The paper considers certain extensions of the system LC introduced in Dunn & Meyer 1997. LC is a structurally free system , but it has combinators as formulas in the place of structural rules. We consider two ways to extend LC with conjunction and disjunction depending on whether they distribute over each other or not. We prove the elimination theorem for the systems. At the end of the paper we give a Routley-Meyer style semantics for the distributive extension, including some (...) new definitions and an algorithm which are used in proving a representation theorem in general, i.e., for arbitrary combinators. (shrink)

Structurally free logic LC was introduced in [4]. A natural extension of LC, in particular, in a sequent formulation, is by conjunction and disjunction that do not distribute over each other. We define a set theoretical semantics for these logics via constructing a representation of a lattice that we extend by intensional operations. Canonically, minimally overlapping filter-ideal pairs are used; this construction avoids the use of an equivalent of the axiom of choice and lends transparency to the structure. We also (...) discuss adapting the present semantics for substructural logics, as well as, the question of cannnonicity of LC+. (shrink)

Nonclassical logics have played an increasing role in recent years in disciplines ranging from mathematics and computer science to linguistics and philosophy. _Generalized Galois Logics_ develops a uniform framework of relational semantics to mediate between logical calculi and their semantics through algebra. This volume addresses normal modal logics such as K and S5, and substructural logics, including relevance logics, linear logic, and Lambek calculi. The authors also treat less-familiar and new logical systems with equal deftness.

Four-valued semantics proved useful in many contexts from relevance logics to reasoning about computers. We extend this approach further. A sequent calculus is defined with logical connectives conjunction and disjunction that do not distribute over each other. We give a sound and complete semantics for this system and formulate the same logic as a tableaux system. Intensional conjunction and its residuals can be added to the sequent calculus straightforwardly. We extend a simplified version of the earlier semantics for this system (...) and prove soundness and completeness. Then, with some modifications to this semantics, we arrive at a mathematically elegant yet powerful semantics that we call generalized Kripke semantics. (shrink)

Kleene algebras and action logic were proposed to be solutions to the finite axiomatization problem of the algebra of regular sets (of strings). They are treated here as nonclassical logics—with Hilbert-style axiomatizations and semantics. We also provide intuitive accounts in terms of information states of the semantics which provide further insights into the formalisms. The three types of "Kripke-style'' semantics which we define develop insights from gaggle theory, and from our four-valued and generalized Kripke semantics for the minimal substructural logic. (...) Soundness and completeness are proven each time. (shrink)

Symmetic combinatory logic with the symmetric analogue of a combinatorially complete base (in the form of symmetric λ-calculus) is known to lack the Church-Rosser property. We prove a much stronger theorem that no symmetric combinatory logic that contains at least two proper symmetric combinators has the Church-Rosser property. Although the statement of the result looks similar to an earlier one concerning dual combinatory logic, the proof is different because symmetric combinators may form redexes in both left and right associated terms. (...) Perhaps surprisingly, we are also able to show that certain symmetric combinatory logics that include just one particular constant are not confluent. This result (beyond other differences) clearly sets apart symmetric combinatory logic from dual combinatory logic, since all dual combinatory systems with a single combinator or a single dual combinator are Church-Rosser. Lastly, we prove that a symmetric combinatory logic that contains the fixed point and the one-place identity combinator has the Church-Rosser property. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix. (shrink)

Sequent calculi constitute an interesting and important category of proof systems. They are much less known than axiomatic systems or natural deduction systems are, and they are much less known than they should be. Sequent calculi were designed as a theoretical framework for investigations of logical consequence, and they live up to the expectations completely as an abundant source of meta-logical results. The goal of this book is to provide a fairly comprehensive view of sequent calculi -- including a wide (...) range of variations. The focus is on sequent calculi for various non-classical logics, from intuitionistic logic to relevance logic, through linear and modal logics. A particular version of sequent calculi, the so-called consecution calculi, have seen important new developments in the last decade or so. The invention of new consecution calculi for various relevance logics allowed the last major open problem in the area of relevance logic to be solved positively: pure ticket entailment is decidable. An exposition of this result is included in chapter 9 together with further new decidability results (for less famous systems). A series of other results that were obtained by J. M. Dunn and me, or by me in the last decade or so, are also presented in various places in the book. Some of these results are slightly improved in their current presentation. Obviously, many calculi and several important theorems are not new. They are included here to ensure the completeness of the picture; their original formulations may be found in the referenced publications. This book contains very little about semantics, in general, and about the semantics of non-classical logic in particular. (shrink)

This book honors J. Michael Dunn, who was a preeminent relevance logician. Dunn's career spanned over 50 years and his research results had an impact on philosophy, mathematics and informatics. Dunn often used algebraic techniques in his research into logics such as relevance, orthomodular and substructural logics. He invented the logic R-mingle and the sequent calculus LR+; he proved crucial theorems about 2-valued first-order logic and non-classical higher-order logics - among many other results. The papers in this volume touch upon (...) topics that Dunn was concerned with. Some authors were students or colleagues of Dunn; some other authors had not met Dunn in person, but share his research interests. None of the articles published here have appeared in print before; indeed, most of the papers were written specifically for this collection. The diversity of the themes of the articles reflects the scope of Dunn's own research in logic. It will also ensure that anybody with an interest in logic - whether a student, a logician or a scholar in another field - will find reading this book a worthwhile endeavor. The editor, Katalin Bimbó was the 14th Ph.D. student of J. Michael Dunn at Indiana University in Bloomington, IN, U.S.A.; currently, she is a professor of philosophy at the University of Alberta in Canada. (shrink)

The implicational fragment of the relevance logic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus (related to (...) $FT_\rightarrow$ ) for the implicational fragment of ticket entailment. (shrink)

The implicational fragment of the relevance logic "ticket entailment" is closely related to the so-called hereditary right maximal terms. I prove that the terms that need to be considered as inhabitants of the types which are theorems of $T_\rightarrow$ are in normal form and built in all but one casefrom B, B' and W only. As a tool in the proof ordered term rewriting systems are introduced. Based on the main theorem I define $FIT_\rightarrow$ - a Fitch-style calculus for the (...) implicational fragment of ticket entailment. (shrink)

Reader-friendly without compromising the precision of exposition, the book includes many new research results not found in the available literature.

It is widely recognized that conceptual and theoretical innovations and the employment of new instruments and experimental techniques are important factors in explaining the growth of scientific knowledge in chemistry. This study examines another dimension of research in chemistry, collaboration and co-authorship. I focus specifically on Theodore Richards’ career and publications. During the period in which Richards worked, co-authorship was beginning to become more common than it had been previously. Richards was the first American chemist to be awarded a Nobel (...) Prize and he was at the forefront in this new trend in chemistry. He collaborated more than was typical for his time, with many scientists, in different sized groups, and he often had persistent collaborative relationships, extending over a number of years. Further, it appears that these collaborations benefited Richards, his collaborators, and the field of chemistry as a whole. (shrink)

A class of ordered relational topological spaces is described, which we call orthomodular spaces. Our construction of these spaces involves adding a topology to the class of orthomodular frames introduced by Hartonas, along the lines of Bimbó's topologization of the class of orthoframes employed by Goldblatt in his representation of ortholattices. We then prove that the category of orthomodular lattices and homomorphisms is dually equivalent to the category of orthomodular spaces and certain continuous frame morphisms, which we call continuous (...) weak p‐morphisms. It is well‐known that orthomodular lattices provide an algebraic semantics for the quantum logic. Hence, as an application of our duality, we develop a topological semantics for using orthomodular spaces and prove soundness and completeness. (shrink)

The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve $\mathbf{t}$ (...) . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper. (shrink)

The implicational fragment of the logic of relevant implication, $R_\to$ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, $T_\to$ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of $T_\to$ to the decidability problem of $R_\to$. The decidability of $T_\to$ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over (...) the base $\{\textsf{B},\textsf{B}',\textsf{I},\textsf{W}\}$. (shrink)

“Doğal din” terimi, bazen doğanın kendisinin ilahi olduğu bir panteistik doktrine atıfta bulunur. “Doğal teoloji” terimi ise aksine, başlangıçta gözlemlenen doğal gerçekler temelinde (ve bazen) Tanrı’nın varlığını savunmaya yönelik projeye atıfta bulunur. Bununla birlikte çağdaş felsefede, hem “doğal din” hem de “doğal teoloji” genel olarak, dinî veya teolojik konuları araştırmak için insana, “doğal” olan bilişsel yetilerini – akıl, algı, içgözlem- kullanma projesini ifade eder. Doğal din veya teoloji, mevcut anlayış üzerine, doğayla ilgili ampirik araştırmalarla sınırlı olmamakla birlikte ayrıca panteistik bir (...) sonuçla da bağlantılı değildir. Bununla birlikte o, özel doğal olmayan yetilere (ESP, telepati, mistik deneyim) veya doğaüstü bilgi kaynaklarına (kutsal metinler, vahiy teolojisi, inanç otoriteleri, doğrudan doğaüstü iletişim) hitap etmekten kaçınır. Genel olarak, doğal din veya teoloji (bundan böyle “doğal teoloji” olarak anılacaktır), diğer felsefi ve bilimsel girişimlerle aynı rasyonel inceleme standartlarına uymayı amaçlamaktadır ve aynı değerlendirme ve eleştiri yöntemlerine tabidir. (shrink)

The dissertation deals with problems in "logic", more precisely, it deals with particular formal systems aiming at capturing patterns of valid reasoning. Sequent calculi were proposed to characterize logical connectives via introduction rules. These systems customarily also have structural rules which allow one to rearrange the set of premises and conclusions. In the "structurally free logic" of Dunn and Meyer the structural rules are replaced by combinatory rules which allow the same reshuffling of formulae, and additionally introduce an explicit marker (...) for the application of the rule---the combinator. The dissertation gives syntactic extensions of such systems introducing conjunction and disjunction . Cut elimination is proved for these systems. ;Syntactic systems as a rule are accompanied by a semantics which interprets them. The extended systems are supplied with semantics using algebraic methods. First, the Lindenbaum algebras are formed, and then these algebras are represented in an algebra of sets. This gives a so called Kripke-style semantics for the systems. The representation for the distributive extension utilizes well-known techniques; it uses sets of prime filters and set theoretical operations on them to deal with conjunction and disjunction, plus a ternary accessibility relation to accommodate the intensional operators as fusion and implication. The representation theorem is proven generally, i.e., for arbitrary sets of combinators using the "squeeze lemma" of Routley and Meyer. ;The non-distributive calculi require a different representation since union and intersection are distributive. Two previous representations for lattices are overviewed, and a modification of the Hartonas-Dunn representation is successfully augmented with the intensional operations. Instead of sets of filters, sets of pairs of minimally inconsistent filters and ideals are used and the representation theorem is proven. The representation results provide soundness and completeness for the systems. The dissertation also investigates properties of dual combinators , proving a series of lemmas about dual combinatory systems not being Church-Rosser and being inconsistent. Combinatorially definable classes of lambda-terms are considered as well. (shrink)

A sequent calculus for the positive fragment of entailment together with the Church constants is introduced here. The single cut rule is admissible in this consecution calculus. A topological dual gaggle semantics is developed for the logic. The category of the topological structures for the logic with frame morphisms is proven to be the dual category of the variety, that is defined by the equations of the algebra of the logic, with homomorphisms. The duality results are extended to the logic (...) of entailment that includes a De Morgan negation. (shrink)

. Relational semantics for nonclassical logics lead straightforwardly to topological representation theorems of their algebras. Ortholattices and De Morgan lattices are reducts of the algebras of various nonclassical logics. We define three new classes of topological spaces so that the lattice categories and the corresponding categories of topological spaces turn out to be dually isomorphic. A key feature of all these topological spaces is that they are ordered relational or ordered product topologies.

Modal ontolojik argüman, Tanrı’nın varlığını sadece bilfiil gerçek olan bu dünyada değil, bütün mümkün dünyalarda göstermeye yönelik bir argümandır. Anselm’in (1033-1109) Proslogion adlı eserinin 3. bölümünde “kendisinden daha büyüğü düşünülemeyen” şeklinde tanımlanan; Tanrı’nın var olmamasının da düşünülemeyeceğini, bu yüzden de varolmamasının imkansızlığı üzerinde kurulu yeni bir argüman bulunduğunu öne süren bazı araştırmacılar, bu argümanı mümkün dünyalar semantiği yardımıyla formüle edip, “modal ontolojik argüman” şeklinde adlandırmışlardır. Çok farklı şekillerde formüle edilmiş bu argüman kabaca Tanrı’nın mümkünse zorunlu olması, dolayısıyla bilfiil gerçek olan (...) bu dünyada da var olması gerektiğini savunur. (shrink)

Urquhart works in several areas of logic where he has proved important results. Our paper outlines his topological lattice representation and attempts to relate it to other lattice representations. We show that there are different ways to generalize Priestley’s representation of distributive lattices—Urquhart’s being one of them, which tries to keep prime filters in the representation. Along the way, we also mention how semi-lattices and lattices figured into Urquhart’s work.

Symmetric generalized Galois logics (i.e., symmetric gGl s) are distributive gGl s that include weak distributivity laws between some operations such as fusion and fission. Motivations for considering distribution between such operations include the provability of cut for binary consequence relations, abstract algebraic considerations and modeling linguistic phenomena in categorial grammars. We represent symmetric gGl s by models on topological relational structures. On the other hand, topological relational structures are realized by structures of symmetric gGl s. We generalize the weak (...) distributivity laws between fusion and fission to interactions of certain monotone operations within distributive super gGl s. We are able to prove appropriate generalizations of the previously obtained theorems—including a functorial duality result connecting classes of gGl s and classes of structures for them. (shrink)

A ternary relation is often used nowadays to interpret an implication connective of a logic, a practice that became dominant in the semantics of relevance logics. This paper examines two early manuscripts --- one by Routley, another by Meyer --- in which they were developing set-theoretic semantics for various relevance logics. A standard presentation of a ternary relational semantics for, let us say, the logic of relevant implication R is quite illuminating, yet the invention of this semantics was fraught with (...) false starts. Meyer's manuscript, in which he builds on some ideas from Routley's manuscript, essentially contains a relational semantics for which R^{ot} is sound and complete. (shrink)

This book celebrates and expands on J. Michael Dunn’s work on informational interpretations of logic. Dunn, in his Ph.D. thesis, introduced a semantics for first-degree entailments utilizing the idea that a sentence can provide positive or negative information about a topic, possibly supplying both or neither. He later published a related interpretation of the logic R-mingle, which turned out to be one of the first relational semantics for a relevance logic. An incompatibility relation between information states lends itself to a (...) definition of negation and it has figured into Dunn's comprehensive investigations into representations of various negations. The informational view of semantics is also a prominent theme in Dunn’s research on other logics, such as quantum logic and linear logic, and led to the encompassing theory of generalized Galois logics. Dunn’s latest work addresses informational interpretations of the ternary accessibility relation and the very nature of information. The book opens with Dunn’s autobiography, followed by a list of his publications. It then presents a series of papers written by respected logicians working on different aspects of information-based logics. The topics covered include the logic R-mingle, which was introduced by Dunn, and its applications in mathematical reasoning as well as its importance in obtaining results for other relevance logics. There are also interpretations of the accessibility relation in the semantics of relevance and other non-classical logics using different notions of information. It also presents a collection of papers that develop semantics for various logics, including certain modal and many-valued logics. The publication of this book is well timed, since we are living in an "information age.” Providing new technical findings, intellectual history and careful expositions of intriguing ideas, it appeals to a wide audience of scholars and researchers. (shrink)

The fixed point combinator (Y) is an important non-proper combinator, which is defhable from a combinatorially complete base. This combinator guarantees that recursive equations have a solution. Structurally free logics (LC) turn combinators into formulas and replace structural rules by combinatory ones. This paper introduces the fixed point and the dual fixed point combinator into structurally free logics. The admissibility of (multiple) cut in the resulting calculus is not provable by a simple adaptation of the similar proof for LC with (...) proper combinators. The novelty of our proof—beyond proving the cut for a newly extended calculus–is that we add a fourth induction to the by-and-large Gentzen-style proof. (shrink)

We define dual and symmetric combinatory calculi (inequational and equational ones), and prove their consistency. Then, we introduce algebraic and set theoretical relational and operational - semantics, and prove soundness and completeness. We analyze the relationship between these logics, and argue that inequational dual logics are the best suited to model computation.

Çağdaş epistemoloji içerisinde, dışsalcı, güvenilirci ve erdeme dayalı epistemolojik tartışmaların bir benzerini, dini epistemoloji içerisinde etkili bir şekilde yer edinen ve reformcu epistemoloji olarak bilinen yaklaşım üzerinden görebilmek mümkündür. Alvin Plantinga, Nicholas Wolterstorff ve William Alston gibi filozofların öne çıktığı bu yaklaşımın ana iddiası, kanıta ya da argümana dayanmaksızın, Tanrı’nın varlığına inanmanın bütünüyle doğru, rasyonel, makul ve güvenilir olacağı şeklindeki bir tezdir. Kanıtın, gerekçelendirme için zorunlu bir koşul olmadığı fikri, algısal deneyimlerin gerekçelendirilmesine benzer şekilde, dini deneyimlerin de öyle olduğu ve (...) dolayısıyla, aynı epistemik statüye sahip olması gerektiğini öne süren, eşitlik tezine dayanır. İlk ortaya çıktığı günden bugüne, üzerinde çokça konuşulan ve tartışılan bir yaklaşım olan reformcu epistemoloji, özellikle son beş yıl içerisinde, oldukça farklı bir hale gelmiştir. Zira bu yaklaşımı, kanıtsalcılıkla uyumlu, erdem epistemolojisiyle bağlantılı ve diğer teistik dinler üzerinden değerlendiren yeni çalışmalar ortaya çıkmıştır. Bu nedenle, bu makalede, reformcu epistemoloji ana hatlarıyla ele alınmıştır. Beraberinde, ona yöneltilen itirazlara değinilmiş ve sonrasında, bahsi geçen yenilik iddiaları soruşturulmaya çalışılmıştır. Son olarak, teistik inancın rasyonalitesiyle ilgili, farklı bir bakış açısı sunulmuştur. (shrink)

Studies of epistemic communities and collaborative research in the social sciences have deepened the understanding of how science works, and more specifically how the social dimensions of scientific practice both enable and impede social scientists in realizing their epistemic goals. Two types of studies of epistemic communities are distinguished: general theories of epistemic communities aim to construct accounts of theoretical change applicable to all social scientific specialties, whereas historical studies emphasize the contingencies that affect specific social scientific disciplines, subfields, or (...) institutions. Although collaborative research plays a less pronounced role in the social sciences than in the natural sciences, collaboration has enabled social scientists to investigate topics that would otherwise be difficult to investigate. (shrink)

Philosophy has much to offer psychiatry, not least regarding ethical issues, but also issues regarding the mind, identity, values, and volition. This has become only more important as we have witnessed the growth and power of the pharmaceutical industry, accompanied by developments in the neurosciences. However, too few practising psychiatrists are familiar with the literature in this area. -/- The Oxford Handbook of Philosophy and Psychiatry offers the most comprehensive reference resource for this area ever published. It assembles challenging and (...) insightful contributions from key philosophers and others to the interactive fields of philosophy and psychiatry. Each contributions is original, stimulating, thorough, and clearly and engagingly written - with no potentially significant philosophical stone left unturned. Broad in scope, the book includes coverage of several areas of philosophy, including philosophy of mind, science, and ethics. For philosophers and psychiatrists, The Oxford Handbook of Philosophy and Psychiatry is a landmark publication in the field - one that will be of value to both students and researchers in this rapidly growing area. (shrink)

Relevance logics are known to be sound and complete for relational semantics with a ternary accessibility relation. This paper investigates the problem of adequacy with respect to special kinds of dynamic semantics (i.e., proper relation algebras and relevant families of relations). We prove several soundness results here. We also prove the completeness of a certain positive fragment of R as well as of the first-degree fragment of relevance logics. These results show that some core ideas are shared between relevance logics (...) and relation algebras. Some details of certain incompleteness results, however, pinpoint where relevance logics and relation algebras diverge. To carry out these semantic investigations, we define a new tableaux formalization and new sequent calculi (with the single cut rule admissible) for various relevance logics. (shrink)

Combinatory logic is known to be related to substructural logics. Algebraic considerations of the latter, in particular, algebraic considerations of two distinct implications, led to the introduction of dual combinators in Dunn & Meyer 1997. Dual combinators are "mirror images" of the usual combinators and as such do not constitute an interesting subject of investigation by themselves. However, when combined with the usual combinators, the whole system exhibits new features. A dual combinatory system with weak equality typically lacks the Church-Rosser (...) property, and in general it is inconsistent. In many subsystems terms "unexpectedly" turn out to be weakly equivalent. The paper is a preliminary attempt to investigate some of these issues, as well as, briefly compare function application in symmetric -calculus and dual combinatory logic. (shrink)

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional (...) inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ LK $$\end{document} without any use of mix. (shrink)

Mitos ve Logos kavramları Antik Yunan uygarlığında söz kavramına karşılık gelen sözcükleri karşılamak için kullanılmıştır. Felsefe tarihinin başlangıcı için yapılan tanımlamalarda ise mitos kavramının yerine logos kavramının tercih edilmesi iki kavram arasında bir farklılığı ortaya koymak için yapılmaktadır. Bu ayrımın nedeni ise mitos’un daha çok dinsel içerikle anılması logos’un ise içerisinde bir tür akılsallık barındırması şeklindeki yorumlarda kendini göstermektedir. Ancak söz konusu ayrımın ilk doğa filozofu/ilk felsefeci olarak nitelendirilen Thales için geçerli olup olmadığı geçmişte olduğu gibi günümüzde de halen tartışılmaktadır. (...) Felsefe tarihi yazımlarında ilk sıraya konan Thales’in anılan kavramları gerçekten farklı şekilde yorumlayıp yorumlamadığı bizce de tartışmaya açık bir husustur. Bu açıdan bakıldığında Thales, söz konusu kavramları iki farklı problemler alanı olarak mı görmüştür? Onun düşüncelerinde kendinden önceki mitolojik mirasın izleri bulunabilir mi? O, katışıksız bir rasyonel midir, yoksa henüz felsefenin yeni filizlendiği bir toplumda bir ara basamağı mı temsil etmektedir? Bu makalenin içerisinde tartışacağımız bu sorular, felsefe tarihinin başlangıcına ilişkin kavramsal arka planın ne olduğu hakkında bize katkı sağlayacaktır. (shrink)

We briefly overview some of the historical landmarks on the path leading to the reduction of the number of logical connectives in classical logic. Relying on the duality inherent in Boolean algebras, we introduce a new operator ( Nallor ) that is the dual of Schönfinkel’s operator. We outline the proof that this operator by itself is sufficient to define all the connectives and operators of classical first-order logic ( Fol ). Having scrutinized the proof, we pinpoint the theorems of (...) Fol that are needed in the proof. Using the insights gained from the proof, we show that there are four binary operators that each can serve as the only undefined logical constant for Fol . Finally, we show that from every n -ary connective that yields a functionally complete singleton set of connectives two Schönfinkel-type operators are definable, and all the latter are so definable. (shrink)