testing correlation

This paper is devoted to the study of a certain statistic,u, defined on samples from a bivariate population with variances ₁₁, ₂₂ and correlation . Let the parameter corresponding tou be . Under binormality assumptions the following is demonstrated. (i) If ₁₁ = ₂₂, then the distribution ofu can be obtained rapidly from theF distribution. Statistical inferences about = may be based onF. (ii) In the general case, allowing for _{11 22}, a certain quantity involvingu,r (sample correlation between the variables) and follows at distribution. Statistical inferences about may be based ont. (iii) In the general case a quantityt may be constructed which involves only the statisticu and only the parameter . If treated like at distributed magnitude,t admits conservative statistical inferences. (iv) TheF distributed quantity mentioned in (i) is equivalent to a certaint distributed quantity as follows from an appropriate transformation of the variable. (v) Three test statistics are given which can be utilized in making statistical inferences about = in the case ₁₁ = ₂₂. A comparison of expected lengths of confidence intervals for obtained from the three test statistics is made. (vi) The use of the formulas derived is illustrated by means of an application to coefficient alpha.This research was supported by the National Institute of Child Health and Human Development, under Research Grant 1 PO1 HD01762.     

On the algebraization of a Feferman's predicate

Summary This paper is devoted to the algebraization of an arithmetical predicate introduced by S. Feferman. To this purpose we investigate the equational class of Boolean algebras enriched with an operation , which translates such predicate, and an operation , which translates the usual predicate Theor. We deduce from the identities of this equational class some properties of and some ties between and ; among these properties, let us point out a fixed-point theorem for a sufficiently large class of - polynomials. The last part of this paper concerns the duality theory for - algebras.Allatum est die 24 Maii 1976     

Some notes on the nature of methodological indeterminacy

This paper is an attempt to extend the meaning of the concept of indeterminacy for the human sciences. The authors do this by coining the term methodological indeterminacy and arguing that indeterminacy is better understood when linked to specific methodological techniques. Paradoxically, while specific research techniques demonstrate that the issue of indeterminacy is complex, yielding the possibility of types and degrees, it does not eliminate the problem of translation first raised by Quine. However, the authors go on to argue that, from a research perspective, indeterminacy can and must be approached in such a way that it is possible to reduce cases of it, even though never completely eliminating it in the human sciences.     

On a certain class ofm-address machines

Conclusion It follows from the proved theorems that ifM =Q, (whereQ={0,q ₁,q ₂,...,q }) is a machine of the classM _F then there exist machinesM _i such thatM _i(1,c)=M (q _i,c) andQ _i={0, 1, 2, ..., +1} (i=1, 2, ..., ).And thus, if the way in which to an initial function of content of memorycC a machine assigns a final onecC is regarded as the only essential property of the machine then we can deal with the machines of the formM ={0, 1, 2, ..., }, and processes (t) (wheret=1,c,cC) only.Such approach can simplify the problem of defining particular machines of the classM _F , composing and simplifying them.Allatum est die 19 Januarii 1970     

Some modifications of Scott's theorem on injective spaces

D. Scott in his paper [5] on the mathematical models for the Church-Curry -calculus proved the following theorem.A topological space X. is an absolute extensor for the category of all topological spaces iff a contraction of X. is a topological space of Scott's open sets in a continuous lattice.In this paper we prove a generalization of this theorem for the category of , -closure spaces. The main theorem says that, for some cardinal numbers , , absolute extensors for the category of , -closure spaces are exactly , -closure spaces of , -filters in , >-semidistributive lattices (Theorem 3.5).If = and = we obtain Scott's Theorem (Corollary 2.1). If = 0 and = we obtain a characterization of closure spaces of filters in a complete Heyting lattice (Corollary 3.4). If = 0 and = we obtain a characterization of closure space of all principial filters in a completely distributive complete lattice (Corollary 3.3).     

La Evolucion de la filosofia de Wittgenstein

Summary The author claims that there is a basic difference between theTractatus and thePhilosophical Investigations; despite Bernstein's and O'Brien's claims to the contrary, there are, indeed, two Wittgensteins. Yet, to ascertain the difference between both we must look at Wittgenstein's conceptions of philosophy rather than at his views on logic and language. Wittgenstein's different, and even divergent, views on logic and language are grounded on his two views on philosophy and not the other way around. At the same time, Wittgenstein's views on philosophy are caused by his ways of conceiving the scope of philosophical activity in regard to language. Both in theTractatus and in thePhilosophical Investigations, Wittgenstein points out what is important in language for philosophy, but in each case he reaches very different conclusions. Now, when all is said, there remains one unifying factor in all of Wittgenstein's investigations: it is the question of the logic of language, which shifts positions from theTractatus to theInvestigations, so that what was earlier a hidden structure becomes later the grammar of its indefinitely complexe uses.     

Parallel action: Concurrent dynamic logic with independent modalities

Regular dynamic logic is extended by the program construct, meaning and executed in parallel. In a semantics due to Peleg, each command is interpreted as a set of pairs (s,T), withT being the set of states reachable froms by a single execution of, possibly involving several processes acting in parallel. The modalities << and [] are given the interpretations<>A is true ats iff there existsT withsRT andA true throughoutT, and[]A is true ats iff for allT, ifsRT thenA is true throughoutT, which make <> and [] no longer interdefinable via negation, as they are in the regular case.We prove that the logic defined by this modelling is finitely axiomatisable and has the finite model property, hence is decidable. This requires the development a new theory of canonical models and filtrations for reachability relations.     

Hick's interpretation of religious pluralism

Conclusion There is no question that Hick's theory rests upon multiple assumptions about a singular, transcendental grounding and the fundamental equality of the various religions that cannot be inductively verified beyond all doubt. That need not mean, however, that the attractiveness of his theory derives solely from the peculiar charm of supposing that the One and the Many are no more at odds in the realm of religion than anywhere else. For Hick's assumptions are not just an exercise in wishful thinking or wild speculation. They are based upon experience from within what he calls the benign circle of faith. Because the reality experienced is ambiguous, acceptance or rejection of his views will, of course, be a matter of choice. And, admittedly, this choice will be dictated not so much by a weighing of empirical evidence that might prove the various religions to be exactly as he sees them, as by a consideration of what we have been surveying in the preceding pages, namely, the import of seeing things as Hick does.

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Power,Freedom, and Individuality: Foucault and Sexual Difference

This paper offers a detailed account of Foucaults ethical and political notion of individuality as presented in his late work, and discusses its relationship to the feminist project of the theory of sexual difference. I argue that Foucaults elaboration of the classical ethos of care for the self opens the way for regarding the I-woman as an ethical, political and aesthetic self-creation. However, it has significant limitations that cannot be ignored. I elaborate on two aspects of Foucaults avoidance of sexual difference as a relevant category for an account of political and ethical individuality, which thus implicitly associates individual agency with men. I argue that Foucault implicitly assumes the existence of an ontological desire to become engaged in political self-creation. However, the ethical position of self-knowledge and desire should be understood as a contingent option that depends on material and historical conditions for its realization. Hence, I argue that a feminist reworking of Foucaults notion of political individuality should add a substantial ethical condition to the imperative of self-knowledge and self-creation – making possible the desiring woman subject.     

On a problem of P(α, δ, π) concerning generalized Alexandroff s cube

Universality of generalized Alexandroff's cube plays essential role in theory of absolute retracts for the category of , -closure spaces. Alexandroff's cube. is an , -closure space generated by the family of all complete filters. in a lattice of all subsets of a set of power .Condition P(, , ) says that is a closure space of all , -filters in the lattice ( ), .Assuming that P (, , ) holds, in the paper [2], there are given sufficient conditions saying when an , -closure space is an absolute retract for the category of , -closure spaces (see Theorems 2.1 and 3.4 in [2]).It seems that, under assumption that P (, , ) holds, it will be possible to givean uniform characterization of absolute retracts for the category of , -closure-spaces.Except Lemma 3.1 from [1], there is no information when the condition P (, , ) holds or when it does not hold.The main result of this paper says, that there are examples of cardinal numbers, , , such that P (, , ) is not satisfied.Namely it is proved, using elementary properties of Lebesgue measure on the real line, that the condition P (, ₁, 2 ) is not satisfied.Moreover it is shown that fulfillment of the condition is essential assumption in, Theorems 2.1 and 3.4 from [1] i.e. it cannot be eliminated.     

“Scientific atheism” in the era of perestrojka

Conclusions It could be argued that some in the military, like certain local officials, are the last holdouts against the reform's ideological thaw toward religion, though Kharev's October–November, 1989, interview inOgonëk makes clear that there are still some higher-level forces in the apparatus who remain opposed to some of the changes. It could be that some of the reformers themselves are concerned about the pace of change. Even in their minds the thaw undoubtedly has limits. They may view the present controversy over restructuring scientific atheism and redefining socialism's attitude toward religion as a necessary and desirable part of the dialectical process. Thus, while encouraging a tactical and more humane detente toward religion, they have not yet moved to eliminate the basis for ideological hostility toward it. They simply want that hostility to be better channeled and controlled, while exploiting the political benefits that their new-found truce with religion offers. Looking at Gorbachevism, Soviet émigré observer Aleksandr Zinov'ev believes that the reform process in the sphere of internal political activity concerns only separate phenomena ... but by no means the very foundations of the social system, the system of power and ideology. That system, he believes, is not yet fundamentally changing in those terms. The final answer remains to be seen.The views expressed are those of the author and are not official views of the U.S. Government nor any department or agency.     

Natural deduction rules for English

A system of natural deduction rules is proposed for an idealized form of English. The rules presuppose a sharp distinction between proper names and such expressions as the c, a (an) c, some c, any c, and every c, where c represents a common noun. These latter expressions are called quantifiers, and other expressions of the form that c or that c itself, are called quantified terms. Introduction and elimination rules are presented for any, every, some, a (an), and the, and also for any which, every which, and so on, as well as rules for some other concepts. One outcome of these rules is that Every man loves some woman is implied by, but does not imply, Some woman is loved by every man, since the latter is taken to mean the same as Some woman is loved by all men. Also, Jack knows which woman came is implied by Some woman is known by Jack to have come, but not by Jack knows that some woman came.     

The transcendentals in the middle ages: An introduction

Although most predicates may be truthfully predicated of only some beings, there are others that seem to apply to every being. The latter, including being itself, were known as the transcendentals in the Middle Ages and gave rise to the much disputed doctrine of the transcendentals. This article explores the main tenets of the doctrine and the difficulties that they face, the reasons why scholastic authors were interested in these issues, and the origins of the doctrine.I am grateful to Jan Aertsen, Kenneth Barber, Michael Gorman, and Scott MacDonald for their suggestions concerning this paper.     

Becoming: A modest proposal

In this paper I attempt a new approach to an old technical term: becoming. I show how the theory that becoming is coming-to-be could be supported by a semantic derivation of the nominalization becoming from its verbal counterpart, by investigating the properties of the present progressive constructions in which becoming as a verbal appears. My theory denies that dates, or qualitative change, play an essential role in the analysis of becoming.