In this paper the PA-completeness of modal logic is studied by syntactical and constructive methods. The main results are theorems on the structure of the PA-proofs of suitable arithmetical interpretationsS of a modal sequentS, which allow the transformation of PA-proofs ofS into proof-trees similar to modal proof-trees. As an application of such theorems, a proof of Solovay's theorem on arithmetical completeness of the modal system G is presented for the class of modal sequents of Boolean combinations of formulas of the (...) form p i,m i=0, 1, 2, ... The paper is the preliminary step for a forthcoming global syntactical resolution of the PA-completeness problem for modal logic. (shrink)

This paper is the second part of the syntactic demonstration of the Arithmetical Completeness of the modal system G, the first part of which is presented in [9]. Given a sequent S so that ⊢GL-LIN S, ⊬G S, and given its characteristic formula H = char(S), which expresses the non G-provability of S, we construct a canonical proof-tree T of ~ H in GL-LIN, the height of which is the distance d(S, G) of S from G. T is the syntactic (...) countermodel of S with respect to Gand is a tool of general interest in Provability Logic, that allows some classification in the set of the arithmetical interpretations. (shrink)

This paper is the first of a series of three articles that present the syntactic proof of the PA-completeness of the modal system G, by introducing suitable proof-theoretic objects, which also have an independent interest. We start from the syntactic PA-completeness of modal system GL-LIN, previously obtained in [7], [8], and so we assume to be working on modal sequents S which are GL-LIN-theorems. If S is not a G-theorem we define here a notion of syntactic metric d(S, G): we (...) calculate a canonical characteristic fomula H of S (char(S)) so that ⊢G ∼ H → (∼S) and ⊢GL-LIN ∼ H, and the complexity σ of ∼ H gives the distance d(S, G) of S from G. Then, in order to produce the whole completeness proof as an induction on this d(S, G), we introduce the tree-interpretation of a modal sequent Q into PA, that sends the letters of Q into PA-formulas describing the properties of a GL-LIN-proof P of Q: It is also a d(*, G)-metric linked interpretation, since it will be applied to a proof-tree T of ∼ H with H = char(S) and σ(∼ H) = d(S, G). (shrink)

This paper is the final part of the syntactic demonstration of the Arithmetical Completeness of the modal system G; in the preceding parts [9] and [10] the tools for the proof were defined, in particular the notion of syntactic countermodel. Our strategy is: PA-completeness of G as a search for interpretations which force the distance between G and a GL-LIN-theorem to zero. If the GL-LIN-theorem S is not a G-theorem, we construct a formula H expressing the non G-provability of S, (...) so that ⊢GL-LIN ∼ H and so that a canonical proof T of ∼ H in GL-LIN is a syntactic countermodel for S with respect to G, which has the height θ(T) equal to the distance d(S, G) of S from G. Then we define the interpretation ξ of S which represents the proof-tree T in PA. By induction on θ(T), we prove that ⊢PA Sξ and d(S, G) > 0 imply the inconsistency of PA. (shrink)

In this paper are studied the properties of the proofs in PRA of provability logic sentences, i.e. of formulas which are Boolean combinations of formulas of the form PIPRA, where h is the Gödel-number of a sentence in PRA. The main result is a Normal Form Theorem on the proof-trees of provability logic sequents, which states that it is possible to split the proof into an arithmetical part, which contains only atomic formulas and has an essentially intuitionistic character, and into (...) a logical part, which is merely instrumental. Moreover, the induction rules which occur in the arithmetical part are implicit. Some applications of the Normal Form Theorem are shown in order to obtain some syntactical results on the PRA-completeness of modal logic. In particular a completeness theorem for Boolean combinations of modalities is given. (shrink)

Панфілова Т. О. Становлення громадянського суспільства у Західній Україні : [монографія] / Т. О. Панфілова ; Національна академія наук України, Інститут українознавства ім. І. Крип'якевича. - Львів, 2015. - 406 с.

10. Catholicity Necessary to Sustain Popular Liberty -- 11. Authority and Liberty -- 12. The Works of Daniel Webster -- 13. Schools of Philosophy -- 14. Liberalism and Socialism -- 15. Civil and Religious Freedom -- 16. Liberalism and Progress -- Part III. Freedom and Communion -- 17. The American Republic -- 18. The Democratic Principle -- Notes.

v. 1. The Universalist years, 1826-29 -- v. 2. The free thought and Unitarian years, 1830-35 -- v. 3. The Transcendentalist years, 1836-38 -- v. 4. The transcendentalist years, 1838-39 -- v. 5. The transcendentalist years, 1840-1841 -- v. 6. Life by communion, 1842 -- v. 7. Life by communion years, 1843-1844.

This work has been selected by scholars as being culturally important, and is part of the knowledge base of civilization as we know it. This work was reproduced from the original artifact, and remains as true to the original work as possible. Therefore, you will see the original copyright references, library stamps (as most of these works have been housed in our most important libraries around the world), and other notations in the work. This work is in the public domain (...) in the United States of America, and possibly other nations. Within the United States, you may freely copy and distribute this work, as no entity (individual or corporate) has a copyright on the body of the work. As a reproduction of a historical artifact, this work may contain missing or blurred pages, poor pictures, errant marks, etc. Scholars believe, and we concur, that this work is important enough to be preserved, reproduced, and made generally available to the public. We appreciate your support of the preservation process, and thank you for being an important part of keeping this knowledge alive and relevant. (shrink)

Two different kinds of stochastic processes in Hilbert space used to introduce spontaneous localization into the quantum evolution are investigated. In the processes of the first type, finite changes of the wave function take place instantaneously with a given mean frequency. The processes of the second type are continuous. It is shown that under a suitable infinite frequency limit the discontinuous process transforms itself into the continuous one.

From explicit pronouncements in the writings of Aquinas, this artide elaborates on how the apprehension of the act of being is accomplished in the human intellect trough its natural habit, the intellectus principiorum. The basic conclusión states that direct reception of truth is implied in the apprehension of the actuality of extramental things.

Contingency is the presence of non-actualized possibility in the world. Fatalism is a view of reality on which there is no contingency. Since it is contingency that permits agency, there has traditionally been much interest in contingency. This interest has long been embarrassed by the contention that simple and plausible assumptions about the world lead to fatalism. I begin with an Aristotelian argument as presented by Richard Taylor. Appreciation of this argument has been stultified by a question pertaining to the (...) source of necessity and possibility and a closely related one regarding the nature of metaphysics itself. Answering these questions reveals the crucial issue here, to wit, necessity and possibility in a temporal world. This issue is investigated through an important later criticism of Taylor by David Foster Wallace. Wallace’s critique is significant because it brings to the fore the pivotal notion for understanding contingency in a temporal world, that of synchronic possibility, the idea that incompatible states of affairs are possible at a single moment. This notion provides the basis of distinguishing two systematic accounts of truth, modality and time: two metaphysics of contingency. On one account, Taylor’s Aristotelian argument is straightforwardly valid and compelling; on the other, it is fallacious. In closing, I present reasons why the former account, supporting the Aristotelian views of time and truth, is correct and make some comments to ameliorate this conclusion. (shrink)

The notion of conceivability has traditionally been regarded as crucial to an account of modal knowledge. Despite its importance to modal epistemology, there is no received explication of conceivability. One purpose of this paper is to argue that the notion is not fruitfully explicated in terms of the imagination. The most natural way of presenting a notion of conceivability qua imaginability is open to cogent criticism. In order to avoid such criticism, an advocate of the modal insightfulness of the imagination (...) must broaden the idea of what it is to be imaginable. I argue that this required broadening renders the imagination idle . Consequently, I distinguish two different accounts of the evidential basis of modal knowledge and present a more general argument that concludes that the very notion of conceivability should be eschewed in modal epistemology. (shrink)

A conjectural inference is proposed, aimed at producing conjectural theorems from formal conjectures assumed as axioms, as well as admitting contradictory statements as conjectural theorems. To this end, we employ Paraconsistent Informational Logic, which provides a formal setting where the notion of conjecture formulated by an epistemic agent can be defined. The paraconsistent systems on which conjectural deduction is based are sequent formulations of the C-systems presented in Carnielli-Marcos [CAR 02b]. Thus, conjectural deduction may also be considered to be a (...) tool for investigating the properties of paraconsistency in general. (shrink)

[kn. 1]. Without special title -- kn. 2. Obrazolohika systemy boh i Boh -- kn. 3. Katehorii︠a︡ obrazu.