Chapin reviewed this 1972 ZEITSCHRIFT paper that proves the completeness theorem for the logic of variable-binding-term operators created by Corcoran and his student John Herring in the 1971 LOGIQUE ET ANALYSE paper in which the theorem was conjectured. This leveraging proof extends completeness of ordinary first-order logic to the extension with vbtos. Newton da Costa independently proved the same theorem about the same time using a Henkin-type proof. This 1972 paper builds on the 1971 “Notes on a (...) Semantic Analysis of Variable Binding Term Operators” (Co-author John Herring), Logique et Analyse 55, 646–57. MR0307874 (46 #6989). A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. Kalish-Montague 1964 proposed using vbtos to formalize definite descriptions “the x: x+x=2”, set abstracts {x: F}, minimization in recursive function theory “the least x: x+x>2”, etc. However, they gave no semantics for vbtos. Hatcher 1968 gave a semantics but one that has flaws described in the 1971 paper and admitted by Hatcher. In 1971 we give a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture, proved in this paper with Hatcher’s help, was proved independently about the same time by Newton da Costa. (shrink)
In this paper I will develop a view about the semantics of imperatives, which I term Modal Noncognitivism, on which imperatives might be said to have truth conditions (dispositionally, anyway), but on which it does not make sense to see them as expressing propositions (hence does not make sense to ascribe to them truth or falsity). This view stands against “Cognitivist” accounts of the semantics of imperatives, on which imperatives are claimed to express propositions, which are then enlisted in (...) explanations of the relevant logico-semantic phenomena. It also stands against the major competitors to Cognitivist accounts—all of which are non-truth-conditional and, as a result, fail to provide satisfying explanations of the fundamental semantic characteristics of imperatives (or so I argue). The view of imperatives I defend here improves on various treatments of imperatives on the market in giving an empirically and theoretically adequate account of their semantics and logic. It yields explanations of a wide range of semantic and logical phenomena about imperatives—explanations that are, I argue, at least as satisfying as the sorts of explanations of semantic and logical phenomena familiar from truth-conditional semantics. But it accomplishes this while defending the notion—which is, I argue, substantially correct—that imperatives could not have propositions, or truth conditions, as their meanings. (shrink)
Anti-exceptionalism about logic is the doctrine that logic does not require its own epistemology, for its methods are continuous with those of science. Although most recently urged by Williamson, the idea goes back at least to Lakatos, who wanted to adapt Popper's falsicationism and extend it not only to mathematics but to logic as well. But one needs to be careful here to distinguish the empirical from the a posteriori. Lakatos coined the term 'quasi-empirical' `for the (...) counterinstances to putative mathematical and logical theses. Mathematics and logic may both be a posteriori, but it does not follow that they are empirical. Indeed, as Williamson has demonstrated, what counts as empirical knowledge, and the role of experience in acquiring knowledge, are both unclear. Moreover, knowledge, even of necessary truths, is fallible. Nonetheless, logical consequence holds in virtue of the meaning of the logical terms, just as consequence in general holds in virtue of the meanings of the concepts involved; and so logic is both analytic and necessary. In this respect, it is exceptional. But its methodologyand its epistemology are the same as those of mathematics and science in being fallibilist, and counterexamples to seemingly analytic truths are as likely as those in any scientic endeavour. What is needed is a new account of the evidential basis of knowledge, one which is, perhaps surprisingly, found in Aristotle. (shrink)
-/- A variable binding term operator (vbto) is a non-logical constant, say v, which combines with a variable y and a formula F containing y free to form a term (vy:F) whose free variables are exact ly those of F, excluding y. -/- Kalish-Montague proposed using vbtos to formalize definite descriptions, set abstracts {x: F}, minimalization in recursive function theory, etc. However, they gave no sematics for vbtos. Hatcher gave a semantics but one that has flaws. We give (...) a correct semantic analysis of vbtos. We also give axioms for using them in deductions. And we conjecture strong completeness for the deductions with respect to the semantics. The conjecture was later proved independently by the authors and by Newton da Costa. -/- The expression (vy:F) is called a variable bound term (vbt). In case F has only y free, (vy:F) has the syntactic propreties of an individual constant; and under a suitable interpretation of the language vy:F) denotes an individual. By a semantic analysis of vbtos we mean a proposal for amending the standard notions of (1) "an interpretation o f a first -order language" and (2) " the denotation of a term under an interpretation and an assignment", such that (1') an interpretation o f a first -order language associates a set-theoretic structure with each vbto and (2') under any interpretation and assignment each vb t denotes an individual. (shrink)
The identity "relation" is misconceived since the syntax of "=" is misconceived as a relative term. Actually, "=" is syncategorematic; it forms (true) sentences with a nonpredicative syntax from pairs of (coreferring) flanking names, much as "&" forms (true) conjunctive sentences from pairs of (true) flanking sentences. In the conaming structure, nothing is predicated of the subject, other than, implicitly, its being so conamed. An identity sentence has both an objectual reading as a necessity about what is named, and (...) also a metalinguistic reading as a contingency about the names. Either way the claim about the subject referent has no extralinguistic content. The necessity of alteridentity (non-self-identity) statements is "lexical", due to contingencies of the names' reference, much like the necessity of analytic statements, due to contingencies of the predicates' sense, and unlike the necessity of logical truths (e.g., self-identities) whose truth is secured by syntax alone. Both alter-identity and analytic sentences are readable as objectual necessities and metalinguistic contingencies. Epistemically, alter-identity statements are not essentially unlike analyticities. "Greece is Hellas"/"g=h" and "Greeks are Hellenes"/"(x)(Gx<=>Hx)" are equally (un)informative; so too for "Azure is cobalt"/"a=c" and "Everything azure is cobalt"/"(x)(Ax<=>Cx)". The real epistemic contrast is between proper names (terms without predicative sense) and terms with a predicative sense (names and predicates of properties). Proper names refer to concrete objects, property names refer to abstract objects. That contrast is metaphysical and thus epistemic. (shrink)
Sentences containing definite descriptions, expressions of the form ‘The F’, can be formalised using a binary quantifier ι that forms a formula out of two predicates, where ιx[F, G] is read as ‘The F is G’. This is an innovation over the usual formalisation of definite descriptions with a term forming operator. The present paper compares the two approaches. After a brief overview of the system INFι of intuitionist negative free logic extended by such a quantifier, which was (...) presented in (Kürbis 2019), INFι is first compared to a system of Tennant’s and an axiomatic treatment of a term forming ι operator within intuitionist negative free logic. Both systems are shown to be equivalent to the subsystem of INFι in which the G of ιx[F, G] is restricted to identity. INFι is then compared to an intuitionist version of a system of Lambert’s which in addition to the term forming operator has an operator for predicate abstraction for indicating scope distinctions. The two systems will be shown to be equivalent through a translation between their respective languages. Advantages of the present approach over the alternatives are indicated in the discussion. (shrink)
This book serves as a concise introduction to some main topics in modern formal logic for undergraduates who already have some familiarity with formal languages. There are chapters on sentential and quantificational logic, modal logic, elementary set theory, a brief introduction to the incompleteness theorem, and a modern development of traditional Aristotelian Logic.
For Kant, ‘reflection’ is a technical term with a range of senses. I focus here on the senses of reflection that come to light in Kant's account of logic, and then bring the results to bear on the distinction between ‘logical’ and ‘transcendental’ reflection that surfaces in the Amphiboly chapter of the Critique of Pure Reason. Although recent commentary has followed similar cues, I suggest that it labours under a blind spot, as it neglects Kant's distinction between ‘pure’ (...) and ‘applied’ general logic. The foundational text of existing interpretations is a passage in Logik Jäsche that appears to attribute to Kant the view that reflection is a mental operation involved in the generation of concepts from non-conceptual materials. I argue against the received view by attending to Kant's division between ‘pure’ and ‘applied’ general logic, identifying senses of reflection proper to each, and showing that none accords well with the received view. Finally, to take account of Kant's notio.. (shrink)
In this paper, I attempt to clarify the heart of Dewey’s philosophy: his method (denotative method (DM) / pattern of inquiry (PI)). Despite the traditional understanding of Dewey as anti-foundationalist, I want to show that Dewey did have metaphysical foundations for his method: the principle of continuity or theory of emergentism. I also argue that Dewey’s metaphysical position is better named as ‘cultural emergentism’, rather than his own term ‘cultural naturalism’. What Dewey called ‘common sense’ in his Logic, (...) Husserl termed as the ‘life-world’ in his Crisis. I compare two perspectives of dealing with the phenomenon and conclude that for Dewey, the difference between natural sciences and the common sense inquiry is that of subject-matter but not of method. Thus, the goal is to find the unified method to be applied in both domains. Whereas Husserl was more pessimistic: for him, the difference was not only in subject-matter, but in the very methods. Following that discussion, I also attempt to reformulate the hard problem of consciousness in Deweyan terms. In the end, I compare Dewey’s DM / PI with Popper’s understandings of scientific method and conclude that there is no significant difference between the two and that Dewey’s method could also be looked at as hypothetic-deductive method, with the only difference in emphases. (shrink)
The title of the present paper might arouse some curiosity among the minds of the readers. The very first question that arises in this respect is whether India produced any logic in the real sense of the term as has been used in the West. This paper is centered only on the three systems of Indian philosophy namely Nyāya, Buddhism and Jainism. We have been talking of Indian philosophy, Indian religion, Indian culture and Indian spirituality, but not that (...) which are of more fundamental concepts for any branch of knowledge whether it is social sciences or humanities. No aspect of human life and the universe has been left unexamined by Indian philosophers, and this leads to a totality of vision in both philosophical and psychological fields. In this paper we will discuss the main thinkers, sources and main concepts related to Indian Logic. (shrink)
If logical truth is necessitated by sheer syntax, mathematics is categorially unlike logic even if all mathematics derives from definitions and logical principles. This contrast gets obscured by the plausibility of the Synonym Substitution Principle implicit in conceptions of analyticity: synonym substitution cannot alter sentence sense. The Principle obviously fails with intercepting: nonuniform term substitution in logical sentences. 'Televisions are televisions' and 'TVs are televisions' neither sound alike nor are used interchangeably. Interception synonymy gets assumed because logical sentences (...) and their synomic interceptions have identical factual content, which seems to exhaust semantic content. However, intercepting alters syntax by eliminating term recurrence, the sole strictly syntactic means of ensuring necessary term coextension, and thereby syntactically securing necessary truth. Interceptional necessity is lexical, a notational artifact. The denial of interception nonsynonymy and the disregard of term recurrence in logic link with many misconceptions about propositions, logical form, conventions, and metalanguages. Mathematics is distinct from logic: its truth is not syntactic; it is transmitted by synonym substitution; term recurrence has no essential role. The '=' of mathematics is an objectual relation between numbers; the '=' of logic marks a syntactic relation of coreferring terms. (shrink)
During the Transfiguration, the apostles on Tabor, “indeed saw the same grace of the Spirit which would later dwell in them”. The light of grace “illuminates from outside on those who worthily approached it and sent the illumination to the soul through the sensitive eyes; but today, because it is confounded with us and exists in us, it illuminates the soul from inward ”. The opposition between knowledge, which comes from outside - a human and purely symbolic knowledge - and (...) “intellectual” knowledge, which comes from within, Meyendorff says what it already exists at Pseudo-Dionysius: “For it is not from without that God stirs them toward the divine. Rather he does so via the intellect and from within and he willingly enlightens them with a ray that is pure and immaterial”. The assertions of the Calabrian philosopher about an “unique knowledge”, common both to the Christians and the Hellenes and pursuing the same goal, the hesychast theologian opposes the reality of the two knowledge, having two distinct purposes and based on two different instruments of perception: “Palamas admitted the authenticity of natural knowledge, however the latter is opposed to the revealed wisdom, that is why it does not provide, by itself, salvation”. Therefore, in the purified human intellect begins to shine of the Trinity light. Purity also depends on the return of the intellect to itself. In this way, we see how the true knowledge of God is an internal meeting or “inner retrieval” of the whole being of man. As well as in the Syrian mystic, on several occasions we have to make the distinction between the contemplative ways of knowledge: intellection illuminated by grace and spiritual vision without any conceptual or symbolic meaning. For example, Robert Beulay shows that, “The term of ‘intellection’ first of all, is employed by John of Dalyatha to be applied to operations caused by grace”. (shrink)
Triadic (systemical) logic can provide an interpretive paradigm for understanding how quantum indeterminacy is a consequence of the formal nature of light in relativity theory. This interpretive paradigm is coherent and constitutionally open to ethical and theological interests. -/- In this statement: -/- (1) Triadic logic refers to a formal pattern that describes systemic (collaborative) processes involving signs that mediate between interiority (individuation) and exteriority (generalized worldview or Umwelt). It is also called systemical logic or the (...) class='Hi'>logic of relatives. The term "triadic logic" emphasizes that this logic involves mediation of dualities through an irreducibly triadic formalism. The term "systemical logic" emphasizes that this logic applies to systems in contrast to traditional binary logic which applies to classes. The term "logic of relatives" emphasizes that this logic is background independent (in the sense discussed by Smolin ). -/- (2) An interpretive paradigm refers to a way of thinking that generates an understanding through concepts, their inter-relationships and their connections with experience. -/- (3) Coherence refers to holistic integrity or continuity in the meaning of concepts that form an interpretation or understanding. -/- (4) Constitutionally open refers to an inherent dependence in principle of an interpretation or understanding on something outside of a specific discipline's discourse or domain of inquiry (epistemic system). Interpretations that are constitutionally open are incomplete in themselves and open to responsive, interdisciplinary discourse and collaborative learning. (shrink)
The paper is about 'absolute logic': an approach to logic that differs from the standard first-order logic and other known approaches. It should be a new approach the author has created proposing to obtain a general and unifying approach to logic and a faithful model of human mathematical deductive process. In first-order logic there exist two different concepts of term and formula, in place of these two concepts in our approach we have just one (...) notion of expression. In our system the set-builder notation is an expression-building pattern. In our system we can easily express second-order, third order and any-order conditions. The meaning of a sentence will depend solely on the meaning of the symbols it contains, it will not depend on external 'structures'. Our deductive system is based on a very simple definition of proof and provides a good model of human mathematical deductive process. The soundness and consistency of the system are proved. We discuss on the completeness of our deductive systems. We also discuss how our system relates to the most know types of paradoxes, from the discussion no specific vulnerability to paradoxes comes out. The paper provides both the theoretical material and a fully documented example of deduction. (shrink)
This paper presents rules of inference for a binary quantifier I for the formalisation of sentences containing definite descriptions within intuitionist positive free logic. I binds one variable and forms a formula from two formulas. Ix[F, G] means ‘The F is G’. The system is shown to have desirable proof-theoretic properties: it is proved that deductions in it can be brought into normal form. The discussion is rounded up by comparisons between the approach to the formalisation of definite descriptions (...) recommended here and the more usual approach that uses a term-forming operator ι, where ιxF means ‘the F’. (shrink)
In this paper, I focus on the important semantic components involved in analogy in hopes of providing an epistemic ground for predicating names of God analogously. To this task, I address a semantic/epistemic problem, which concludes that the doctrine of analogy lacks epistemological grounding insofar as it presupposes a prior understanding of God in order to sufficiently alter a given concept to be proportionate to God. In hopes of avoiding this conclusion, I introduce Aquinas’s specifically semantic aspects that follow after (...) the real distinction between a thing’s esse and its essence or form in the context of analogy and show that the ratio of a term can be altered in a way proportionate to a consideration of the mode of being of God. (shrink)
This article discusses a relation between the formal science of logical semantics and some monotheistic, polytheistic and Trinitarian Christian notions. This relation appears in the use of the existential quantifier and of logical-modal notions when some monotheistic and polytheistic concepts and, principally, the concept of Trinity Dogma are analyzed. Thus, some presupposed modal notions will appear in some monotheistic propositions, such as the notion of “logically necessary”. From this, it will be shown how the term “God” is a polysemic (...)term and is often treated as both subject and predicate. This will make it clear that there is no plausible intellectual justification for believing that the term “God” can only be used as a name and never as a predicate, and vice versa. After that analysis, I will show that the conjunction of the “Trinity Dogma” with some type of “monotheistic position” would necessarily imply some class of absurdity and/or semantic “oddity”. (shrink)
Interpretations of Hegel’s social and political thought tend to present Hegel as critic of modern individualism and defender of institutionalism or proto-communitarianism. Yet Hegel has praise for the historically emancipatory role of individualism and gives a positive role to individuals in his discussion of ethics and the state. Drawing on Hegel’s analysis of the category of ‘individual’ in his Logic, this chapter shows that Hegel criticizes the conception of ‘individual’ as a simple and argues instead that it is a (...)term in need of specification or completion. Hegel’s revisionary logic of the category of ‘individual’ is both interesting in itself and useful as an interpretative tool, because it shows the consistency of his various statements about individuals in his practical philosophy. (shrink)
A principle, according to which any scientific theory can be mathematized, is investigated. That theory is presupposed to be a consistent text, which can be exhaustedly represented by a certain mathematical structure constructively. In thus used, the term “theory” includes all hypotheses as yet unconfirmed as already rejected. The investigation of the sketch of a possible proof of the principle demonstrates that it should be accepted rather a metamathematical axiom about the relation of mathematics and reality. Its investigation needs (...) philosophical means. Husserl’s phenomenology is what is used, and then the conception of “bracketing reality” is modelled to generalize Peano arithmetic in its relation to set theory in the foundation of mathematics. The obtained model is equivalent to the generalization of Peano arithmetic by means of replacing the axiom of induction with that of transfinite induction. A comparison to Mach’s doctrine is used to be revealed the fundamental and philosophical reductionism of Husserl’s phenomenology leading to a kind of Pythagoreanism in the final analysis. Accepting or rejecting the principle, two kinds of mathematics appear differing from each other by its relation to reality. Accepting the principle, mathematics has to include reality within itself in a kind of Pythagoreanism. These two kinds are called in paper correspondingly Hilbert mathematics and Gödel mathematics. The sketch of the proof of the principle demonstrates that the generalization of Peano arithmetic as above can be interpreted as a model of Hilbert mathematics into Gödel mathematics therefore showing that the former is not less consistent than the latter, and the principle is an independent axiom. An information interpretation of Hilbert mathematics is involved. It is a kind of ontology of information. Thus the problem which of the two mathematics is more relevant to our being is discussed. An information interpretation of the Schrödinger equation is involved to illustrate the above problem. (shrink)
Here I review Robert Trivers' 2011 book _The Folly of Fools_, in which he advocates the evolutionary theory of deceit and self-deception that he pioneered in his famous preface to Richard Dawkins' _Selfish Gene_. Although the book contains a wealth of interesting discussion on topics ranging from warfare to immunology, I find it lacking on two major fronts. First, it fails to give a proper argument for its central thesis--namely, that self-deception evolved to facilitate deception of others. Second, the book (...) lacks conceptual clarity with respect to the focal term "self-deception.". (shrink)
In the 17th century, Hobbes stated that we reason by addition and subtraction. Historians of logic note that Hobbes thought of reasoning as “a ‘species of computation’” but point out that “his writing contains in fact no attempt to work out such a project.” Though Leibniz mentions the plus/minus character of the positive and negative copulas, neither he nor Hobbes say anything about a plus/minus character of other common logical words that drive our deductive judgments, words like ‘some’, ‘all’, (...) ‘if’, and ‘and’, each of which actually turns out to have an oppositive, character that allows us, “in our silent reasoning,” to ignore its literal meaning and to reckon with it as one reckons with a plus or a minus operator in elementary algebra or arithmetic. These ‘logical constants’ of natural language figure crucially in our everyday reasoning. Because Hobbes and Leibniz did not identify them as the plus and minus words we reason with, their insight into what goes on in ‘ratiocination’ did not provide a guide for a research program that could develop a +/- logic that actually describes how we reason deductively. I will argue that such a +/- logic provides a way back from modern predicate logic—the logic of quantifiers and bound variables that is now ‘standard logic’—to an Aristotelian termlogic of natural language that had been the millennial standard logic. (shrink)
Corcoran, John. 2005. Meanings of word: type-occurrence-token. Bulletin of Symbolic Logic 11(2005) 117. -/- Once we are aware of the various senses of ‘word’, we realize that self-referential statements use ambiguous sentences. If a statement is made using the sentence ‘this is a pronoun’, is the speaker referring to an interpreted string, a string-type, a string-occurrence, a string-token, or what? The listeners can wonder “this what?”. -/- John Corcoran, Meanings of word: type-occurrence-token Philosophy, University at Buffalo, Buffalo, NY 14260-4150 (...) E-mail: corcoran@buffalo.edu The four-letter written-English expression ‘word’, which plays important roles in applications and expositions of logic and philosophy of logic, is ambiguous (multisense, or polysemic) in that it has multiple normal meanings (senses, or definitions). Several of its meanings are vague (imprecise, or indefinite) in that they admit of borderline (marginal, or fringe) cases. This paper juxtaposes, distinguishes, and analyses several senses of ‘word’ focusing on a constellation of senses analogous to constellations of senses of other expression words such as ‘expression’, ‘symbol’, ‘character’, ‘letter’, ‘term’, ‘phrase’, ‘formula’, ‘sentence’, ‘derivation’, ‘paragraph’, and ‘discourse’. Consider, e.g., the word ‘letter’. In one sense there are exactly twenty-six letters (letter-types or ideal letters) in the English alphabet and there are exactly four letters in the word ‘letter’. In another sense, there are exactly six letters (letter-repetitions or letter-occurrences) in the word-type ‘letter’. In yet another sense, every new inscription (act of writing or printing) of ‘letter’ brings into existence six new letters (letter-tokens or ink-letters) and one new word that had not previously existed. The number of letter-occurrences (occurrences of a letter-type) in a given word-type is the same as the number of letter-tokens (tokens of a letter-type) in a single token of the given word. Many logicians fail to distinguish “token” from “occurrence” and a few actually confuse the two concepts. Epistemological and ontological problems concerning word-types, word-occurrences, and word-tokens are described in philosophically neutral terms. This paper presents a theoretical framework of concepts and principles concerning logicography, including use of English in logic. The framework is applied to analytical exposition and critical evaluation of classic passages in the works of philosophers and logicians including Boole, Peirce, Frege, Russell, Tarski, Church and Quine. This paper is intended as a philosophical sequel to Corcoran et al. “String Theory”, Journal of Symbolic Logic 39(1974) 625-637. https://www.academia.edu/s/cdfa6c854e?source=link -/- . (shrink)
Hilbert's ε-calculus is based on an extension of the language of predicate logic by a term-forming operator εx. Two fundamental results about the ε-calculus, the first and second epsilon theorem, play a rôle similar to that which the cut-elimination theorem plays in sequent calculus. In particular, Herbrand's Theorem is a consequence of the epsilon theorems. The paper investigates the epsilon theorems and the complexity of the elimination procedure underlying their proof, as well as the length of Herbrand disjunctions (...) of existential theorems obtained by this elimination procedure. (shrink)
ABSTRACT: In antiquity we encounter a distinction of two types of hypothetical syllogisms. One type are the ‘mixed hypothetical syllogisms’. The other type is the one to which the present paper is devoted. These arguments went by the name of ‘wholly hypothetical syllogisms’. They were thought to make up a self-contained system of valid arguments. Their paradigm case consists of two conditionals as premisses, and a third as conclusion. Their presentation, either schematically or by example, varies in different authors. For (...) instance, we find ‘If (it is) A, (it is) B; if (it is) B, (it is) C; therefore, if (it is) A, (it is) C’. The main contentious point about these arguments is what the ancients thought their logical form was. Are A, B, C schematic letters for terms or propositions? Is ‘is’, where it occurs, predicative, existential, or veridical? That is, should ‘A esti’ be translated as ‘it is an A’, ‘A exists’, ‘As exist’ or ‘It is true/the case that A’? If A, B, C are term letters, and ‘is’ is predicative, are the conditionals quantified propositions or do they contain designators? If one cannot answer these questions, one can hardly claim to know what sort of arguments the wholly hypothetical syllogisms were. In fact, all the above-mentioned possibilities have been taken to describe them correctly. In this paper I argue that it would be mistaken to assume that in antiquity there was one prevalent understanding of the logical form of these arguments - even if the ancients thought they were all talking about the same kind of argument. Rather, there was a complex development in their understanding, starting from a term-logical conception and leading to a propositional-logical one. I trace this development from Aristotle to Philoponus and set out the deductive system on which the logic of the wholly hypothetical syllogisms was grounded. (shrink)
Since Mates’ seminal Stoic Logic there has been uncertainty and debate about how to treat the term anapodeiktos when used of Stoic syllogisms. This paper argues that the customary translation of anapodeiktos by ‘indemonstrable’ is accurate, and it explains why this is so. At the heart of the explanation is an argument that, contrary to what is commonly assumed, indemonstrability is rooted in the generic account of the Stoic epistemic notion of demonstration. Some minor insights into Stoic (...) class='Hi'>logic ensue. (shrink)
Corcoran, J. 2007. Psychologism. American Philosophy: an Encyclopedia. Eds. John Lachs and Robert Talisse. New York: Routledge. Pages 628-9. -/- Psychologism with respect to a given branch of knowledge, in the broadest neutral sense, is the view that the branch is ultimately reducible to, or at least is essentially dependent on, psychology. The parallel with logicism is incomplete. Logicism with respect to a given branch of knowledge is the view that the branch is ultimately reducible to logic. Every branch (...) of knowledge depends on logic. Psychologism is found in several fields including history, political science, economics, ethics, epistemology, linguistics, aesthetics, mathematics, and logic. Logicism is found mainly in branches of mathematics: number theory, analysis, and, more rarely, geometry. Although the ambiguous term ‘psychologism’ has senses with entirely descriptive connotations, it is widely used in senses that are derogatory. No writers with any appreciation of this point will label their own views as psychologistic. It is usually used pejoratively by people who disapprove of psychologism. The term ‘scientism’ is similar in that it too has both pejorative and descriptive senses but its descriptive senses are rarely used any more. It is almost a law of linguistics that the negative connotations tend to drive out the neutral and the positive. Dictionaries sometimes mark both words with a usage label such as “Usually disparaging”. In this article, the word is used descriptively mainly because there are many psychologistic views that are perfectly respectable and even endorsed by people who would be offended to have their views labeled psychologism. A person who subscribes to logicism is called a logicist, but there is no standard word for a person who subscribes to psychologism. ‘Psychologist’, which is not suitable, occurs in this sense. ‘Psychologician’, with stress on the second syllable as in ‘psychologist’, has been proposed. In the last century, some of the most prominent forms of psychologism pertained to logic; the rest of this article treats only such forms. Psychologism in logic is very “natural”. After all, logic studies reasoning, which is done by the mind, whose nature and functioning is studied in psychology—using the word ‘psychology’ in its broadest etymological sense. (shrink)
If novels can be arguments, that fact should shape logic or argumentation studies as well as literary studies. Two senses the term ‘narrative argument’ might have are (a) a story that offers an argument, or (b) a distinctive argument form. I consider whether there is a principled way of extracting a novel’s argument in sense (a). Regarding the possibility of (b), Hunt’s view is evaluated that many fables and much fabulist literature inherently, and as wholes, have an analogical (...) argument structure. I argue that a better account is that some novels inherently exhibit a transcendental argument structure. (shrink)
Free logics aim at freeing logic from existence assumptions by making them explicit, e.g., by adding an existence premisse to the antecedence of the classical axiom-schema of Universal Instantiation. Their historical development was motivated by the problem of empty singular terms, and that one of simple statements containing at least one such singular term: what is the referential status of such singular terms and what truth-value, if any, do such statemants have? Free logics can be classified with regard (...) to their respective answers to these problems. Negative free logics assume that non-existent objects cannot have any properties at all; hence, in particular, they cannot be self-identical or rotate. Positive free logics believe that non-existents can be self-identical according to the Leibnizian concept of identity. Neutral free logics think that statements of self-identity are truth-valueless because of the Fregean principle of compositionality. Since only in negative free logic, but not in positive free logic, two statements of the forms "a = a" and "E!a" are logically equivalent, one also can define only for NFL, but not for PFL, existence by self-identity. (shrink)
Recent work in formal semantics suggests that the language system includes not only a structure building device, as standardly assumed, but also a natural deductive system which can determine when expressions have trivial truth‐conditions (e.g., are logically true/false) and mark them as unacceptable. This hypothesis, called the ‘logicality of language’, accounts for many acceptability patterns, including systematic restrictions on the distribution of quantifiers. To deal with apparent counter‐examples consisting of acceptable tautologies and contradictions, the logicality of language is often paired (...) with an additional assumption according to which logical forms are radically underspecified: i.e., the language system can see functional terms but is ‘blind’ to open class terms to the extent that different tokens of the same term are treated as if independent. This conception of logical form has profound implications: it suggests an extreme version of the modularity of language, and can only be paired with non‐classical—indeed quite exotic—kinds of deductive systems. The aim of this paper is to show that we can pair the logicality of language with a different and ultimately more traditional account of logical form. This framework accounts for the basic acceptability patterns which motivated the logicality of language, can explain why some tautologies and contradictions are acceptable, and makes better predictions in key cases. As a result, we can pursue versions of the logicality of language in frameworks compatible with the view that the language system is not radically modular vis‐á‐vis its open class terms and employs a deductive system that is basically classical. (shrink)
Semanticists face substitution challenges even outside of contexts commonly recognized as opaque. Jennifer M. Saul has drawn attention to pairs of simple sentences - her term for sentences lacking a that-clause operator - of which the following are typical: -/- (1) Clark Kent went into the phone booth, and Superman came out. (1*) Clark Kent went into the phone booth, and Clark Kent came out. -/- (2) Superman is more successful with women than Clark Kent. (2*) Superman is more (...) successful with women than Superman. -/- She challenges us to explain why the upper and lower sentences in each pair differ, or at least appear to differ, in their truth-values and hence truth-conditions. This appearance of substitution failure is inherently puzzling. Moreover, it is taken by Saul to generate a dilemma for anyone hostile to direct reference accounts of that-clause constructions. Direct reference theorists regard the appearance of substitution failure in that-clause contexts as mere appearance, to be dealt with pragmatically rather than semantically. -/- Critics of such accounts need to say something about simple-sentence cases. If they choose to allow that intuitions of substitution failure can be over-ridden and explained away pragmatically in simple-sentence cases but not in that-clause cases, they lay themselves open to the charge of operating a double standard. But if they do not choose this option, they must offer a semantic explanation of apparent substitution failure in simple-sentence cases - no easy task, it turns out. Other respondents to Saul's challenge have sought to provide elaborate semantic treatments. In contrast, this paper proposes a far simpler pragmatic explanation of intuitions of substitution failure in simple sentences, an explanation that deploys no more resources than are to be found in Grice's 'Logic and Conversation'. Ironically, this proposal turns out to be incompatible with a direct reference perspective. So if it is, as I maintain, the most plausible treatment of simple-sentence cases available, Saul's initial thought gets turned around 180 degrees: the phenomenon she has drawn attention to ends up representing a challenge to supporters of direct reference theories. (shrink)
Monk’s ‘The Temptations of Phenomenology’ examines what the term ‘Phänomenologie’ meant for Wittgenstein. Contesting various other scholars, Monk claims that Wittgenstein’s relation to ‘Phänomenologie’ began and ended during 1929. Monk only partially touches on the question of Wittgenstein’s relation to the phenomenological movement during this time. Though Monk does not mention this, 1929 was also the year in which Ryle and Carnap turned their critical attention toward Heidegger. Wittgenstein also expressed his sympathy for Heidegger in 1929. Furthermore, though in (...) 1929 Wittgenstein agrees with the early Husserl on relating logic and science to phenomenology, it is not clear that they mean the same thing by either logic or phenomenology, or that they agree on what the relation between the two should be. (shrink)
For the OK, there is in fact no opposition between the logical and the material or the spiritual: reality is a formless logical substance. Representation is morphogenesis and the terms 'material' and 'spiritual' only denote categories of morphogenesis. Our constant experience shows us that spiritual and material interact. The border between understanding and becoming, between meaning and act, which seems trivial to us, is elusive when we try to approach it. For example: when the subject follows the object of his (...) attention with his gaze, where to draw the line between material and spiritual ? Representation is morphogenesis and the OK uses the term 'horizon', not to separate the material from the spiritual, but to distinguish the formless from the representable, the unspeakable from the speakable. For the OK the speakable is not 'something else' than the unspeakable, it is a mode of order: The rainbow is not 'an other thing' than rain, light, sun, the walk that brought me here and now, my sense of vision, my culture etc ... all this unspeakable set of logical interdependencies from which emerges, for me subject, the meaning of the rainbow. And above all, this horizon is not 'in my mind'. The ordaining of the unspeakable is not carried out 'by my understanding'. The emergence of the meaning from the unspeakable logic is not happening 'in' and 'by' my mind. On the contrary, it is a process of pure logic*, a progressive aglomeration of interdependance, which makes the subject emerge as a representation of the world and of his own existence. The logical transcends the Existing. This is the key to understanding this article: We have not left the Garden of Eden. The representation is not 'overhanging' the world, it is 'the' world. Representation is not "grasping form" but "morphogenesis", not as "attribution of form" but as "necessary ordering mode of logical reality". Common sense (including science and logic) does not represent reality, it emerges from it in the form of modes of order of reality, of logic. My representation of this white stone is not 'something else' than the unspeakable logic from which the meaning of 'white stone' emerges for me, it is only a mode of order. There are no "other things" than the things which represent themselves in me and which are modes of order. The Existing is representation. Existence is a mode of order *. The amorphous logical "substance" precedes (transcends) its representation. The logical transcends logic. (shrink)
In this study, I intend to show how and why, in the Port-Royal Logic, a singular term can reveal the nature of the logical judgment in the handbook. As I argue, the treatment given to one of thee singular terms, namely, the defined descriptions, in the terminology introduced by Russell, leads to an opening to langage that sounds unexpected and unjustified. Considering the privilege of thinking over langage and also that judgment is the mental act that defines (...) class='Hi'>logic, however, we may understand how the authors regard langage, in relation to the epistemic constituents, namely, the mental acts within the terms. In doing so we are compelled to recognize the implications of this step towards pragmatism in fixing the meaning of defined descriptions to the nature of judgment in the handbook. This opening to langage reveals the conception of judgment as a twofold mental act: a formal and a practical (moral and theological) one. (shrink)
Given its centrality to the intellectual thought processes through which the great structures of logic, nature, and spirit are unfolded it is clear that mediation is vital to the very possibility of Hegel’s encyclopaedic philosophy. Yet Hegel gives little specific explanation of the concept of mediation. Surprisingly, it has been the subject of even less attention by scholars of Hegel. Nevertheless it is casually used in discussions of Hegel and post- Hegelian philosophy as though its meaning were simple and (...) straightforward. In these discussions mediation is the thesis that meanings are not atomic in that the independence of something is inseparable from its relation to something else. Hence being is mediated by nothing, the particular by the universal, the individual by society. But does Hegel ever explain mediation in a way which justifies such use of the concept? The same easy employment of mediation is found in Theodor Adorno whose works are replete with the use of this concept and, indeed, acknowledgements of its Hegelian origin. But the concept of mediation in Adorno’s negative dialectic is operative in an entirely different context from that of Hegel. How, it might be asked, can a concept be so adaptable? I want to argue that mediation is, in fact, an equivocal term which in both Hegel and Adorno covers a variety of entirely different conceptual relations. Furthermore, as propounded by both Hegel and Adorno it lacks the rigour which could allow the particular conclusions which the concept allegedly facilitates. (shrink)
Synthetic biology is a field of research that concentrates on the design, construction, and modification of new biomolecular parts and metabolic pathways using engineering techniques and computational models. By employing knowledge of operational pathways from engineering and mathematics such as circuits, oscillators, and digital logic gates, it uses these to understand, model, rewire, and reprogram biological networks and modules. Standard biological parts with known functions are catalogued in a number of registries (e.g. Massachusetts Institute of Technology Registry of Standard (...) Biological Parts). Biological parts can then be selected from the catalogue and assembled in a variety of combinations to construct a system or pathway in a microbe. Through the innovative re-engineering of biological circuits and the optimization of certain metabolic pathways, biological modules can be designed to reprogram organisms to produce products or behaviors. Synthetic biology is what is known as a “platform technology”. That is, it generates highly transferrable theoretical models, engineering principles, and know-how that can be applied to create potential products in a wide variety of industries. Proponents suggest that applications of synthetic biology may be able to provide scientific and engineered solutions to a multitude of worldwide problems from health to energy. Synthetic biology research has already been successful in constructing microbial products which promise to offer cheaper pharmaceuticals such as the antimalarial synthetic drug artemisinin, engineered microbes capable of cleaning up oil spills, and the engineering of biosensors that can detect the presence of high concentrations of arsenic in drinking water. One of the potential benefits of synthetic biology research is in its application to biofuel production. It is this application which is the focus of this entry. The term “biofuel” has referred generally to all liquid fuels that are sourced from plant or plant byproducts and are used for energy necessary for transportation vehicles (Thompson 2012). Biofuels that are produced using synthetic biological techniques re-engineer microbes into biofuel factories are a subset of these. (shrink)
In the 20th century, the term “media logic” was introduced to denote the influence of independent mass media on political systems and other institutions. In recent years the idea has been reworked and labeled “mediatization” to widen the framework by including new media and new areas of application. In Section Two the paper discusses different conceptualizations. It is argued that even if they bring new insights, they cannot be unified into one concept, and that they also lack a (...) consistent definition of digital media. Section three provides a definition of digital media in order to identify new trajectories made possible by these media, which have led into a new media matrix built around the internet and mobile devices. It will be argued that the new media matrix cannot be understood from a point of view defined by the framework of 20th century mass media because digital media open new trajectories and because in the new matrix the previously existing media have had to transform themselves. (shrink)
This paper is aimed at understanding one central aspect of Bolzano's views on deductive knowledge: what it means for a proposition and for a term to be known a priori. I argue that, for Bolzano, a priori knowledge is knowledge by virtue of meaning and that Bolzano has substantial views about meaning and what it is to know the latter. In particular, Bolzano believes that meaning is determined by implicit definition, i.e. the fundamental propositions in a deductive system. I (...) go into some detail in presenting and discussing Bolzano's views on grounding, a priori knowledge and implicit definition. I explain why other aspects of Bolzano's theory and, in particular, his peculiar understanding of analyticity and the related notion of Ableitbarkeit might, as it has invariably in the past, mislead one to believe that Bolzano lacks a significant account oï a priori knowledge. Throughout the paper, I point out to the ways in which, in this respect, Bolzano's antagonistic relationship to Kant directly shaped his own views. (shrink)
In this essay, I argue that Frege plagiarized the Stoics --and I mean exactly that-- on a large scale in his work on the philosophy of logic and language as written mainly between 1890 and his death in 1925 (much of which published posthumously) and possibly earlier. I use ‘plagiarize' (or 'plagiarise’) merely as a descriptive term. The essay is not concerned with finger pointing or casting moral judgement. The point is rather to demonstrate carefully by means of (...) detailed evidence that there are numerous (over a hundred) and extensive parallels both in formulation and --more importantly-- in content between the Stoics and Frege, parallels so plentiful that one would be hard pressed to brush them off as coincidence. These parallels include several that appear to occur in no other modern works that were published before Frege’s own and were accessible to him. Additionally, a cluster of corroborating historical data is adduced to support the suggestion, showing how easy it would have to been for Frege to plagiarize the Stoics. This (first) part of the essay is easy to read and vaguely entertaining, or so I hope. (shrink)
The subject of the first section is Carnapian modal logic. One of the things I will do there is to prove that certain description principles, viz. the ''self-predication principles'', i.e. the principles according to which a descriptive term satisfies its own descriptive condition, are theorems and that others are not. The second section will be devoted to Carnapian modal arithmetic. I will prove that, if the arithmetical theory contains the standard weak principle of induction, modal truth collapses to (...) truth. Then I will propose a different formulation of Carnapian modal arithmetic and establish that it is free of collapse. Noteworthy is that one can retain the standard strong principle of induction. I will occupy myself in the third section with Carnapian epistemic logic and arithmetic. Here too it is claimed that the standard weak principle of induction is invalid and that the alternative principle is valid. In the fourth and last section I will get back to the self-predication principles and I will point to some of the consequences if one adds them to Carnapian Epistemic arithmetic. The interaction of self-predication principles and the strong principle of induction results in a collapse of de re knowability. (shrink)
This article explores the usefulness of interdisciplinarity as method of enquiry by proposing an investigation of the concept of information in the light of semiotics. This is because, as Kull, Deacon, Emmeche, Hoffmeyer and Stjernfelt state, information is an implicitly semiotic term (Biological Theory 4(2):167–173, 2009: 169), but the logical relation between semiosis and information has not been sufficiently clarified yet. Across the history of cybernetics, the concept of information undergoes an uneven development; that is, information is an ‘objective’ (...) entity in first order cybernetics, and becomes a ‘subjective’ entity in second order cybernetics. This contradiction relegates the status of information to that of a ‘true’ or ‘false’ formal logic problem. The present study proposes that a solution to this contradiction can be found in Deely’s reconfiguration of Peirce’s ‘object’ (as found in his triadic model of semiosis) into ‘thing’ and ‘object’ (Deely 1981). This ontology allows one to argue that information is neither ‘true’ nor ‘false’, and to suggest that, when considered in light of its workability, information can be both true and false, and as such it constitutes an organism’s purely objective reality (Deely 2009b). It is stated that in the process of building such a reality, information is ‘motivated’ by environmental, physiological, emotional (including past feelings and expectations) constraints which are, in turn, framed by observership. Information is therefore found in the irreducible cybersemiotic process that links at once all these conditions and that is simultaneously constrained by them. The integration of cybernetics’ and semiotics’ understanding of information shows that history is the analytical principle that grants scientific rigour to interdisciplinary investigations. As such, in any attempt to clarify its epistemological stance (e.g. the semiotic aspect of information), it is argued that biosemiotics does not need only to acknowledge semiotics (as it does), but also cybernetics in its interdisciplinary heritage. (shrink)
ABSTRACT This part of the series has a dual purpose. In the first place we will discuss two kinds of theories of proof. The first kind will be called a theory of linear proof. The second has been called a theory of suppositional proof. The term "natural deduction" has often and correctly been used to refer to the second kind of theory, but I shall not do so here because many of the theories so-called are not of the second (...) kind--they must be thought of either as disguised linear theories or theories of a third kind (see postscript below). The second purpose of this part is 25 to develop some of the main ideas needed in constructing a comprehensive theory of proof. The reason for choosing the linear and suppositional theories for this purpose is because the linear theory includes only rules of a very simple nature, and the suppositional theory can be seen as the result of making the linear theory more comprehensive. CORRECTION: At the time these articles were written the word ‘proof’ especially in the phrase ‘proof from hypotheses’ was widely used to refer to what were earlier and are now called deductions. I ask your forgiveness. I have forgiven Church and Henkin who misled me. (shrink)
After coining the term “philopsychy” to describe a “soul-loving” approach to philosophical practice, especially when it welcomes a creative synthesis of philosophy and psychology, this article identifies a system of geometrical figures (or “maps”) that can be used to stimulate reflection on various types of perspectival differences. The maps are part of the author’s previously established mapping methodology, known as the Geometry of Logic. As an illustration of how philosophy can influence the development of psychology, Immanuel Kant’s table (...) of twelve categories and Carl Jung’s theory of psychological types are shown to share a common logical structure. Just as Kant proposes four basic categories, each expressed in terms of three subordinate categories, Jung proposes four basic personality functions, each having three possible manifestations. The concluding section presents four scenarios illustrating how such maps can be used in philosophical counseling sessions to stimulate philopsychic insight. (shrink)
This self-contained one page paper produces one valid two-premise premise-conclusion argument that is a counterexample to the entire three traditional rules of distribution. These three rules were previously thought to be generally applicable criteria for invalidity of premise-conclusion arguments. No longer can a three-term argument be dismissed as invalid simply on the ground that its middle is undistributed, for example. The following question seems never to have been raised: how does having an undistributed middle show that an argument's conclusion (...) does not follow from its premises? This result does nothing to vitiate the theories of distribution developed over the period beginning in medieval times. What it does vitiate is many if not all attempts to use distribution in tests of invalidity outside of the standard two-premise categorical arguments—where they were verified on a case-by-case basis without further theoretical grounding. In addition it shows that there was no theoretical basis for many if not all claims of fundamental status of rules of distribution. These results are further support for approaching historical texts using mathematical archeology. (shrink)
This document diagrams the forms OOA, OOE, OOI, and OOO, including all four figures. Each form and figure has the following information: (1) Premises as stated: Venn diagram showing what the premises say; (2) Purported conclusion: diagram showing what the premises claim to say; (3) Relation of premises to conclusion: intended to describe how the premises and conclusion relate to each other, such as validity or contradiction. Used in only a few examples; (4) Distribution: intended to create a system in (...) which each syllogism has a unique code. In each premise and conclusion, the terms are each assigned a one or a zero, based on whether the term is distributed; (5) Rules: lists the rules of the syllogism and shows whether that particular syllogism follows, violates, or is unaffected by, each rule. (shrink)
Corcoran reviews Boute’s 2013 paper “How to calculate proofs”. -/- There are tricky aspects to classifying occurrences of variables: is an occurrence of ‘x’ free as in ‘x + 1’, is it bound as in ‘{x: x = 1}’, or is it orthographic as in ‘extra’? The trickiness is compounded failure to employ conventions to separate use of expressions from their mention. The variable occurrence is free in the term ‘x + 1’ but it is orthographic in that (...) class='Hi'>term’s quotes name ‘‘{x: x = 1}’’. The term has no quotes, the term’s name has one set of quotes, and the name of the term’s name has two sets of quotes. The trickiness is further compounded by failure to explicitly distinguish a variable’s values from it substituents. The variable ranges over its values but its occurrences are replaced by occurrences of its substituents. In arithmetic the values are numbers not numerals but the substituents are numerals not numbers. See https://www.academia.edu/s/1eddee0c62?source=link -/- Raymond Boute tries to criticize Daniel Velleman for mistakes in this area. However, Corcoran finds mistakes in Boute’s handling of the material. The reader is invited to find mistakes in Corcoran’s handling of this tricky material. -/- The paper and the review treat other issues as well. -/- Acknowledgements: Raymond Boute, Joaquin Miller, Daniel Velleman, George Weaver, and others. (shrink)
The effective functioning of a modern enterprise necessitates the awareness of the owners and managers of its strategic orientations, the state of the internal and external environment, its competitive advantages, entrepreneurial potential and development prospects. A well-built business model of the enterprise helps to face these urgent and permanent challenges. However, the formation of a real business model of the enterprise necessitates mastering of theoretical and methodological bases and implementation of long-term practical measures in the field of assessment, shaping, (...) organization and control. This study offers a comprehensive business model of the enterprise that systematically identifies key factors which influence business activity of the enterprise and provide value logic for doing business in the global economic environment. (shrink)
According to the scientific "justificationist" method, knowledge consisted of proven sentences. Classical intellectuals (or "rationalists," in the narrow sense of the term) have accepted extremely varied - and powerful "proofs", through revelation, intellectual intuition, experience. These, with the help of logic, have allowed them to prove any kind of scientific statement. Classical empiricists accepted as axioms only a relatively small set of "factual propositions" that expressed "hard facts". The value of their truth has been established by experience and (...) has been the empirical basis of science. DOI: 10.13140/RG.2.2.28722.25288. (shrink)
Abstract: Four main forms of Doomsday Argument (DA) exist—Gott’s DA, Carter’s DA, Grace’s DA and Universal DA. All four forms use different probabilistic logic to predict that the end of the human civilization will happen unexpectedly soon based on our early location in human history. There are hundreds of publications about the validity of the Doomsday argument. Most of the attempts to disprove the Doomsday Argument have some weak points. As a result, we are uncertain about the validity of (...) DA proofs and rebuttals. In this article, a meta-DA is introduced, which uses the idea of logical uncertainty over the DA’s validity estimated based on a virtual prediction market of the opinions of different scientists. The result is around 0.4 for the validity of some form of DA, and even smaller for “Strong DA”, which predicts the end of the world in the near term. We discuss many examples of the validity of the DA in real life as an instrument to prove it “experimentally”. We also show that DA becomes strongest if it is based on the idea of the “natural reference class” of observers, that is, the observers who know about the DA (i.e. a Self-Referenced DA). Such a DA predicts that there is a high probability of a global catastrophe with human extinction in the 21st century, which aligns with what we already know based on analysis of different technological risks. (shrink)
ABSTRACT: GEORGES CANGUILHEM’S BIOPHILOSOPHY The eminent French biologist and historian of biology, François Jacob, once notoriously declared «On n’interroge plus la vie dans les laboratoires»: laboratory research no longer inquires into the notion of “Life”. Certain influential French philosophers of science of the mid‐century such as Georges Canguilhem would disagree, or at least seek to resist some of Jacob’s diagnosis. Not by imposing a different kind of research program in laboratories, but by an unusual combination of historical and philosophical inquiry (...) into the foundations of the life sciences. Canguilhem speaks of «defending vitalist biology» and declares that Life cannot be grasped by logic. Is this history and philosophy of biology? Is it vitalism? It definitely is a different project from current philosophy of biology. One short‐lived term for it was “biophilosophy”. In this paper I explore the content of this term as it relates to the above questions. (shrink)
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