Do Goedel's incompleteness theorems set absolute limits on the ability of the brain to express and communicate mental concepts verifiably?

Classical interpretations of Goedel's formal reasoning imply that the truth of some arithmetical propositions of any formal mathematical language, under any interpretation, is essentially unverifiable. However, a language of general, scientific, discourse cannot allow its mathematical propositions to be interpreted ambiguously. Such a language must, therefore, define mathematical truth verifiably. We consider a constructive interpretation of classical, Tarskian, truth, and of Goedel's reasoning, under which any formal system of Peano Arithmetic is verifiably complete. We show how some paradoxical concepts of Quantum mechanics can be expressed, and interpreted, naturally under a constructive definition of mathematical truth.Comment: 73 pages; this is an updated version of the NQ essay; an HTML version is available at http://alixcomsi.com/Do_Goedel_incompleteness_theorems.ht

Ordinary Truth in Tarski and Næss

Alfred Tarski seems to endorse a partial conception of truth, the T-schema, which he believes might be clarified by the application of empirical methods, specifically citing the experimental results of Arne Næss (1938a). The aim of this paper is to argue that Næss’ empirical work confirmed Tarski’s semantic conception of truth, among others. In the first part, I lay out the case for believing that Tarski’s T-schema, while not the formal and generalizable Convention-T, provides a partial account of truth that may be buttressed by an examination of the ordinary person’s views of truth. Then, I address a concern raised by Tarski’s contemporaries who saw Næss’ results as refuting Tarski’s semantic conception. Following that, I summarize Næss’ results. Finally, I will contend with a few objections that suggest a strict interpretation of Næss’ results might recommend an overturning of Tarski’s theory

The Principle Of Excluded Middle Then And Now: Aristotle And Principia Mathematica

The prevailing truth-functional logic of the twentieth century, it is argued, is incapable of expressing the subtlety and richness of Aristotle's Principle of Excluded Middle, and hence cannot but misinterpret it. Furthermore, the manner in which truth-functional logic expresses its own Principle of Excluded Middle is less than satisfactory in its application to mathematics. Finally, there are glimpses of the "realism" which is the metaphysics demanded by twentieth century logic, with the remarkable consequent that Classical logic is a particularly inept instrument to analyze those philosophies which stand opposed to the "realism" it demands

e-Learning Nudism: Stripping Context from Content

Educational economics plays an increasing role in university development. In order to attract students well developed curricula are needed and they are expected to contain a fair amount of digital resources, which are much more expensive to create than sheets of paper in the old days. The flip side is: they can be sold. Whereas hand-outs remained an obscure asset, suitably organized electronic courseware promises to become a major business. As "Learning Management Systems" offer comprehensive services to entire universities at substantial costs, university administrators try to channel traditional teaching into new formats, hoping to serve more students at lesser expense. One catchword, capturing those concerns, is "learning object". A learning object is the equivalent of a chunk of beef, registered according to some classificatory scheme, marked by a stamp of approval by some authority, deep-frozen and waiting for delivery. Here is a more respectable description. Learning objects are digital entities designed to be used (and re-used) in learning activities.[1] They are supposed to be independent of specific educational settings, disengaged from more comprehensive courses. Information pertaining to their educational, technical and legal status is to be captured by meta-data accompanying the objects. Learning object repositories (LORs) collect those molecular units and offer facilities for search and peer evaluation

Information enforcement in learning with graphics : improving syllogistic reasoning skills

This thesis is an investigation into the factors that contribute to good choices among graphical systems used in teaching, and the feasibility of implementing teaching software that uses this knowledge.The thesis describes a mathematical metric derived from a cognitive theory of human diagram processing. The theory characterises differences among representations by their ability to express information. The theory provides the factors and relationships needed to build the metric. It says that good representations are easily processed because they are more vivid, more tractable and less expressive, than poor representations.The metric is applied to abstract systems for teaching and learning syllogistic reasoning, TARSKI'S WORLD, EULER CIRCLES, VENN DIAGRAMS and CARROLL'S GAME OF LOGIC. A rank ordering reflects the value of each system predicted by the theory and the metric. The theory, the metric and the systems are then tested in empirical studies. Five studies involving sixty-eight learners, examined the benefit of software based on these abstract systems.Studies showed the theory correctly predicted learners' success with the circle systems and poorer performance with TARSKI'S WORLD. The metric showed small but clear differences in expressivity between the circle systems. Differences between results of the learners using the circle systems contradicted the predictions of the metric.Learners with mathematical training were better equipped and more successful at learning syllogistic reasoning with the systems. Performance of learners without mathematical training declined after using the software systems. Diagrams drawn by learners together with video footage collected during problem solving, led to a catalogue of errors, misconceptions and some helpful strategies for learning from graphical systems.A cognitive style test investigated the poor performance of non-mathematically trained learners. Learners with mathematics training showed serialist and versatile learning styles while learners without this training showed a holist learning style. This is consistent with the hypothesis that non-mathematically trained learners emphasise the use of semantic cues during learning and problem solving.A card-sorting task investigated learners' preferences for parts of the graphical lexicon used in the diagram systems. Preferences for the EULER lexicon increased difficulty in explaining the system's poor results in earlier studies. Video footage of learners using the systems in the final study illustrated useful learning strategies and improved performance with EULER while individual instruction was available.Further work describes a preliminary design for an adaptive syllogism tutor and other related work

Stalnaker and Field on truth and intentionality.

In a series of publications, Robert Stalnaker and Hartry Field have undertaken a dispute about what is the correct way to explain intentionality naturalistically. Field wishes to assimilate mental intentionality to linguistic intentionality and to explain both kinds of intentionality using Tarskian truth theory plus the causal theory of reference. Stalnaker wishes to subsume mental intentionality under the notion of indication and to explain it on the model of measurement theory, leaving linguistic intentionality to be explained derivatively. I attempt to adjudicate their dispute, paying particular attention to the question whether Tarskian truth theory has a role to play in explaining intentionality naturalistically. The first half of the dissertation examines Tarski\u27s theory. I argue that Field and Stalnaker are incorrect when they agree that Tarski\u27s reduction of the notion of truth is defective and I explain Tarski\u27s \u27structural\u27 notion of truth. In the second half of the dissertation, I argue that Stalnaker\u27s criticisms of Field mostly do not stand up to scrutiny, but that Field\u27s theory is unsatisfactory nonetheless; I also argue that certain criticisms of Stalnaker do more damage than he thinks. I conclude that neither has solved \u27the problem of intentionality\u27, but that Stalnaker is right that truth theory as such will not have a role in the correct solution

On the Relation of Informal to Formal Logic

The distinction between formal and informal logic is clarified as a prelude to considering their ideal relation. Aristotle\u27s syllogistic describes forms of valid inference, and is in that sense a formal logic. Yet the square of opposition and rules of middle term distribution of positive or negative propositions in an argument\u27s premises and conclusion are standardly received as devices of so-called informal logic and critical reasoning. I propose a more exact criterion for distinguishing between formal and informal logic, and then defend a model for fruitful interaction between informal and formal methods of investigating and critically assessing the logic of arguments

Metaontological Studies relating to the Problem of Universals

My dissertation deals with metaontology or metametaphysics. This is the subdiscipline of philosophy that is concerned with the investigation of metaphysical concepts, statements, theories and problems on the metalevel. It analyses the meaning of metaphysical statements and theories and discusses how they are to be justified. The name "metaontology" is recently coined, but the task of metaontology is the same as Immanuel Kant already dealt with in his Critique of Pure Reason. As methods I use both historical research and logical (or rather semantical) analysis. In order to understand clearly what metaphysical terms or theories mean or should mean we must both look at how they have been characterized in the course of the history of philosophy and then analyse the meanings that have historically been given to them with the methods of modern formal semantics. Metaontological research would be worthless if it could not in the end be applied to solving some substantive ontological questions. In the end of my dissertation, therefore, I give arguments for a solution to the substantively ontological problem of universals, a form of realism about universals called promiscuous realism. To prepare the way for that argument, I argue that the metaontological considerations most relevant to the problem of universals are considerations concerning ontological commitment, as the American philosophers Quine and van Inwagen have argued, not those concerning truthmakers as such philosophers as the Australian realist D. M. Armstrong have argued or those concerning verification conditions as such philosophers as Michael Dummett have argued. To justify this conclusion, I go first through well-known objections to verificationism, and show that they apply also to current verificationist theories such as Dummett's theory and Field's deflationist theory of truth. In the process I also respond to opponents of metaphysics who try to show with the aid of verificationism or structuralism that metaphysical questions would be meaningless or illegitimate in some other way. Having justified the central role of ontological commitment, I try to develop a detailed theory of it. The core of my work is a rigorous formal development of a theory of ontological commitment. I construct it by combining Alonzo Church's theory of ontological commitment with Tarski's theory of truth.Väitöskirjani käsittelee metaontologiaa eli metametafysiikkaa. Tämä on se metafilosofian osa-alue, joka tutkii metafyysisten väitteiden ja termien merkitystä ja sitä, miten metafyysiset väitteet ja teoriat voitaisiin oikeuttaa. Metafysiikka tai ontologia on taas tiede, joka tutkii olevaa yleensä tai kaikkeutta kokonaisuutena. Menetelminä käytän sekä historiallista tutkimusta että loogista (tai pikemminkin semanttista) analyysiä. On olemassa kolme pääasiallista teoriaa siitä, mikä on metaontologian keskeisin käsite. Sellaiset filosofit kuin australialainen Armstrong ovat väittäneet, että se on totuustekijöiden (truthmakers) käsite. Sellaiset anti-realistiset filosofit kuin englantilainen filosofi Michael Dummett ovat taas väittäneet että se on todennettavuusehtojen (verification conditions) käsite. Argumentoin näitä kahta käsitystä vastaan ja kolmannen puolesta, jonka mukaan keskeisin käsite on ontologisten sitoumusten käsite, kuten amerikkalainen filosofi Quine on väittänyt. Argumentoin, että Quinen ontologisten sitoumusten teoria voidaan erottaa hänen muista ontologisista näkemyksistään, kuten hänen semanttisesta holismistaan, ontologisesta relativismistaan tai strukturalismistaan, mitkä ovat mielestäni virheellisiä. Väitöskirjani ydin on täsmällinen teoria ontologisista sitoumuksista, jonka rakennan yhdistämällä Alonzo Churchin teoriaa ontologisista sitoumuksista Alfred Tarskin totuusteoriaan. Metaontologinen tutkimus olisi arvotonta, ellei sitä voisi lopulta käyttää substantiivisten ontologisten kysymysten ratkaisemiseen. Käsittelen siksi väitöskirjani loppupuolella yhtä perinteistä ontologian ongelmaa, universaalien ongelmaa. Jo Aristoteles määritteli teoksessaan Tulkinnasta universaalien olevan olioita, jotka (Lauri Carlsonin käännöksen mukaan) luonnostaan predikoidaan (sanotaan) monesta. Universaaliongelma koskee sitä, ovatko tällaiset universaalit vain kielellisiä ilmauksia, kuten yleisnimet, verbit ja adjektiivit, tai ihmismielestä riippuvia olioita, kuten yleiskäsitteet, vai voidaanko myös sanoa, että maailmassa itsessään olevia olioita voidaan predikoida jostakin. Realistin mukaan vastaus on myöntävä. Esitän väitöskirjan lopussa alustavan argumentin universaaleja koskevan realismin puolesta

With reference to truth : studies in referential semantics

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Linguistics and Philosophy, 1982.MICROFICHE COPY AVAILABLE IN ARCHIVES AND HUMANITIESVita.Includes bibliographical references.by Douglas Fillmore Cannon.Ph.D