SUMMARY1 The rational soul becomes the constant and dimensionless Euclidean point in all experience - defining the situations in which it finds itself, but itself undefined and undefinable in any situation. It is in nature but not of nature. Just as the dimensionless Euclidean point can occupy infinite positions on a line and yet remain unaltered, so the immortal, active intellect remains unaffected by the world in which it finds itself. It is not influenced by age, sense data, sickness or (...) death. It is the knower whose nature is pure knowing or reason. This is in great part the basic assumption that runs through the history of Aristotelean and modern symbolic logics24.The rational soul grasps identity, reasons consistently, and knows objectively for though it is in the world it is not of the world. Thus it is unnecessary and impossible to relate its truth to time or place or condition. As the source of understanding it need not be understood. It understands in that it knows. It is this which places the objective knower and observer outside of all systems of knowledge that it creates in philosophy, science and theology, and frees it from subjective interpretation. It is the absolute observer beyond self-observation.252 Language, values, and meanings reflect the absolute viewpoint of the dimensionless observer beyond the necessity of observation. The task of this observer is to know trie world just as the task of the animal and plant soul is motion and reproduction. The rational soul performs its task beyond nature as surely as animals and plants perform their tasks in nature. This is why it is possible to discuss the necessary evolution of living organisms in nature, but unnecessary to discuss the necessary or unnecessary nature of the evolution of this discussion. This is one of the reasons why determinism or predestination which seem so clearly applicable to nature, need not be applied to theories of determinism and predestination. These are principles grasped in reality beyond nature and totally inapplicable to the non-natural spirit that comprehends their essences. Thus self-reflexive questions about the questioner are as irrelevant as would be questions historically of “in what sense might the definition of dimensionless point itself be dimensioned or dimension-less?”26 Questions, on the other hand, that relate to the observed world are important for they can be truly answered (where questions as to the nature of the questioner cannot). In this view reality is not nature, but rather it is from reality that the objective observer, philosopher, scientist, and theologian understands and controls nature.3 Logic, the science of reason, is the science of all sciences. It reflects the pure and innate rational nature of the knowing intellect or soul. It is pure form beyond nature. Its realm is that of first principles or axioms. These too are in the world but not of it, as with Plato's eidos and Aristotle's universals. They are clear, true and self-evident. This explains the on going use of axiomatized revelations in studies in the natural and social sciences, the arts and the humanities. In symbolic logics, natural languages are in and of the world, but meta-languages are the constructs of pure intellectual activity and thus not natural. The abstract tends to become the more real, the less it can be found in nature. Rational souls, however, share a common reality that is as universal for Frege as it was for Aristotle.274 The occidental object tradition in general breaks out of the nature continuum by withdrawing into reality where the intellect knows and manipulates the world in philosophy, science, and theology from its position in and understanding of reality. Nature is a sub-set of reality and not vice versa. Axioms are the laws and determinants of nature and logic, but they are not nature.28 This means that at the basis of the occidental continuum of nature there is a qualitative and irreducible dichotomy of reality vs. nature. This has been most useful in the history of occidental science, but it produces a world view in which nature may lose the race with reality. SUMMARY IIIn the first portion of my paper, I discussed the implications of the Euclidean egg for philosophy and science as nature is understood and manipulated from reality. I shall now turn to what I see as the implications of the three-legged chicken view that the human intellect is in and of nature and that reality is a perspective in nature and is not simply the other of an irreducible qualitative dichotomy. Although there are an endless number of such implications that I could explore, the following, it seems to me are the most important:1 In the natural continuum there is no “dimensionless real” which can serve either as the locus of an external observer or a fixed position in nature. There is no perspective which is not an aspect of that upon which it is a perspective. The Chinese sage, for example, is one who sees himself/herself as one with or within natural process. This is expressed in a number of ways. Fung Yu-lan speaks of the difference between having no knowledge and having no-knowledge.34 Within such a continuum there can be no abstracting out of the absolute and unchanging. The nameless is the source of the named. First principles are the process itself and these principles are the alternation of opposites:“The Nameless is the origin of Heaven and Earth;The Named is the mother of all things.Therefore let there always be non-being so we may see their subtlety, And let there always be being so we may see their outcome. The two are the same.“352 An empty mind is not a blank mind. Where Aristotle viewed the mind as a blank tablet at birth to be filled by sense experience, Hsun Tzu speaks of its emptiness. It is the emptiness of the Tao itself:“The mind is constantly storing up things, and yet it is said to be empty. The mind is constantly marked by diversity, and yet it is said to be unified. The mind is constantly moving, and yet it is said to be still. Inevitably it will see things for itself. And although the objects it perceives may be many and diverse, if its acuity is of the highest level, it cannot become divided within itself.”36The nature in which and of which the mind is both perspective and aspect remains a totality. Being a totality, it is empty of itself just as any part of a whole, while a part of the whole, is empty of the whole. My ideas of myself are both of myself and empty of myself. This point can be illustrated with the simple example of knowing and speaking about two hands – one of which is mine, and the second of which belongs to someone else. 1 can objectify the second hand and fill my blank mind with information and observations about it, but I continue to remain outside of it –the arbiter in my statements of its nature and reality. However, in knowing my own hand, the hand known is also the knower and what I know represents a perspective upon an aspective of a totality of which “I–my–hand” are aspects. Being an aspect of the whole is being empty of the whole. If mind is in and of nature, any perspective on nature is an aspect of the whole. A blank mind is an unnatural mind outside of nature, filling up with impressions of nature. An empty mind is nature viewing nature as aspect and perspective. The occidental would be ashamed to reach old age and still have a blank mind. The Chinese sage would be ashamed to reach old age and still not recognize that the mind is empty.3 If mind is in nature and like nature, then language too reflects natural flow and change and process. It is one thing for me to talk about myself and another for me to talk about someone else. In the case of the latter, I may insist that objectively this other subject fits into this or that category and even administer tests to determine that it does do indeed so. But when I talk about myself, the speaker and the object of speech are one and the same. The terms I select do represent a dichotomy of sorts, but it is one of aspect and perspective and not subject and object. When I say “this is my hand” for example, the words create distinctions which are aspects of awareness but do not create an / that is or is not my hand. Language does not model or even represent reality as opposed to nature. It makes reality possible as something that nature says or thinks. Reality through language is an aspect of nature and thus a perspective upon itself as aspect. This is an important distinction that translators often miss.4 Logic in this natural continuum is concerned with distinctions in a very different sense. As a consequence of the unity of nature and mind, three kinds of distinction in experience are manifested: (a) Those distinctions which come into being as aspects of the changing world.The 10,000 things are in this sense aspects and not separate “identities” as some philosophers and scientists presuppose.37 (b) Those distinctions which come into being as aspects of the human mind. These are impressions of and thoughts about the infinite distinctions of which the restless mind is capable, (c) Those distinctions which come into being by way of the mind's use of symbols. These are the distinctions which permit us to stabilize and store our experiences, preserve the answers to our questions, and name the world into compartments and relationships.Implicit in these three assumptions which connect the distinguishing mind to the unity of a nature of'10,000‘ aspects, is an inferential process of aspect to perspective to knowledge to understanding to attitude to action/response. The reasoning mind synthesizes some aspect of experience in nature with a perspective upon this aspect. This synthesis, in turn, is an aspect of mind and the reasoning process proceeds from it to a perspective upon it which is knowledge and understanding. Knowledge and understanding in turn are aspects of thinking and the reasoning process proceeds from them to a perspective upon them which is an attitude. Attitude in turn is an aspect of thinking and the reasoning process proceeds from it to a perspective upon it which is reflected in the reasoning individual's actions and responses. This inferential process is a continuum which proceeds from aspects of experience to perspectives in thought to action and response. In this sequence an aspect is always connected by a perspective which becomes an aspect, until action/non-action, response occurs. This is as applicable to the details of life as it is to mastering life as a whole. When this inference process is incomplete many problems may result, ranging from action without understanding to the confusion of aspect and perspective with one another or to “obsession with distinctions.”38In practice this is a logic of paradox. The human being is aware that he or she is a living-dying aspect of nature, but an aspect which is also a perspective upon nature's aspects of living and dying. This paradox is the process around which a profound logic of aspect/perspective is developed by the sage. This view of logic within a continuum does not mean that there is no principle of identity, contradiction or excluded middle present – as some Chinese and European philosophers assume.These principles are also implicit in the knowing of my hand versus the knowing of the hand of another. However, the content, meaning, and application of these principles may vary just as when I say propriety, I mean propriety, and when the Chinese says li he means li but the context, meaning and application of the term may differ in each case.5 Aspect logic as it relates mind to nature in a cosmological totality is a series of syntheses in a cyclic continuum organizing thinking in phases. Each phase is an aspect/perspective synthesis which becomes an aspect of a further aspect/perspective synthesis. This progression might be described thusly: The synthesis of nature/human which is Tao (non-being) translates into the aspects of being (the 10,000 things), and as such is a synthesis of aspect and perspective. This synthesis of aspect and perspective translates into an aspect of ideas and as such is a synthesis of perception and conception. This synthesis of perception and conception translates into an aspect of knowledge, and as such is a synthesis of knowledge and understanding (understanding is a perspective upon knowledge) i.e., no-knowledge. This synthesis of knowledge and understanding translates into an aspect of purpose, and as such is a synthesis of purpose and action. The perspective here on action is aesthetic and/or moral. This synthesis of purpose and actions translates again into an aspect of nature and as such is a synthesis beyond individual aspects and perspective, i.e. Tao. Thus there is a cyclic dynamic in the system which remains open to further development.Tai Chen in his Inquiry Into Goodness connects this cycle of syntheses with wisdom and benevolence:Because there are mind and intelligence, the superior man knows that the perfection of the Way of Man depends completely upon nature. When he looks at the signs of the natural (Cheng), he knows the beginning of things; when he cherishes the virtue of supreme illumination, (Shen-ming), he knows the end of things. Proceeding from mind and intelligence to the attainment of the virtue of supreme intelligence, [the superior man] in his treatment of things in the world will cause the world to return to [the principles of] benevolence; and in his treatment of the capability inherent in things, he will cause the world to follow wisdom (Chih).39. (shrink)

After a discussion of the different treatments in the literature of vacuous descriptions, the notion of descriptor is slightly generalized to function descriptor Ⅎ $\overset \rightarrow \to{y}$, so as to form partial functions φ = Ⅎ $y.A$ which satisfy $\forall \overset \rightarrow \to{x}z\leftrightarrow y=z))$. We use logic with existence predicate, as introduced by D. S. Scott, to handle partial functions, and prove that adding function descriptors to a theory based on such a logic is conservative. For theories with quantification over (...) functions, the situation is different: there the addition of Ⅎ yields new theorems in the Ⅎ-free fragment, but an axiomatisation is easily given. The proofs are syntactical. (shrink)

This paper is an investigation of the general logic of "identifications", claims such as 'To be a vixen is to be a female fox', 'To be human is to be a rational animal', and 'To be just is to help one's friends and harm one's enemies', many of which are of great importance to philosophers. I advocate understanding such claims as expressing higher-order identity, and discuss a variety of different general laws which they might be thought to obey. [New version: (...) Nov. 4th, 2016]. (shrink)

A categorical, higher dimensional algebra and generalized topos framework for Łukasiewicz–Moisil Algebraic–Logic models of non-linear dynamics in complex functional genomes and cell interactomes is proposed. Łukasiewicz–Moisil Algebraic–Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of non-linear dynamic systems with n-state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable ‘next-state functions’ is extended to a Łukasiewicz–Moisil (...) Topos with an n-valued Łukasiewicz–Moisil Algebraic Logic subobject classifier description that represents non-random and non-linear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen’s (M,R)-systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occurring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, non-commutative quantum logics are considered in the context of an emerging Quantum Relational Biology. (shrink)

External obstacles to properly understanding Wittgenstein’s philosophy of mathematics are not lacking, either. For one thing, there is the piecemeal way that his mathematical manuscripts have been made available. The editors of Remarks on the Foundations of Mathematics write that “the time has not yet come to print the whole of Wittgenstein’s MSS on these... topics”. One wonders what sorts of reasons there could be for that editorial choice.

There are many wonderful puzzles concerning Principia Mathematica, but none are more striking than those arising from the crisis that befell Whitehead in November of 1910. Volume 1 appeared in December of 1910. Volume 2 on cardinal numbers and Russell's relation arithmetic might have appeared in 1911 but for Whitehead's having halted the printing. He discovered that inferences involving the typically ambiguous notation ‘Nc‘α’ for the cardinal number of α might generate fallacies. When the volume appeared in 1912, it was (...) extensively emended by Whitehead and accompanied by a Prefatory Statement of Symbolic Conventions. This paper endeavors to recover from Whitehead's bad emendations—including his bewildering thesis that since ‘‘α’ is ‘true whenever significant,’ ‘α is to be accepted. It is supposedly a fallacy to apply Modus Ponens and infer Nc‘α from ‘α and‘‘α. (shrink)

Frege's philosophical writings, including the “logistic project,” acquire a new insight by being confronted with Kant's criticism and Wittgenstein's logical and grammatical investigations. Between these two points a non-formalist history of logic is just taking shape, a history emphasizing the Greek and Kantian inheritance and its aftermath. It allows us to understand the radical change in rationality introduced by Gottlob Frege's syntax. This syntax put an end to Greek categorization and opened the way to the multiplicity of expressions producing their (...) own intelligibility. This article is based on more technical analyses of Frege which Claude Imbert has previously offered in other writings. (shrink)

Frege’s _Basic Law V_ is inconsistent. The reason often given is that it posits the existence of an injection from the larger collection of first-order concepts to the smaller collection of objects. This article explains what is right and what is wrong with this diagnosis.

Consider the following sentence schema:This sentence entails that ϕ.Call a sentence which is obtained from this schema by the substitution of an arbitrary, contingent sentence, s, for ϕ, the sentence CS. Thus, This sentence entails that s.Now ask the following question: Is CS true?One sentence classically entails a second if and only if it is impossible for both the first to be true and the second to be false. Thus ‘Xanthippe is a mother’ entails ‘Xanthippe is female’ if and only (...) if it is impossible for both ‘Xanthippe is a mother’ to be true and ‘Xanthippe is female’ to be false. CS makes a claim about a purported entailment. Thus, CS is true if and only if it is impossible for both the sentence it mentions as entailing a second to be true and the sentence it mentions as being entailed by the first to be false. In other words, CS is true if and only if it is impossible for both CS to be true and s to be false. In yet other words, CS is false if and only if it is possible for both CS to be true and s to be false. (shrink)

The ArgumentRecent studies in the philosophy of mathematics have increasingly stressed the social and historical dimensions of mathematical practice. Although this new emphasis has fathered interesting new perspectives, it has also blurred the distinction between mathematics and other scientific fields. This distinction can be clarified by examining the special interaction of thebodyandimagesof mathematics.Mathematics has an objective, ever-expanding hard core, the growth of which is conditioned by socially and historically determined images of mathematics. Mathematics also has reflexive capacities unlike those of (...) any other exact science. In no other exact science can the standard methodological framework usedwithinthe discipline also be used to study the nature of the discipline itself.Although it has always been present in mathematical research, reflexive thinking has become increasingly central to mathematics over the past century. Many of the images of the discipline have been dictated by the increase in reflexive thinking which has also determined a great portion of the contemporary philosophy and historiography of mathematics. (shrink)

The concept of quantity (Größe) plays a key role in Frege's theory of real numbers. Typically enough, he refers to this theory as ?theory of quantity? (?Größenlehre?) in the second volume of his opus magnum Grundgesetze der Arithmetik (Frege 1903). In this essay, I deal, in a critical way, with Frege's treatment of the concept of quantity and his approach to analysis from the beginning of his academic career until Frege 1903. I begin with a few introductory remarks. In Section (...) 2, I first analyze Frege's use of the term ?source of knowledge? (?Erkenntnisquelle?) with particular emphasis on the logical source of knowledge. The analysis includes a brief comparison between Frege and Kant's conceptions of logic and the logical source of knowledge. In a second step, I examine Frege's theory of quantity in Rechnungsmethoden, die sich auf eine Erweiterung des Größenbegriffes gründen (Frege 1874). Section 3 contains a couple of critical observations on Frege's comments on Hankel's theory of real numbers in Die Grundlagen der Arithmetik (Frege 1884). In Section 4, I consider Frege's discussion of the concept of quantity in Frege 1903. Section 5 is devoted to Cantor's theory of irrational numbers and the critique deployed by Frege. In Section 6, I return to Frege's own constructive treatment of analysis in Frege 1903 and succinctly describe what I take to be the quintessence of his account. (shrink)

I argue that three main interpretations of the aim of Russell’s early logicism in The Principles of Mathematics (1903) are mistaken, and propose a new interpretation. According to this new interpretation, the aim of Russell’s logicism is to show, in opposition to Kant, that mathematical propositions have a certain sort of complete generality which entails that their truth is independent of space and time. I argue that on this interpretation two often-heard objections to Russell’s logicism, deriving from Gödel’s incompleteness theorem (...) and from the non-logical character of some of the axioms of Principia Mathematica respectively, can be seen to be inconclusive. I then proceed to identify two challenges that Russell’s logicism, as presently construed, faces, but argue that these challenges do not appear unanswerable. (shrink)

The paper offers a historical survey of the emergence of logical formalization in twentieth-century analytically oriented philosophy of religion. This development is taken to have passed through three main ?stages?: a pioneering stage in the late nineteenth and early twentieth centuries (led by Frege and Russell), a stage of crisis in the 1920s and early 1930s (occasioned by Wittgenstein, logical positivists such as Carnap, and neo-Thomists such as Maritain), and a stage of rehabilitation in the 1930s, 1940s, and 1950s (led (...) by the Cracow Circle and Quine). (shrink)

We re-examine the problem of existential import by using classical predicate logic. Our problem is: How to distribute the existential import among the quantified propositions in order for all the relations of the logical square to be valid? After defining existential import and scrutinizing the available solutions, we distinguish between three possible cases: explicit import, implicit non-import, explicit negative import and formalize the propositions accordingly. Then, we examine the 16 combinations between the 8 propositions having the first two kinds of (...) import, the third one being trivial and rule out the squares where at least one relation does not hold. This leads to the following results: (1) three squares are valid when the domain is non-empty; (2) one of them is valid even in the empty domain: the square can thus be saved in arbitrary domains and (3) the aforementioned eight propositions give rise to a cube, which contains two more (non-classical) valid squares and several hexagons. A classical solution to the problem of existential import is thus possible, without resorting to deviant systems and merely relying upon the symbolism of First-order Logic (FOL). Aristotle’s system appears then as a fragment of a broader system which can be developed by using FOL. (shrink)

I offer you some theories of intellectual obligations and rights (virtue Ethics): initially, RBT (a Right to Believe Truth, if something is true it follows one has a right to believe it), and, NDSM (one has no right to believe a contradiction, i.e., No right to commit Doxastic Self-Mutilation). Evidence for both below. Anthropology, Psychology, computer software, Sociology, and the neurosciences prove things about human beliefs, and History, Economics, and comparative law can provide evidence of value about theories of rights. (...) However, insofar as we have methods within Philosophy to help us formulate precise concepts of 'belief' and 'rights', methods that also help us to prove links (or absence thereof) amongst families of concepts of rights and belief, our discipline is in and of itself capable of sound reasoning about issues as puzzling as the following. Suppose a Jane who does not believe in God yet who believes she ought to so believe: Jane is undergoing doxastic moral regret (moral regret for lack of faith). We have all known such Janes and perhaps at one time or another even been one. Paradox: given RBT and NDSM, Jane as described not only does not exist, Jane cannot exist. Thus, to enrich the ways in which Philosophy need not get all its evidence from other academic disciplines, I present a brief introduction to what I call Neutral Universal Frames (NUFs). NUFs solve hard puzzles about interactions among modal concepts of belief and rights, concepts that occur in RBT, NDSM and the description of our Janes. NUFs for theories precisely articulated via any two or more modal concepts are a powerful and immensely general set of tools enabling us to define rich theories of truth ("models on frames") to test philosophical theories for internal consistency and to prove the existence of connections (or absence thereof) amongst alternative articulations of philosophical theories. NUFs thereby add to the constructive knowledge producing way Logics intersect with Philosophy of Religion. And we will soon see why Jane, be she named 'Jane' or known simply as you: cannot exist. Read on at your own risk. (shrink)

To define new property terms, we combine already familiar ones by means of certain logical operations. Given suitable constraints, these operations may presumably include the resources of first-order logic: truth-functional sentence connectives and quantification over objects. What is far less clear is whether we can also use modal operators for this purpose. This paper clarifies what is involved in this question, and argues in favor of modal property definitions.

Many commentators have attempted to say, more clearly than Wittgenstein did in his Tractatus logico-philosophicus, what sort of things the ‘simple objects’ spoken of in that book are. A minority approach, but in my view the correct one, is to reject all such attempts as misplaced. The Tractarian notion of an object is categorially indeterminate: in contrast with both Frege's and Russell's practice, it is not the logician's task to give a specific categorial account of the internal structure of elementary (...) propositions or atomic facts, nor, correlatively, to give an account of the forms of simple objects. The few commentators who have hitherto maintained this view have mainly devoted themselves to establishing that this was Wittgenstein's intention, and do not much address the question why Wittgenstein held that it is not the logician's business to say what the objects are. The present paper means to fill this lacuna by placing this view in the context of the Tractatus's treatment of logic generally, and in particular by connecting it with Wittgenstein's treatment of generality and with his reaction to Russell's approach to logical form. (shrink)

The paper surveys two contrasting views of first‐order analyses of classical theistic doctrines about the existence and nature of God. On the first view, first‐order logic provides methods for the adequate analysis of these doctrines, for example by construing ‘God’ as a singular term or as a monadic predicate, or by taking it to be a definite description. On the second view, such analyses are conceptually inadequate, at least when the doctrines in question are viewed against the background of classical (...) theism’s doctrine of divine simplicity, for first‐order analyses presuppose an ontological complexity on the part of the propositions which they analyze, which fits ill with this doctrine. (shrink)

It is argued that the finitist interpretation of wittgenstein fails to take seriously his claim that philosophy is a descriptive activity. Wittgenstein's concentration on relatively simple mathematical examples is not to be explained in terms of finitism, But rather in terms of the fact that with them the central philosophical task of a clear 'ubersicht' of its subject matter is more tractable than with more complex mathematics. Other aspects of wittgenstein's philosophy of mathematics are touched on: his view that mathematical (...) propositions are 'grammatical propositions' (and so, Strictly speaking, Not genuine propositions at all) and his view that in mathematics 'everything is algorithm, Nothing meaning'. His views on consistency and his anti-Foundationalism are linked with this central thesis. (shrink)

The reflections recorded in this paper arise from three moments in the theory of definition and of conceptual analysis. The moments are: Frege’s review of Husserl’s Philosophy of Arithmetic, the discussion there of the paradox of analysis, and the division that Frege marks, ensuing upon his distinction of Sinn/sense from Bedeutung/reference, between two different conceptions of definition; Leibniz’s still serviceable account of a distinction between the clarity and the distinctness of ideas---a distinction that prompts the suggestion that the guiding purpose (...) of lexical definition is Leibnizian clarity whereas that of real definition is inseparable from the pursuit of Leibnizian distinctness; Leibniz’s speculations concerning the limit or terminus of analysis. The apparent failure of these speculations, casting doubt as it does upon the aspirations that give rise to them, points to the long-standing need to reconfigure the philosophical business of enquiry into concepts. (shrink)

In this paper, I argue that there are universals. I begin (Sect. 1) by proposing a sufficient condition for a thing’s being a universal. I then argue (Sect. 2) that some truths exist necessarily. Finally, I argue (Sects. 3 and 4) that these truths are structured entities having constituents that meet the proposed sufficient condition for being universals.

In 1880, when Oliver Wendell Holmes (later to be a Justice of the U.S. Supreme Court) criticized the logical theology of law articulated by Christopher Columbus Langdell (the first Dean of Harvard Law School), neither Holmes nor Langdell was aware of the revolution in logic that had begun, the year before, with Frege's Begriffsschrift. But there is an important element of truth in Holmes's insistence that a legal system cannot be adequately understood as a system of axioms and corollaries; and (...) this element of truth is not obviated by the more powerful logical techniques that are now available. (shrink)

If quantum mechanics (QM) is to be taken as an atomistic theory with the elementary particles as atoms (an ATEP), then the elementary particlcs must be individuals. There must then be, for each elementary particle a, a property being identical with a that a alone has. But according to QM, elementary particles of the same kind share all physical properties. Thus, if QM is an ATEP, identity is a metaphysical but not a physical property. That has unpalatable consequences. Dropping the (...) assumption that QM is an ATEP makes it possible to replace the assumption that elementary particles are individuals with the assumption that there are various kinds of elementary stuff that have smallest quantities — the smallest quantity of light, for example, is a photon. The problems about identity disappear, and the explanatory virtues of an ATEP are maintained. (shrink)

The paper describes the evolution of russell's theory of judgment between 1910 and 1913, With especial reference to his recently published "theory of knowledge" (1913). Russell abandoned the book and with it the theory of judgment as a result of wittgenstein's criticisms. These criticisms are examined in detail and found to constitute a refutation of russell's theory. Underlying differences between wittgenstein's and russell's views on logic are broached more sketchily.

In this paper we define intensional models for the classical theory of types, thus arriving at an intensional type logic ITL. Intensional models generalize Henkin's general models and have a natural definition. As a class they do not validate the axiom of Extensionality. We give a cut-free sequent calculus for type theory and show completeness of this calculus with respect to the class of intensional models via a model existence theorem. After this we turn our attention to applications. Firstly, it (...) is argued that, since ITL is truly intensional, it can be used to model ascriptions of propositional attitude without predicting logical omniscience. In order to illustrate this a small fragment of English is defined and provided with an ITL semantics. Secondly, it is shown that ITL models contain certain objects that can be identified with possible worlds. Essential elements of modal logic become available within classical type theory once the axiom of Extensionality is given up. (shrink)

The paper critically scrutinizes the widespread idea that Russell subscribes to a "Universalist Conception of Logic." Various glosses on this somewhat under-explained slogan are considered, and their fit with Russell's texts and logical practice examined. The results of this investigation are, for the most part, unfavorable to the Universalist interpretation.

A number of authors in favor of a unitary account of singular descriptions have alleged that the unitary account can be extrapolated to account for plural definite descriptions. In this paper I take a closer look at this suggestion. I argue that while the unitary account is clearly onto something right, it is in the end empirically inadequate. At the end of the paper I offer a new partitive account of plural definite descriptions that avoids the problems with both the (...) unitary account and standard Russellian analyses. (shrink)

A novel conceptual framework is introduced for the Complexity Levels Theory in a Categorical Ontology of Space and Time. This conceptual and formal construction is intended for ontological studies of Emergent Biosystems, Super-complex Dynamics, Evolution and Human Consciousness. A claim is defended concerning the universal representation of an item’s essence in categorical terms. As an essential example, relational structures of living organisms are well represented by applying the important categorical concept of natural transformations to biomolecular reactions and relational structures that (...) emerge from the latter in living systems. Thus, several relational theories of living systems can be represented by natural transformations of organismic, relational structures. The ascent of man and other living organisms through adaptation, is viewed in novel categorical terms, such as variable biogroupoid representations of evolving species. Such precise but flexible evolutionary concepts will allow the further development of the unifying theme of local-to-global approaches to highly complex systems in order to represent novel patterns of relations that emerge in super- and ultra-complex systems in terms of compositions of local procedures. Solutions to such local-to-global problems in highly complex systems with ‘broken symmetry’ might be possible to be reached with the help of higher homotopy theorems in algebraic topology such as the generalized van Kampen theorems (HHvKT). Categories of many-valued, Łukasiewicz-Moisil (LM) logic algebras provide useful concepts for representing the intrinsic dynamic ‘asymmetry’ of genetic networks in organismic development and evolution, as well as to derive novel results for (non-commutative) Quantum Logics. Furthermore, as recently pointed out by Baianu and Poli (Theory and applications of ontology, vol 1. Springer, Berlin, in press), LM-logic algebras may also provide the appropriate framework for future developments of the ontological theory of levels with its complex/entangled/intertwined ramifications in psychology, sociology and ecology. As shown in the preceding two papers in this issue, a paradigm shift towards non-commutative, or non-Abelian, theories of highly complex dynamics—which is presently unfolding in physics, mathematics, life and cognitive sciences—may be implemented through realizations of higher dimensional algebras in neurosciences and psychology, as well as in human genomics, bioinformatics and interactomics. (shrink)

Kripke's lectures, published as Wittgenstein on Rules and Private Language , posed a sceptical problem about following a rule, which he cautiously attributed to Wittgenstein. He briefly noticed an analogy between his new kind of scepticism and Goodman's riddle of induction. ‘Grue’, he said, could be used to formulate a question not about induction but about meaning: the problem would not be Goodman's about induction—‘Why not predict that grass, which has been grue in the past, will be grue in the (...) future?’—but Wittgenstein's about meaning: ‘Who is to say that in the past I did not mean grue by “green”, so that now I should call the sky, not the grass, “green”?’. (shrink)

In this paper I show that one of the most fruitful ways of employing paradoxes has been as a philosophical method that forces us to reconsider basic assumptions. After a brief discussion of recent understandings of the notion of paradoxes, I show that Zeno of Elea was the inventor of paradoxes in this sense, against the background of Heraclitus’ and Parmenides’ way of argumentation: in contrast to Heraclitus, Zeno’s paradoxes do not ask us to embrace a paradoxical reality; and in (...) contrast to Parmenides, Zeno shows common assumptions to be internally problematic, not just in light of Eleatic positions. (shrink)

The use of the symbol ∨for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol ∨ in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel. We show that the origin of the symbol ∨ for disjunction can be traced to Whitehead and (...) Russell’s pre-Principia work in formal logic. Because of Principia’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of ∨ in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed. (shrink)

In this article I describe the contributions made by Samuel Alexander to the issue of relations which so vexed Bertrand Russell and F. H. Bradley in the late nineteenth and early twentieth centuries. I provide a novel understanding of Alexander’s position concerning relations and describe the way in which he viewed his position as superior to those of Bradley and Russell. I offer, therefore, a more complete picture of a philosophical debate central to the relevant period, through the introduction of (...) a third character whose views have been largely omitted from available narratives. (shrink)

The paper provides a proof theoretic characterization of the Russellian theory of definite descriptions (RDD) as characterized by Kalish, Montague and Mar (KMM). To this effect three sequent calculi are introduced: LKID0, LKID1 and LKID2. LKID0 is an auxiliary system which is easily shown to be equivalent to KMM. The main research is devoted to LKID1 and LKID2. The former is simpler in the sense of having smaller number of rules and, after small change, satisfies cut elimination but fails to (...) satisfy the subformula property. In LKID2 an additional analysis of different kinds of identities leads to proliferation of rules but yields the subformula property. This refined proof theoretic analysis leading to fully analytic calculus with constructive proof of cut elimination is the main contribution of the paper. (shrink)

Standard histories of mathematics and of analytic philosophy contend that work on the foundations of mathematics was motivated by a crisis such as the discovery of paradoxes in set theory or the discovery of non-Euclidean geometries. Recent scholarship, however, casts doubt on the standard histories, opening the way for consideration of an alternative motive for the study of the foundations of mathematics—unification. Work on foundations has shown that diverse mathematical practices could be integrated into a single framework of axiomatic systems (...) and that much of mathematics could be expressed in a single language. The new framework was the product of an interdisciplinary coalition whose ideas resemble those later adopted by the Vienna Circle and logical empiricists. (shrink)

In a recent book, Robert Trueman develops a version of the identity theory of truth, the theory that true propositions are not in some kind of correspondence with, but are rather identical with, facts. He claims that this theory ‘collapses the gap between mind and world’. Whether it does so will obviously depend on how the theory is to be understood, which in turn depends on the argumentative route to it. Trueman’s route is clear, rigorous, and free of extravagant assumptions. (...) Perhaps because of those merits it seems obvious that it falls short of the claim he makes for it. But there are difficult questions about the nature of the shortfall and about what in the character of Trueman’s philosophical approach prevents him from appreciating it. The paper explores those questions through a comparison with Moore’s ‘original’ identity theory and the Idealist philosophy he directed it against. (shrink)

The Hilbert program was actually a specific approach for proving consistency, a kind of constructive model theory. Quantifiers were supposed to be replaced by ε-terms. εxA(x) was supposed to denote a witness to ∃xA(x), or something arbitrary if there is none. The Hilbertians claimed that in any proof in a number-theoretic system S, each ε-term can be replaced by a numeral, making each line provable and true. This implies that S must not only be consistent, but also 1-consistent. Here we (...) show that if the result is supposed to be provable within S, a statement about all Pi-0-2 statements that subsumes itself within its own scope must be provable, yielding a contradiction. The result resembles Gödel's but arises naturally out of the Hilbert program itself. (shrink)

Conceptual engineers have made hay over the differences of their metaphilosophy from those of conceptual analysts. In this article, I argue that the differences are not as great as conceptual engineers have, perhaps rhetorically, made them seem. That is, conceptual analysts asking ‘What is X?’ questions can do much the same work that conceptual engineers can do with ‘What is X for?’ questions, at least if conceptual analysts self-understand their activity as a revisionary enterprise. I show this with a study (...) of Russell's metaphilosophy, which was just such a revisionary conception of conceptual analysis. (shrink)

In this paper I discuss Ernst Zermelo’s ideas concerning the possibility of developing a system of infinitary logic that, in his opinion, should be suitable for mathematical inferences. The presentation of Zermelo’s ideas is accompanied with some remarks concerning the development of infinitary logic. I also stress the fact that the second axiomatization of set theory provided by Zermelo in 1930 involved the use of extremal axioms of a very specific sort.1.

The paper reviews some conceptions of logical form in the light of Andrea Iacona’s book Logical Form. I distinguish the following: logical form as schematization of natural language, provided by, for example, Aristotle’s syllogistic; the relevance to logical form of formal languages like those used by Frege and Russell to express and prove mathematical theorems; Russell’s mid-period conception of logical form as the structural cement binding propositions; the conceptions of logical form discussed by Iacona; and logical form regarded as an (...) empirical hypothesis about the psychology of language processing, as in the Discourse Representation Theory tradition. Whereas neither schematization, nor the use of special languages for mathematics, raise general methodological or empirical difficulties, other conceptions of logical form raise at least apparent problems. (shrink)