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Predication and Ontology*
- Karel Lambert
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- Canadian Journal of Philosophy / Volume 17 / Issue 3 / September 1987
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- 01 January 2020, pp. 603-614
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It is an historical fact that one of Russell's greatest philosophical contributions was to highlight the role that premises about logical form play in ontological arguments. A pair of quotations will introduce his point that great metaphysical systems are often not only based on, but are debased by, the belief that certain statements of philosophical discourse are logically subject-predicate in form.
Speaking of Hegel's Absolute Idealism, Russell wrote in Our Knowledge of The Extemal World:
Mr. Bradley has worked out a theory according to which, in all judgment, we are ascribing a predicate to Reality as a whole; and this theory is derived from Hegel. Now the traditional logic holds that every proposition ascribes a predicate to a subject, and from this it easily follows that there can be only one subject, the Absolute, for if there were two, the proposition that there were two would not ascribe a predicate to either. Thus Hegel's doctrine, that philosophical propositions must be of the form, “the Absolute is such and such,” depends on the traditional belief in the universality of the subject-predicate form. This belief, being traditionaL scarcely self-conscious, and not supposed to be important, operates underground, and is assumed in arguments which, like the refutation of relations, appear at first sight to e;tablish its truth.
Contributors
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- By Rose Teteki Abbey, K. C. Abraham, David Tuesday Adamo, LeRoy H. Aden, Efrain Agosto, Victor Aguilan, Gillian T. W. Ahlgren, Charanjit Kaur AjitSingh, Dorothy B E A Akoto, Giuseppe Alberigo, Daniel E. Albrecht, Ruth Albrecht, Daniel O. Aleshire, Urs Altermatt, Anand Amaladass, Michael Amaladoss, James N. Amanze, Lesley G. Anderson, Thomas C. Anderson, Victor Anderson, Hope S. Antone, María Pilar Aquino, Paula Arai, Victorio Araya Guillén, S. Wesley Ariarajah, Ellen T. Armour, Brett Gregory Armstrong, Atsuhiro Asano, Naim Stifan Ateek, Mahmoud Ayoub, John Alembillah Azumah, Mercedes L. García Bachmann, Irena Backus, J. Wayne Baker, Mieke Bal, Lewis V. Baldwin, William Barbieri, António Barbosa da Silva, David Basinger, Bolaji Olukemi Bateye, Oswald Bayer, Daniel H. Bays, Rosalie Beck, Nancy Elizabeth Bedford, Guy-Thomas Bedouelle, Chorbishop Seely Beggiani, Wolfgang Behringer, Christopher M. Bellitto, Byard Bennett, Harold V. Bennett, Teresa Berger, Miguel A. Bernad, Henley Bernard, Alan E. Bernstein, Jon L. Berquist, Johannes Beutler, Ana María Bidegain, Matthew P. Binkewicz, Jennifer Bird, Joseph Blenkinsopp, Dmytro Bondarenko, Paulo Bonfatti, Riet en Pim Bons-Storm, Jessica A. Boon, Marcus J. Borg, Mark Bosco, Peter C. Bouteneff, François Bovon, William D. Bowman, Paul S. Boyer, David Brakke, Richard E. Brantley, Marcus Braybrooke, Ian Breward, Ênio José da Costa Brito, Jewel Spears Brooker, Johannes Brosseder, Nicholas Canfield Read Brown, Robert F. Brown, Pamela K. Brubaker, Walter Brueggemann, Bishop Colin O. Buchanan, Stanley M. Burgess, Amy Nelson Burnett, J. Patout Burns, David B. Burrell, David Buttrick, James P. Byrd, Lavinia Byrne, Gerado Caetano, Marcos Caldas, Alkiviadis Calivas, William J. Callahan, Salvatore Calomino, Euan K. Cameron, William S. Campbell, Marcelo Ayres Camurça, Daniel F. Caner, Paul E. Capetz, Carlos F. Cardoza-Orlandi, Patrick W. Carey, Barbara Carvill, Hal Cauthron, Subhadra Mitra Channa, Mark D. Chapman, James H. Charlesworth, Kenneth R. Chase, Chen Zemin, Luciano Chianeque, Philip Chia Phin Yin, Francisca H. Chimhanda, Daniel Chiquete, John T. Chirban, Soobin Choi, Robert Choquette, Mita Choudhury, Gerald Christianson, John Chryssavgis, Sejong Chun, Esther Chung-Kim, Charles M. A. Clark, Elizabeth A. Clark, Sathianathan Clarke, Fred Cloud, John B. Cobb, W. Owen Cole, John A Coleman, John J. Collins, Sylvia Collins-Mayo, Paul K. Conkin, Beth A. Conklin, Sean Connolly, Demetrios J. Constantelos, Michael A. Conway, Paula M. Cooey, Austin Cooper, Michael L. Cooper-White, Pamela Cooper-White, L. William Countryman, Sérgio Coutinho, Pamela Couture, Shannon Craigo-Snell, James L. Crenshaw, David Crowner, Humberto Horacio Cucchetti, Lawrence S. Cunningham, Elizabeth Mason Currier, Emmanuel Cutrone, Mary L. Daniel, David D. Daniels, Robert Darden, Rolf Darge, Isaiah Dau, Jeffry C. Davis, Jane Dawson, Valentin Dedji, John W. de Gruchy, Paul DeHart, Wendy J. 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- Edited by Daniel Patte, Vanderbilt University, Tennessee
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- The Cambridge Dictionary of Christianity
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- 05 August 2012
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- 20 September 2010, pp xi-xliv
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Explaining away Singular Non-existence Statements1
- Karel Lambert
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- Dialogue: Canadian Philosophical Review / Revue canadienne de philosophie / Volume 1 / Issue 4 / March 1963
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- 09 June 2010, pp. 381-389
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Nowhere in philosophy is there a more persistent kind of nuisance statement than the singular negative existential. Examples of this sort of statement are the descriptional statement
(1) The teacher at Sleepy Hollow doesn't exist,
and the referential statement
(2) Ichabod Crane doesn't exist.
Statements denying existence to a specific object have bothered philosophers from Parmenides to Ryle and Quine. Their high nuisance value rests in a simple paradox they generate. The paradox may be phrased as follows.
4 - The Hilbert-Bernays Theory of Definite Descriptions
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 44-68
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Summary
INTRODUCTION
The theory of definite descriptions developed by David Hilbert and Paul Bernays has original and revised versions. The original, and most distinctive version (hereafter H-BTDD), received its most explicit statement in the first edition of their treatise on the foundations of mathematics. The account that follows relies primarily on this source. This version of the theory is briefly discussed by Rudolf Carnap, and more fully but informally by G. T. Kneebone and Stephen Kleene (among others). The later theory was Fregeian in spirit, and thus is not distinctive. Moreover, newer versions of the theory (hereafter Neo-HBTDD theories), though more in the spirit of the original theory, converge on but cannot be identified with that species called free definite description theory.
Given its essentially mathematical goal H-BTDD might be thought to be of limited interest outside logic where the canons of reasoning in any discipline are of concern. But caution in this regard is dictated by the fact that there are modifications of H-BTDD where the goal, in part at least, is to provide a treatment of definite descriptions more in keeping with the needs of general philosophy. Moreover, inspired by many of Russell's remarks about (logically proper) names, it is hard to resist to thinking of H-BTDD as a theory of (logically proper) definite descriptions.
Hilbert and Bernays note that it is often convenient to introduce into a piece of mathematical reasoning about a specific mathematical object – for instance, a number, a function or a set – an expression referring to that object by means of some uniquely identifying phrase.
Frontmatter
- Karel Lambert, University of California, Irvine
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- Free Logic
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5 - Foundations of the Hierarchy of Positive Free Definite Description Theories
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 69-91
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2 - Existential Import, ‘E!’ and ‘The’
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 16-32
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Introduction
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp ix-xii
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Summary
In one way or another the nine chapters of this book all have to do with free logic. Most are updated revisions in and adaptations of previously published papers. The exceptions are Chapters 4 and 5, though Chapter 4 contains a revised segment from a recently published paper.
Chapter 1 began as an invited address to the Western Division of the American Philosophical Association Meetings in 1991, and at the request of the organizers of those meetings was subsequently published in slightly revised form in Philosophical Studies, 65 (1992), pp. 153–167. It is a critical analysis of Russell's famous theory of definite descriptions of which there are two quite distinct versions. The defining feature of either version is that definite descriptions are not singular terms. That the essence of Russell's theory has to do with logical grammar was stressed in my ‘Explaining away singular existence statements’, Dialogue, 1 (1963), pp. 381–389, and later, independently, by David Kaplan in ‘What is Russell's theory of descriptions?’ Physics, History and Logic (eds. W. Yourgrau and A. Breck), Plenum Press, New York (1970), pp. 277–288.
6 - Predication and Extensionality
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 92-106
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INTRODUCTION
Quine's concept of predication is intimately related to his notion of referential opacity. To treat the position of a singular term in a sentence as “purely referential”, and hence the sentence as “referentially transparent”, is to treat the very same sentence as a predication. “Predication”, he says, “joins a general term and a singular term to form a sentence that is true or false according as the general term is true or false of the object, if any, to which the singular term refers.” The conception of predication expressed in the quoted passage is not restricted to its author. Among others who hold it are many free logicians.
Quine has also written that “so long merely as the predicated general term is true of the object named by the singular term … the substitution of a new singular term that names the same object leaves the predication true.” So if a sentence is a predication, it satisfies the substitutivity of identity. Moreover, “in an opaque construction you … cannot in general supplant a general term by a coextensive term (one true of the same objects) … without disturbing the truth-value of the containing sentence. Such a failure is one of the failures of extensionality.”
The theory of predication under consideration is non-extensional precisely in the sense that it does not satisfy the principle that co-extensive general terms substitute for each other salva veritate. The proof of this fact is the first order of business.
7 - Nonextensionality
- Karel Lambert, University of California, Irvine
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- Free Logic
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INTRODUCTION
Consider the statement
(1) Necessarily 9 = 9.
(1) is believed by most philosophical logicians to be non-extensional according to each of two prominent conceptions of extensionality. First, it fails the salva veritate substitution test when ‘9’ is replace by the co-referential singular term ‘the number of planets’. Second, in possible world semantics, its truth is regarded as dependent on the reference-in-all-possible-worlds of ‘9’, or, alterntively, on the intension of ‘9’ conceived as a function from possible worlds to individuals.
More generally, a statement is SV-Extensional according to the salva veritate substitution conception just in case singular terms co-referential with a statement's constituent singular terms, predicates co extensive with the statement's constituent predicates, and statements co-valent with a statement's constituent statement(s), substitute in that statement salva veritate. A statement is not SV- Extensional if there is at least one failure of salva veritate substitution. (This is close to the characterization in Quine's Word and Object.)
A statement is TD-Extensional according to the truth-value dependence conception just in case its truth value depends only on the extensions simpliciter, if any at all, of its constituent singular terms, predicates and statements. A statement is not TD-Extensional if entities other than extensional entities areinvolved in the computation of its truth value.
1 - Russell's Version of the Theory of Definite Descriptions
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 1-15
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INTRODUCTION
It is mildly ironic that the title of this chapter is an unfulfilled (or improper) definite description because Russell really had two versions of the theory of definite descriptions. The two versions differ in primary goals, character and philosophical strength.
The first version of Russell's theory of definite descriptions was developed in his famous essay of 1905, ‘On Denoting’. Its primary goal was to ascertain the logical form of natural language statements containing denoting phrases. The class of such statements included statements with definite descriptions, a species of denoting phrase, such as ‘The Prime Minister of England in 1904 favored retaliation’ and ‘The gold mountain is gold’. So the theory of definite descriptions contained in what Russell himself regarded as his finest philosophical essay is a theory about how to paraphrase natural language statements containing definite descriptions into an incompletely specified formal language about propositional functions. Russell used this version of his theory to disarm arguments such as Meinong's arguments for beingless objects. Such reasoning, he said, is the product of a mistaken view about the logical form of statements containing definite descriptions.
The second and later version is presented in that epic work of 1910, Principia Mathematica (hereafter usually Principia). Its primary goal. in contrast to the first version, was to provide a foundation for mathematics, indeed, to reduce all of mathematics to logic. In chapter *14 Russell introduces a special symbol, the inverted iota, and uses it to make singular term-like expressions out of quasi-statements.
9 - Logical Truth and Microphysics
- Karel Lambert, University of California, Irvine
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LOGICAL TRUTH AND TRUTH-VALUE GAPS
Elementary microphysical statements can be neither true nor false without violating the classical codification of statement logic. The existence of such a possibility depends upon a revision in the standard explication of logical truth, a revision more harmonious with the idea of argument validity as merely truth-preserving. The revision in question, in turn, depends upon Bas van Fraassen's investigations into the semantical foundations of positive free logic, a species of logical system whose philosophical significance was first made plain in Henry Leonard's pioneering study of 1956, ‘The Logic of Existence’.
Students who steadfastly refuse to accept an argument as valid unless all of the component statements are in fact true frustrate teachers of logic. “Now look!” the teacher may heatedly emphasize, “the validity of an argument has to do with its form alone. So to say that an argument is valid is to say only that if its premises were true its conclusion would also be true!” But, then, not only is the argument from the pair of false statements
Jim Thorpe was Russian
and
If Jim Thorpe was Russian, he was a Bolshevik
to the false statement
Jim Thorpe was a Bolshevik
valid, but so is the argument from the pair of (allegedly) truth-valueless statements
The Queen of the United States dreamed she was being led down a bridal path by a gorilla
and
If the Queen of the United States dreamed she was being led down a bridal path by a gorilla, she desires to marry a man named ‘Harry’
to the (allegedly) truth-valueless statement
The Queen of the United States desires to marry a man named ‘Harry’.
8 - The Philosophical Foundations of Free Logic
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 05 June 2012
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- 31 October 2002, pp 122-175
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Summary
WHAT FREE LOGIC IS AND WHAT IT ISN'T
On page 149 of the second edition of their book, Deductive Logic, Hugues Leblanc and William Wisdom say the following about the origins of free logic.
Presupposition-free logic (known for short as free logic) grew out of two papers published simultaneously: Hintikka's ‘Existential Presuppositions and Existential Commitments’, The Journal of Philosophy. Vol. 56. 1959. pp. 125–137, and Leblanc and (Theodore) Hailperin's ‘Nondesignating Singular Terms’. The Philosophical Review. Vol. 68, 1959. pp. 129–136. Both made use of the identity sign ‘=’. In a later paper Karel Lambert devised a free logic without ‘=’ … (See Lambert's ‘Existential Import Revisited’, Notre Dame Journal of Formal Logic, vol. 4, (1963), pp. 288–292).
This account is in one respect misleading, and in another inaccurate. It is misleading because Rolf Schock, independently either of Hintikka or of Leblanc and Hailperin, was developing a version of free logic in the early 1960s quite different in character from those mentioned by Leblanc and Wisdom. Moreover Schock's ideas were in certain respects more fully developed because he also supplied models for his own systematization. The writers mentioned by Leblanc and Wisdom had suggested – in print at any rate – only informally the semantical bases for their formulations. (Apparently Schock's ideas were known to some European scholars in the early 1960s but oddly were not disseminated. One can find an account of Schock's pioneering efforts in his 1968 book, Logics Without Existence Assumptions.
Contents
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 05 June 2012
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- 31 October 2002, pp vii-viii
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3 - The Reduction of Two Paradoxes and the Significance Thereof
- Karel Lambert, University of California, Irvine
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- Free Logic
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- 31 October 2002, pp 33-43
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Summary
THE CONJECTURE
David Kaplan once suggested to me that the pair of self-contradictory statements:
(1) The round square both is and isn't a round square,
and
(2) The class of all classes not members of themselves both is and isn't a member of itself
“ought to have the same father”. But apparently they don't despite their family resemblance. Russell deduced (1) from a principle he presumed correctly to be a key ingredient of Meinong's theory of objects. That principle says:
MP The object that is so and so is (a) so and so.
On the other hand, Russell deduced (2) from a seemingly unrelated but no less fundamental principle in Frege's version of set theory, the principle of set abstraction. That principle, a version of the principle of comprehension, (in effect) says:
FP Everything is such that it is a member of the class of so and sos if and only if it is (a) so and so.
The lack of common ancestry between MP and FP, and hence between their respective consequences (1) and (2), enabled Russell to treat the theory of objects and the theory of sets (or classes) very differently. He thought (1) “demolished” the theory of objects, but he didn't think (2) destroyed the theory of classes. Russell's attitude was wrong, because Kaplan's suspicion of the common kinship of (1) and (2) is justified, and the proof of this fact is the next order of business.
Free Logic
- Selected Essays
- Karel Lambert
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- 05 June 2012
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- 31 October 2002
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Free logic is an important field of philosophical logic that first appeared in the 1950s. J. Karel Lambert was one of its founders and coined the term itself. The essays in this collection (written over a period of 40 years) explore the philosophical foundations of free logic and its application to areas as diverse as the philosophy of religion and computer science. Amongst the applications on offer are those to the analysis of existence statements, to definite descriptions and to partial functions. The volume contains a proof that free logics of any kind are non-extensional and then uses that proof to show that Quine's theory of predication and referential transparency must fail. The purpose of this collection is to bring an important body of work to the attention of a new generation of professional philosophers, computer scientists and mathematicians.
Susan Haack. Deviant logic. Some philosophical issues. Cambridge University Press, New York and London1974, xiv + 191 pp.
- Karel Lambert
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- The Journal of Symbolic Logic / Volume 43 / Issue 2 / June 1978
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- 12 March 2014, pp. 377-379
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- June 1978
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Universally free logic and standard quantification theory
- Robert K. Meyer, Karel Lambert
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- The Journal of Symbolic Logic / Volume 33 / Issue 1 / 26 April 1968
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- 12 March 2014, pp. 8-26
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- 26 April 1968
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Interest has steadily increased among logicians and philosophers in versions of quantification theory which meet the following criteria: (1) no existence assumptions are made with respect to individual constants, and (2) theorems are valid in every domain including the empty domain. Logics meeting the former of these criteria are called free logics by Lambert and have been investigated in a series of papers by him and by van Fraassen, and by Leblanc and Thomason.1
Although it is natural to impose (2) in the presence of (1), the criteria are independent.2 Hence we baptize logics which meet both criteria universally free.