En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un indi... more En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un individuo o una comunidad) en un determinado momento T. Muchas argumentaciones paradójicas tienen lugar en el desarrollo histórico y práctico de las ciencias. Algunas suponen grandes sorpresas acompañadas de profundas crisis, como ocurre con las llamadas antinomias. Solventar, y eventualmente resolver, una paradoja en este sentido supone avances revolucionarios que se obtienen al precio de rechazar creencias previamente asumidas o tenidas por verdaderas por la comunidad científica. El concepto de paradoja que se propone es relativo a sujetos y dinámico en el tiempo. Una argumentación que resulta paradójica para X pudiera no ser paradójica para Y, simplemente porque X e Y no tienen por qué compartir las mismas creencias. Asimismo, X puede descubrir que tiene una paradoja en T, y dejar de tenerla ulteriormente en virtud de un cambio de sus creencias.
We present a memorial summary of the professional life and contributions to logic of John Corcora... more We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications. Courtesy of Lynn Corcoran
Abstract: One of the multiple meanings of the word ‘information ’ is given implicitly in the post... more Abstract: One of the multiple meanings of the word ‘information ’ is given implicitly in the postulates and conditions of information-theoretic logic (I-T-L). The tradition of looking at logical phenomena from an informational stance goes back as far as the XIX century. Logicians such as Boole, De Morgan, Jevons, and Venn already suggested that deducing is a sort of unpacking the information already contained in given premises. In the XX century this tradition is recovered by Carnap and Bar Hillel, Cohen and Nagel, and more recently by Corcoran. John Corcoran has articulated a specific information-theoretic viewpoint of logic with its own particular characteristics. I intend to explain the basic ideas of I-T-L by motivating their philosophical underpinnings. One desideratum is to complement and to shed light on some of the philosophical shortcomings of the nowadays paradigmatic model-theoretic concept of logical consequence. Another is to provide a brief sample of questions to be ne...
THEORIA. An International Journal for Theory, History and Foundations of Science, 2018
I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practic... more I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.
The President of the International Court of Justice (ICJ), Judge Abdulqawi Yusuf, announced the C... more The President of the International Court of Justice (ICJ), Judge Abdulqawi Yusuf, announced the Court’s decision to ‘clearly define rules’ regulating the ‘extrajudicial activities’ of ICJ judges, such as taking up appointments on arbitration tribunals, in his speech to the United Nations General Assembly on 25 October 2018.1 As a result, sitting ICJ Judges can no longer accept any new appointments to serve as arbitrators in investor-state proceedings and can participate by exception only in inter-state adjudication processes. While President Yusuf justified this decision in light of the ‘ever-increasing workload of the Court’,2 it was arguably also made in response to a legal and ethical dilemma referred to by critics of investment arbitration as ‘moonlighting’: the situation where sitting ICJ judges, in addition to their full-time employment at the Court, perform other functions, including acting as investment arbitrators or sitting on annulment committees at the International Centre for the Settlement of Investment Disputes (ICSID). Recent statistics3 show that seven of those currently on the bench and 13 former ICJ judges were involved as adjudicators in investor-state dispute settlement (ISDS) in at least 90 cases.4 The decision to bar current ICJ judges from serving as arbitrators in investorstate proceedings was aimed at placing the impartiality and independence of judges ‘beyond reproach’, as pointed out by President Yusuf during his speech. However, this was the solution for only one of the dilemmas that investment arbitrators face in the course of arbitral proceedings – others includes issue conflicts, such as multiple hatting. What makes these dilemmas more difficult to resolve in investment arbitration are two factors: First, unlike the regulation of judicial conduct in national legal systems, there is a plurality of rules originating from different legal orders and giving rise to different legal hierarchies that could shape or direct the actions of investment arbitrators. Second, in contrast with domestic legal systems, there is no entity at the international level akin to a judicial council that can clarify the possible contradictions between such rules so that arbitrators get considerable leeway. It is in this context that Key Duties of International Investment Arbitrators by Katia Fach Gomez offers a valuable approach to understanding the current ethical and legal dilemmas of investment arbitrators. This work breaks down the plurality of rules, including those originating
The aim of this paper is to discuss some of the subtleties in the distinct way Stephen Read respo... more The aim of this paper is to discuss some of the subtleties in the distinct way Stephen Read responds, in his paper “The Truth-Schema and the Liar” to the Liar paradox using an improved formulation of Tarski’s T-scheme. Examining the orthodox and the new account of the Liar shows two different ways of tackling the problem of modelling the underlying logic of Liar-type discourse. Specific discussion is provided of Read’s key notions of saying that and strict implication, which he uses to articulate his new intensional proposal.
Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-... more Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: (1) terminological and conceptual preliminaries, (2) Tarski's definition of the concept of logical consequence, (3) Tarski's discussion of omega-incomplete theories, and (4) concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P (the premise-set) and a single proposition c (the conclusio...
This article discusses two coextensive concepts of logical consequence that are implicit in the t... more This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ‘say’ something), and they have subject matter (they are ‘about’ something). The asymmetry between establishing validity and establishing invalidity has long been noted:
Each science has its own domain of investigation, but one and the sam e science can be formalized... more Each science has its own domain of investigation, but one and the sam e science can be formalized in diOE erent languages with diOE erent universes of discourse. The concept of the dom ain of a science and the concept of the universe of discours e of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppos itions and im plications of choices of domain and universe , this article discusses the treatm ent of omega argum ents in three very diOEerent form alizations of arithmetic. In Peano' s formalization the domain is a restricted class of individuals, while the universe of discours e is the unrestric ted class of all individu als. In Go $ del' s form alization the domain is a restricted class of individu als as in Peano' s form alization, but the universe of discours e coincides with the domain. In W hitehead-Russell' s formalization the dom ain is a class of logical notions in Tarski' s sense , that are necessarily not individu als, whereas the universe of discours e is the unrestricted class of individu als as in Peano' s formalization. The present approac h emphasizes the viewpoint that the universe of discours e of a given discours e is importan t in determ ining which propositi ons are expressed by which sentences.
tripleC: Communication, Capitalism & Critique. Open Access Journal for a Global Sustainable Information Society, 2009
One of the multiple meanings of the word ‘information’ is given implicitly in the postulates and ... more One of the multiple meanings of the word ‘information’ is given implicitly in the postulates and conditions of information-theoretic logic (I-T-L). The tradition of looking at logical phenomena from an informational stance goes back as far as the XIX century. Logicians such as Boole, De Morgan, Jevons, and Venn already suggested that deducing is a sort of unpacking the information already contained in given premises. In the XX century this tradition is recovered by Carnap and Bar Hillel, Cohen and Nagel, and more recently by Corcoran. John Corcoran has articulated a specific information-theoretic viewpoint of logic with its own particular characteristics. I intend to explain the basic ideas of I-T-L by motivating their philosophical underpinnings. One desideratum is to complement and to shed light on some of the philosophical shortcomings of the nowadays paradigmatic model-theoretic concept of logical consequence. Another is to provide a brief sample of questions to be newly address...
Taking as starting point Kuhn's analysis of science textbooks and its application to Sinnott and ... more Taking as starting point Kuhn's analysis of science textbooks and its application to Sinnott and Dunn's (1925), it will be discussed the problem of the existence of laws in biology. In particular, it will be showed, in accordance with the proposals of Darden (1991) and Schaffner (1980, 1986, 1993), the relevance of the exemplars, diagrammatically or graphically represented, in the way in which is carried out the teaching and learning process of classical genetics, inasmuch as the information contained in them, indispensable for the right development of that process, exceeds the information contained in the "laws" linguistically articulated and presented in the textbooks. However, it will be maintained that the information is implicit in the law that according to the structuralist concept of fundamental law and the reconstruction of genetics presented by Balzer & Dawe (1990), and later developed by Balzer & Lorenzano (1997) and Lorenzano (1995, 2000, 2002a) could be considered the fundamental law of classical genetics, the law of matching, clearly identified in this paper.
Fitchs problem and the" knowability paradox" involve a couple of argumentations that ar... more Fitchs problem and the" knowability paradox" involve a couple of argumentations that are to each other in the same relation as Cantors uncollected multitudes theorem and Russells paradox. The authors exhibit the logical nature of the theorem and of the paradox ...
This article may be used for research, teaching, and private study purposes. Any substantial or s... more This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Fitchs problem and the" knowability paradox" involve a couple of argumentations that ar... more Fitchs problem and the" knowability paradox" involve a couple of argumentations that are to each other in the same relation as Cantors uncollected multitudes theorem and Russells paradox. The authors exhibit the logical nature of the theorem and of the paradox ...
Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teache... more Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, and collaborator—on his eightieth birthday.
This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege–Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.
Platonism and logicism are distinct tendencies in the philosophy of mathematics, which, neverthel... more Platonism and logicism are distinct tendencies in the philosophy of mathematics, which, nevertheless, are often either not distinguished or else discussed as though one contains the other. Kleene (1977) takes platonism to include logicism. Myhill (1972) takes logicism to include platonism, as do many writers who distinguish only three tendencies in modern philosophy of mathematics. In part I, the two tendencies are distinguished, they are shown to have separate histories, and they are shown to be independent in various senses. Part II treats varieties of logicism. Part III considers the mutual opposition of a logicism held by Russell and others versus a platonism held by Gödel and others. A neutral stance, respecting the coherence and plausibility of each tendency, is maintained throughout.
This issue poses the question: what information really is. We assume that information has some pl... more This issue poses the question: what information really is. We assume that information has some place or other in reality, in particular, we assume there is a cohesive and coherent account of informational phenomena, able to coherently set up facts, contents and values regarding information. In our current information era it seems natural to assume without further critical reflection a disunited class of uses of “information”. The point of this issue of tripleC is setting up a cohesive account of information in complex contemporary open societies and scientific communities.
There are at least three dimensions in our plea for such cohesive account of information: (a) from a conceptual point of view, there is a plethora of seemingly incompatible notions of “information”, (b) from a societal viewpoint, information can acritically postulate a new infinite realm of merchandise which does not foster a more cohesive society but instead a growing inequality, (c) a coherent unified approach to both the manifest image and the scientific image of information is still lacking.
En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un indi... more En este artículo una paradoja es un tipo de argumentación con respecto a un sujeto X (sea un individuo o una comunidad) en un determinado momento T. Muchas argumentaciones paradójicas tienen lugar en el desarrollo histórico y práctico de las ciencias. Algunas suponen grandes sorpresas acompañadas de profundas crisis, como ocurre con las llamadas antinomias. Solventar, y eventualmente resolver, una paradoja en este sentido supone avances revolucionarios que se obtienen al precio de rechazar creencias previamente asumidas o tenidas por verdaderas por la comunidad científica. El concepto de paradoja que se propone es relativo a sujetos y dinámico en el tiempo. Una argumentación que resulta paradójica para X pudiera no ser paradójica para Y, simplemente porque X e Y no tienen por qué compartir las mismas creencias. Asimismo, X puede descubrir que tiene una paradoja en T, y dejar de tenerla ulteriormente en virtud de un cambio de sus creencias.
We present a memorial summary of the professional life and contributions to logic of John Corcora... more We present a memorial summary of the professional life and contributions to logic of John Corcoran. We also provide a full list of his many publications. Courtesy of Lynn Corcoran
Abstract: One of the multiple meanings of the word ‘information ’ is given implicitly in the post... more Abstract: One of the multiple meanings of the word ‘information ’ is given implicitly in the postulates and conditions of information-theoretic logic (I-T-L). The tradition of looking at logical phenomena from an informational stance goes back as far as the XIX century. Logicians such as Boole, De Morgan, Jevons, and Venn already suggested that deducing is a sort of unpacking the information already contained in given premises. In the XX century this tradition is recovered by Carnap and Bar Hillel, Cohen and Nagel, and more recently by Corcoran. John Corcoran has articulated a specific information-theoretic viewpoint of logic with its own particular characteristics. I intend to explain the basic ideas of I-T-L by motivating their philosophical underpinnings. One desideratum is to complement and to shed light on some of the philosophical shortcomings of the nowadays paradigmatic model-theoretic concept of logical consequence. Another is to provide a brief sample of questions to be ne...
THEORIA. An International Journal for Theory, History and Foundations of Science, 2018
I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practic... more I discuss Putnam’s conception of logical truth as grounded in his picture of mathematical practice and ontology. i begin by comparing Putnam’s 1971 Philosophy of Logic with Quine’s homonymous book. Next, Putnam’s changing views on modality are surveyed, moving from the modal pre-formal to the de-modalized formal characterization of logical validity. Section three suggests a complementary view of Platonism and modalism underlying different stages of a dynamic mathematical practice. The final section argues for the pervasive platonistic conception of the working mathematician.
The President of the International Court of Justice (ICJ), Judge Abdulqawi Yusuf, announced the C... more The President of the International Court of Justice (ICJ), Judge Abdulqawi Yusuf, announced the Court’s decision to ‘clearly define rules’ regulating the ‘extrajudicial activities’ of ICJ judges, such as taking up appointments on arbitration tribunals, in his speech to the United Nations General Assembly on 25 October 2018.1 As a result, sitting ICJ Judges can no longer accept any new appointments to serve as arbitrators in investor-state proceedings and can participate by exception only in inter-state adjudication processes. While President Yusuf justified this decision in light of the ‘ever-increasing workload of the Court’,2 it was arguably also made in response to a legal and ethical dilemma referred to by critics of investment arbitration as ‘moonlighting’: the situation where sitting ICJ judges, in addition to their full-time employment at the Court, perform other functions, including acting as investment arbitrators or sitting on annulment committees at the International Centre for the Settlement of Investment Disputes (ICSID). Recent statistics3 show that seven of those currently on the bench and 13 former ICJ judges were involved as adjudicators in investor-state dispute settlement (ISDS) in at least 90 cases.4 The decision to bar current ICJ judges from serving as arbitrators in investorstate proceedings was aimed at placing the impartiality and independence of judges ‘beyond reproach’, as pointed out by President Yusuf during his speech. However, this was the solution for only one of the dilemmas that investment arbitrators face in the course of arbitral proceedings – others includes issue conflicts, such as multiple hatting. What makes these dilemmas more difficult to resolve in investment arbitration are two factors: First, unlike the regulation of judicial conduct in national legal systems, there is a plurality of rules originating from different legal orders and giving rise to different legal hierarchies that could shape or direct the actions of investment arbitrators. Second, in contrast with domestic legal systems, there is no entity at the international level akin to a judicial council that can clarify the possible contradictions between such rules so that arbitrators get considerable leeway. It is in this context that Key Duties of International Investment Arbitrators by Katia Fach Gomez offers a valuable approach to understanding the current ethical and legal dilemmas of investment arbitrators. This work breaks down the plurality of rules, including those originating
The aim of this paper is to discuss some of the subtleties in the distinct way Stephen Read respo... more The aim of this paper is to discuss some of the subtleties in the distinct way Stephen Read responds, in his paper “The Truth-Schema and the Liar” to the Liar paradox using an improved formulation of Tarski’s T-scheme. Examining the orthodox and the new account of the Liar shows two different ways of tackling the problem of modelling the underlying logic of Liar-type discourse. Specific discussion is provided of Read’s key notions of saying that and strict implication, which he uses to articulate his new intensional proposal.
Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-... more Tarski's 1936 paper, “On the concept of logical consequence”, is a rather philosophical, non-technical paper that leaves room for conflicting interpretations. My purpose is to review some important issues that explicitly or implicitly constitute its themes. My discussion contains four sections: (1) terminological and conceptual preliminaries, (2) Tarski's definition of the concept of logical consequence, (3) Tarski's discussion of omega-incomplete theories, and (4) concluding remarks concerning the kind of conception that Tarski's definition was intended to explicate. The third section involves subsidiary issues, such as Tarski's discussion concerning the distinction between material and formal consequence and the important question ofthe criterion for distinguishing between logical and non-logical terms.§1. Preliminaries. In this paper an argument is a two-part system composed of a set of propositions P (the premise-set) and a single proposition c (the conclusio...
This article discusses two coextensive concepts of logical consequence that are implicit in the t... more This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they ‘say’ something), and they have subject matter (they are ‘about’ something). The asymmetry between establishing validity and establishing invalidity has long been noted:
Each science has its own domain of investigation, but one and the sam e science can be formalized... more Each science has its own domain of investigation, but one and the sam e science can be formalized in diOE erent languages with diOE erent universes of discourse. The concept of the dom ain of a science and the concept of the universe of discours e of a formalization of a science are distinct, although they often coincide in extension. In order to analyse the presuppos itions and im plications of choices of domain and universe , this article discusses the treatm ent of omega argum ents in three very diOEerent form alizations of arithmetic. In Peano' s formalization the domain is a restricted class of individuals, while the universe of discours e is the unrestric ted class of all individu als. In Go $ del' s form alization the domain is a restricted class of individu als as in Peano' s form alization, but the universe of discours e coincides with the domain. In W hitehead-Russell' s formalization the dom ain is a class of logical notions in Tarski' s sense , that are necessarily not individu als, whereas the universe of discours e is the unrestricted class of individu als as in Peano' s formalization. The present approac h emphasizes the viewpoint that the universe of discours e of a given discours e is importan t in determ ining which propositi ons are expressed by which sentences.
tripleC: Communication, Capitalism & Critique. Open Access Journal for a Global Sustainable Information Society, 2009
One of the multiple meanings of the word ‘information’ is given implicitly in the postulates and ... more One of the multiple meanings of the word ‘information’ is given implicitly in the postulates and conditions of information-theoretic logic (I-T-L). The tradition of looking at logical phenomena from an informational stance goes back as far as the XIX century. Logicians such as Boole, De Morgan, Jevons, and Venn already suggested that deducing is a sort of unpacking the information already contained in given premises. In the XX century this tradition is recovered by Carnap and Bar Hillel, Cohen and Nagel, and more recently by Corcoran. John Corcoran has articulated a specific information-theoretic viewpoint of logic with its own particular characteristics. I intend to explain the basic ideas of I-T-L by motivating their philosophical underpinnings. One desideratum is to complement and to shed light on some of the philosophical shortcomings of the nowadays paradigmatic model-theoretic concept of logical consequence. Another is to provide a brief sample of questions to be newly address...
Taking as starting point Kuhn's analysis of science textbooks and its application to Sinnott and ... more Taking as starting point Kuhn's analysis of science textbooks and its application to Sinnott and Dunn's (1925), it will be discussed the problem of the existence of laws in biology. In particular, it will be showed, in accordance with the proposals of Darden (1991) and Schaffner (1980, 1986, 1993), the relevance of the exemplars, diagrammatically or graphically represented, in the way in which is carried out the teaching and learning process of classical genetics, inasmuch as the information contained in them, indispensable for the right development of that process, exceeds the information contained in the "laws" linguistically articulated and presented in the textbooks. However, it will be maintained that the information is implicit in the law that according to the structuralist concept of fundamental law and the reconstruction of genetics presented by Balzer & Dawe (1990), and later developed by Balzer & Lorenzano (1997) and Lorenzano (1995, 2000, 2002a) could be considered the fundamental law of classical genetics, the law of matching, clearly identified in this paper.
Fitchs problem and the" knowability paradox" involve a couple of argumentations that ar... more Fitchs problem and the" knowability paradox" involve a couple of argumentations that are to each other in the same relation as Cantors uncollected multitudes theorem and Russells paradox. The authors exhibit the logical nature of the theorem and of the paradox ...
This article may be used for research, teaching, and private study purposes. Any substantial or s... more This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae, and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand, or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material.
Fitchs problem and the" knowability paradox" involve a couple of argumentations that ar... more Fitchs problem and the" knowability paradox" involve a couple of argumentations that are to each other in the same relation as Cantors uncollected multitudes theorem and Russells paradox. The authors exhibit the logical nature of the theorem and of the paradox ...
Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teache... more Dedicated to Professor Roberto Torretti, philosopher of science, historian of mathematics, teacher, friend, and collaborator—on his eightieth birthday.
This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory. In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege–Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.
Platonism and logicism are distinct tendencies in the philosophy of mathematics, which, neverthel... more Platonism and logicism are distinct tendencies in the philosophy of mathematics, which, nevertheless, are often either not distinguished or else discussed as though one contains the other. Kleene (1977) takes platonism to include logicism. Myhill (1972) takes logicism to include platonism, as do many writers who distinguish only three tendencies in modern philosophy of mathematics. In part I, the two tendencies are distinguished, they are shown to have separate histories, and they are shown to be independent in various senses. Part II treats varieties of logicism. Part III considers the mutual opposition of a logicism held by Russell and others versus a platonism held by Gödel and others. A neutral stance, respecting the coherence and plausibility of each tendency, is maintained throughout.
This issue poses the question: what information really is. We assume that information has some pl... more This issue poses the question: what information really is. We assume that information has some place or other in reality, in particular, we assume there is a cohesive and coherent account of informational phenomena, able to coherently set up facts, contents and values regarding information. In our current information era it seems natural to assume without further critical reflection a disunited class of uses of “information”. The point of this issue of tripleC is setting up a cohesive account of information in complex contemporary open societies and scientific communities.
There are at least three dimensions in our plea for such cohesive account of information: (a) from a conceptual point of view, there is a plethora of seemingly incompatible notions of “information”, (b) from a societal viewpoint, information can acritically postulate a new infinite realm of merchandise which does not foster a more cohesive society but instead a growing inequality, (c) a coherent unified approach to both the manifest image and the scientific image of information is still lacking.
This article discusses two coextensive concepts of logical consequence that are implicit in the t... more This article discusses two coextensive concepts of logical consequence that are implicit in the two fundamental logical practices of establishing validity and invalidity for premise-conclusion arguments. The premises and conclusion of an argument have information content (they 'say' something), and they have subject matter (they are 'about' something). The asymmetry between establishing validity and establishing invalidity has long been noted: validity is established through an information-processing procedure exhibiting a step-by-step deduction of the conclusion from the premise-set. Invalidity is established by exhibiting a countermodel satisfying the premises but not the conclusion. The process of establishing validity focuses on information content; the process of establishing invalidity focuses on subject matter. Corcoran's information-theoretic concept of logical consequence corresponds to the former. Tarski's model-theoretic concept of logical consequence formulated in his famous 1936 no-countermodels definition corresponds to the latter. Both are found to be indispensable for understanding the rationale of the deductive method and each complements the other. This study discusses the ontic question of the nature of logical consequence and the epistemic question of the human capabilities presupposed by practical applications of these two concepts as they make validity and invalidity accessible to human knowledge.
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This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory.
In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege–Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.
Drafts by Jose Sagüillo
Books by Jose Sagüillo
There are at least three dimensions in our plea for such cohesive account of information: (a) from a conceptual point of view, there is a plethora of seemingly incompatible notions of “information”, (b) from a societal viewpoint, information can acritically postulate a new infinite realm of merchandise which does not foster a more cohesive society but instead a growing inequality, (c) a coherent unified approach to both the manifest image and the scientific image of information is still lacking.
This paper discusses the history of the confusion and controversies over whether the definition of consequence presented in the 11-page 1936 Tarski consequence-definition paper is based on a monistic fixed-universe framework—like Begriffsschrift and Principia Mathematica. Monistic fixed-universe frameworks, common in pre-WWII logic, keep the range of the individual variables fixed as ‘the class of all individuals’. The contrary alternative is that the definition is predicated on a pluralistic multiple-universe framework—like the 1931 Gödel incompleteness paper. A pluralistic multiple-universe framework recognizes multiple universes of discourse serving as different ranges of the individual variables in different interpretations—as in post-WWII model theory.
In the early 1960s, many logicians—mistakenly, as we show—held the ‘contrary alternative’ that Tarski 1936 had already adopted a Gödel-type, pluralistic, multiple-universe framework. We explain that Tarski had not yet shifted out of the monistic, Frege–Russell, fixed-universe paradigm. We further argue that between his Principia-influenced pre-WWII Warsaw period and his model-theoretic post-WWII Berkeley period, Tarski’s philosophy underwent many other radical changes.
There are at least three dimensions in our plea for such cohesive account of information: (a) from a conceptual point of view, there is a plethora of seemingly incompatible notions of “information”, (b) from a societal viewpoint, information can acritically postulate a new infinite realm of merchandise which does not foster a more cohesive society but instead a growing inequality, (c) a coherent unified approach to both the manifest image and the scientific image of information is still lacking.