We show that van Lambalgen's Theorem fails with respect to recursive randomness and Schnorr randomness for some real in every high degree and provide a full characterization of the Turing degrees for which van Lambalgen's Theorem can fail with respect to Kurtz randomness. However, we also show that there is a recursively random real that is not Martin-Löf random for which van Lambalgen's Theorem holds with respect to recursive randomness.

Van Lambalgen's Theorem plays an important role in algorithmic randomness, especially when studying relative randomness. In this paper we extend van Lambalgen's Theorem by considering the join of infinitely many reals which are random relative to each other. In addition, we study computability of the reals in the range of Omega operators. It is known that $\Omega^{\phi'}$ is high. We extend this result to that $\Omega^{\phi^{(n)}}$ is $\textrm{high}_n$ . We also prove that there exists A such (...) that, for each n , the real $\Omega^A_M$ is $\textrm{high}_n$ for some universal Turing machine M by using the extended van Lambalgen's Theorem. (shrink)

In this article we provide a mathematical model of Kant?s temporal continuum that satisfies the (not obviously consistent) synthetic a priori principles for time that Kant lists in the Critique of pure Reason (CPR), the Metaphysical Foundations of Natural Science (MFNS), the Opus Postumum and the notes and frag- ments published after his death. The continuum so obtained has some affinities with the Brouwerian continuum, but it also has ‘infinitesimal intervals’ consisting of nilpotent infinitesimals, which capture Kant’s theory of rest (...) and motion in MFNS. While constructing the model, we establish a concordance between the informal notions of Kant?s theory of the temporal continuum, and formal correlates to these notions in the mathematical theory. Our mathematical reconstruction of Kant?s theory of time allows us to understand what ?faculties and functions? must be in place for time to satisfy all the synthetic a priori principles for time mentioned. We have presented here a mathematically precise account of Kant?s transcendental argument for time in the CPR and of the rela- tion between the categories, the synthetic a priori principles for time, and the unity of apperception; the most precise account of this relation to date. We focus our exposition on a mathematical analysis of Kant’s informal terminology, but for reasons of space, most theorems are explained but not formally proven; formal proofs are available in (Pinosio, 2017). The analysis presented in this paper is related to the more general project of developing a formalization of Kant’s critical philosophy (Achourioti & van Lambalgen, 2011). A formal approach can shed light on the most controversial concepts of Kant’s theoretical philosophy, and is a valuable exegetical tool in its own right. However, we wish to make clear that mathematical formalization cannot displace traditional exegetical methods, but that it is rather an exegetical tool in its own right, which works best when it is coupled with a keen awareness of the subtleties involved in understanding the philosophical issues at hand. In this case, a virtuous ?hermeneutic circle? between mathematical formalization and philosophical discourse arises. (shrink)

We present a faithful axiomatization of von Mises' notion of a random sequence, using an abstract independence relation. A byproduct is a quantifier elimination theorem for Friedman's "almost all" quantifier in terms of this independence relation.

We present a critical discussion of the claim (most forcefully propounded by Chaitin) that algorithmic information theory sheds new light on Godel's first incompleteness theorem.

Although Kant (1998) envisaged a prominent role for logic in the argumentative structure of his Critique of Pure Reason, logicians and philosophers have generally judged Kantgeneralformaltranscendental logics is a logic in the strict formal sense, albeit with a semantics and a definition of validity that are vastly more complex than that of first-order logic. The main technical application of the formalism developed here is a formal proof that Kants logic is after all a distinguished subsystem of first-order logic, namely what (...) is known as geometric logic. (shrink)

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for (...) which van Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

In this paper we provide a mathematical model of Kant’s temporal continuum that yields formal correlates for Kant’s informal treatment of this concept in theCritique of Pure Reasonand in other works of his critical period. We show that the formal model satisfies Kant’s synthetic a priori principles for time and that it even illuminates what “faculties and functions” must be in place, as “conditions for the possibility of experience”, for time to satisfy such principles. We then present a mathematically precise (...) account of Kant’s transcendental theory of time—the most precise account to date.Moreover, we show that the Kantian continuum which we obtain has some affinities with the Brouwerian continuum but that it also has “infinitesimal intervals” consisting of nilpotent infinitesimals; these allow us to capture Kant’s theory of rest and motion in theMetaphysical Foundations of Natural Science.While our focus is on Kant’s theory of time the material in this paper is more generally relevant for the problem of developing a rigorous theory of the phenomenological continuum, in the tradition of Whitehead, Russell, and Weyl among others. (shrink)

We aim to show that Kant’s theory of time is consistent by providing axioms whose models validate all synthetic a priori principles for time proposed in the Critique of Pure Reason. In this paper we focus on the distinction between time as form of intuition and time as formal intuition, for which Kant’s own explanations are all too brief. We provide axioms that allow us to construct ‘time as formal intuition’ as a pair of continua, corresponding to time as ‘inner (...) sense’ and the external representation of time as a line Both continua are replete with infinitesimals, which we use to elucidate an enigmatic discussion of ‘rest’ in the Metaphysical foundations of natural science. Our main formal tools are Alexandroff topologies, inverse systems and the ring of dual numbers. (shrink)

Interpretation is the process whereby a hearer reasons to an interpretation of a speaker's discourse. The hearer normally adopts a credulous attitude to the discourse, at least for the purposes of interpreting it. That is to say the hearer tries to accommodate the truth of all the speaker's utterances in deriving an intended model. We present a nonmonotonic logical model of this process which defines unique minimal preferred models and efficiently simulates a kind of closed-world reasoning of particular interest for (...) human cognition. Byrne's “suppression” data (Byrne, 1989) are used to illustrate how variants on this logic can capture and motivate subtly different interpretative stances which different subjects adopt, thus indicating where more fine-grained empirical data are required to understand what subjects are doing in this task. We then show that this logical competence model can be implemented in spreading activation network models. A one pass process interprets the textual input by constructing a network which then computes minimal preferred models for (3-valued) valuations of the set of propositions of the text. The neural implementation distinguishes easy forward reasoning from more complex backward reasoning in a way that may be useful in explaining directionality in human reasoning. (shrink)

We sketch four applications of Marr's levels‐of‐analysis methodology to the relations between logic and experimental data in the cognitive neuroscience of language and reasoning. The first part of the paper illustrates the explanatory power of computational level theories based on logic. We show that a Bayesian treatment of the suppression task in reasoning with conditionals is ruled out by EEG data, supporting instead an analysis based on defeasible logic. Further, we describe how results from an EEG study on temporal prepositions (...) can be reanalyzed using formal semantics, addressing a potential confound. The second part of the article demonstrates the predictive power of logical theories drawing on EEG data on processing progressive constructions and on behavioral data on conditional reasoning in people with autism. Logical theories can constrain processing hypotheses all the way down to neurophysiology, and conversely neuroscience data can guide the selection of alternative computational level models of cognition. (shrink)

We review briefly the attempts to define random sequences. These attempts suggest two theorems: one concerning the number of subsequence selection procedures that transform a random sequence into a random sequence; the other concerning the relationship between definitions of randomness based on subsequence selection and those based on statistical tests.

Gaisi Takeuti has recently proposed a new operation on orthomodular lattices L, ⫫: $\scr{P}\rightarrow L$ . The properties of ⫫ suggest that the value of ⫫ $$ corresponds to the degree in which the elements of A behave classically. To make this idea precise, we investigate the connection between structural properties of orthomodular lattices L and the existence of two-valued homomorphisms on L.

We reformulate slightly Russell's notion of typicality, so as to eliminate its circularity and make it applicable to elements of any first‐order structure. We argue that the notion parallels Martin‐Löf (ML) randomness, in the sense that it uses definable sets in place of computable ones and sets of “small” cardinality (i.e., strictly smaller than that of the structure domain) in place of measure zero sets. It is shown that if the domain M satisfies, then there exist typical elements and only (...) non‐typical ones. In particular this is true for the standard model of second‐order arithmetic. By allowing parameters in the defining formulas, we are led to relative typicality, which satisfies most of van Lambalgen's axioms for relative randomness. However van Lambalgen's theorem is false for relative typicality. The class of typical reals is incomparable (with respect to ⊆) with the classes of ML‐random, Schnorr random and computably random reals. Also the class of typical reals is closed under Turing degrees and under the jump operation (both ways). (shrink)

This reply to Oaksford and Chater’s ’s critical discussion of our use of logic programming to model and predict patterns of conditional reasoning will frame the dispute in terms of the semantics of the conditional. We begin by outlining some common features of LP and probabilistic conditionals in knowledge-rich reasoning over long-term memory knowledge bases. For both, context determines causal strength; there are inferences from the absence of certain evidence; and both have analogues of the Ramsey test. Some current work (...) shows how a combination of counting defeaters and statistics from network monitoring can provide the information for graded responses from LP reasoning. With this much introduction, we then respond to O&C’s specific criticisms and misunderstandings. (shrink)

Sieg has proposed axioms for computability whose models can be reduced to Turing machines. This lecture will investigate to what extent these axioms hold for reasoning. In particular we focus on the requirement that the configurations that a computing agent (whether human or machine) operates on must be ’immediately recognisable’. If one thinks of reasoning as derivation in a calculus, this requirement is satisfied; but even in contexts which are only slightly less formal, the requirement cannot be met. Our main (...) example will be the Wason selection task, a propositional reasoning task in which in a typical (undergraduate) subject group only around 5% arrive at the answer dictated by classical logic. The instructions for this task (as well as other standard tasks in the psychology of reasoning, such as syllogisms) do not contain any ’immediately recognisable’ configurations. The subject must try to find an interpretation of the task by making the various elements in the instructions cohere, in effect solving a difficult constraint satisfaction problem, which has no unique solution. The subject has given a complete interpretation of the task if she can formulate the problem posed in the task as a theorem to be proved. The complexity of such theorems can be quite high; e.g. for the propositional Wason selection task the theorem can be in Σ1 3 . This sounds implausible, but we’ll present experimental data confirming this point. (shrink)

We review the various explanations that have been offered toaccount for subjects'' behaviour in Wason ''s famous selection task. Weargue that one element that is lacking is a good understanding ofsubjects'' semantics for the key expressions involved, and anunderstanding of how this semantics is affected by the demands the taskputs upon the subject''s cognitive system. We make novel proposals inthese terms for explaining the major content effects of deonticmaterials. Throughout we illustrate with excerpts from tutorialdialogues which motivate the kinds of (...) analysis proposed. Our long termgoal is an integration of the various insights about conditionalreasoning on offer from different cognitive science methodologies. Thepurpose of this paper is to try to draw the attention of logicians andsemanticists to this area, since we believe that empirical investigationof the cognitive processes involved could benefit from semanticanalyses. (shrink)

We investigate the role of continuous reductions and continuous relativization in the context of higher randomness. We define a higher analogue of Turing reducibility and show that it interacts well with higher randomness, for example with respect to van Lambalgen’s theorem and the Miller–Yu/Levin theorem. We study lowness for continuous relativization of randomness, and show the equivalence of the higher analogues of the different characterizations of lowness for Martin-Löf randomness. We also characterize computing higher [Formula: see text]-trivial sets (...) by higher random sequences. We give a separation between higher notions of randomness, in particular between higher weak 2-randomness and [Formula: see text]-randomness. To do so we investigate classes of functions computable from Kleene’s [Formula: see text] based on strong forms of the higher limit lemma. (shrink)

This essay attempts to develop a psychologically informed semantics of perception reports, whose predictions match with the linguistic data. As suggested by the quotation from Miller and Johnson-Laird, we take a hallmark of perception to be its fallible nature; the resulting semantics thus necessarily differs from situation semantics. On the psychological side, our main inspiration is Marr's (1982) theory of vision, which can easily accomodate fallible perception. In Marr's theory, vision is a multi-layered process. The different layers have filters of (...) different gradation, which makes vision at each of them approximate. On the logical side, our task is therefore twofold - to formalise the layers and the ways in which they may refine each other, and. (shrink)

We study randomness beyond${\rm{\Pi }}_1^1$-randomness and its Martin-Löf type variant, which was introduced in [16] and further studied in [3]. Here we focus on a class strictly between${\rm{\Pi }}_1^1$and${\rm{\Sigma }}_2^1$that is given by the infinite time Turing machines introduced by Hamkins and Kidder. The main results show that the randomness notions associated with this class have several desirable properties, which resemble those of classical random notions such as Martin-Löf randomness and randomness notions defined via effective descriptive set theory such as${\rm{\Pi (...) }}_1^1$-randomness. For instance, mutual randoms do not share information and a version of van Lambalgen’s theorem holds.Towards these results, we prove the following analogue to a theorem of Sacks. If a real is infinite time Turing computable relative to all reals in some given set of reals with positive Lebesgue measure, then it is already infinite time Turing computable. As a technical tool towards this result, we prove facts of independent interest about random forcing over increasing unions of admissible sets, which allow efficient proofs of some classical results about hyperarithmetic sets. (shrink)

We study inversions of the jump operator on ${\mathrm{\Pi }}_{1}^{0}$ classes, combined with certain basis theorems. These jump inversions have implications for the study of the jump operator on the random degrees—for various notions of randomness. For example, we characterize the jumps of the weakly 2-random sets which are not 2-random, and the jumps of the weakly 1-random relative to 0′ sets which are not 2-random. Both of the classes coincide with the degrees above 0′ which are not 0′-dominated. A (...) further application is the complete solution of [24, Problem 3.6.9]: one direction of van Lambalgen's theorem holds for weak 2-randomness, while the other fails. Finally we discuss various techniques for coding information into incomplete randoms. Using these techniques we give a negative answer to [24, Problem 8.2.14]: not all weakly 2-random sets are array computable. In fact, given any oracle X, there is a weakly 2-random which is not array computable relative to X. This contrasts with the fact that all 2-random sets are array computable. (shrink)

It is well known that, in classical theory, Liouville's theorem shows that if an ensemble of systems is distributed over a small element of volume in phase space, the ensemble fills a region of equal volume at all later instants of time. In quantum mechanics, the uncertainty principle is associated with the products of the errors in conjugate coordinates and momenta, and such products can be interpreted in terms of volume elements in phase space. Comparison of these two facts (...) leads to the assertion in the interesting volume on “Statistical Mechanics” recently published by J. E. and M. G. Mayer that “The Liouville theorem is essential for the complete understanding of the uncertainty principle. (shrink)

In 1969, De Jongh proved the “maximality” of a fragment of intuitionistic predicate calculus forHA. Leivant strengthened the theorem in 1975, using proof-theoretical tools (normalisation of infinitary sequent calculi). By a refinement of De Jongh's original method (using Beth models instead of Kripke models and sheafs of partial combinatory algebras), a semantical proof is given of a result that is almost as good as Leivant's. Furthermore, it is shown thatHA can be extended to Higher Order Heyting Arithmetic+all trueΠ 2 (...) 0 -sentences + transfinite induction over primitive recursive well-orderings. As a corollary of the proof, maximality of intuitionistic predicate calculus is established wrt. an abstract realisability notion defined over a suitable expansion ofHA. (shrink)

ABSTRACTOaksford and Chater critiqued the logic programming approach to nonmonotonicity and proposed that a Bayesian probabilistic approach to conditional reasoning provided a more empirically adequate theory. The current paper is a reply to Stenning and van Lambalgen's rejoinder to this earlier paper entitled ‘Logic programming, probability, and two-system accounts of reasoning: a rejoinder to Oaksford and Chater’ in Thinking and Reasoning. It is argued that causation is basic in human cognition and that explaining how abnormality lists are created in (...) LP requires causal models. Each specific rejoinder to the original critique is then addressed. While many areas of agreement are identified, with respect to the key differences, it is concluded the current evidence favours the Bayesian approach, at least for the moment. (shrink)

Luitzen Egburtus Jan Brouwer founded a school of thought whose aim was to include mathematics within the framework of intuitionistic philosophy; mathematics was to be regarded as an essentially free development of the human mind. What emerged diverged considerably at some points from tradition, but intuitionism has survived well the struggle between contending schools in the foundations of mathematics and exact philosophy. Originally published in 1981, this monograph contains a series of lectures dealing with most of the fundamental topics such (...) as choice sequences, the continuum, the fan theorem, order and well-order. Brouwer's own powerful style is evident throughout the work. (shrink)

In a brief but remarkable section of the Inquiry into the Human Mind, Thomas Reid argued that the visual field is governed by principles other than the familiar theorems of Euclid—theorems we would nowadays classify as Riemannian. On the strength of this section, he has been credited by Norman Daniels, R. B. Angell, and others with discovering non-Euclidean geometry over half a century before the mathematicians—sixty years before Lobachevsky and ninety years before Riemann. I believe that Reid does indeed have (...) a fair claim to have discovered a non-Euclidean geometry. However, the real basis of Reid’s geometry of visibles is subtle and easy to misidentify. My main aim in what follows is to make clear what the real basis is, separating it from several fallacious or irrelevant considerations on which Reid may seem to be relying. A secondary aim is to air the worry that Reid’s case for his geometry can succeed only at the cost of compromising the achievement for which Reid is best known—his direct realist theory of perception. (shrink)

Brouwer's demonstration of his Bar Theorem gives rise to provocative questions regarding the proper explanation of the logical connectives within intuitionistic and constructivist frameworks, respectively, and, more generally, regarding the role of logic within intuitionism. It is the purpose of the present note to discuss a number of these issues, both from an historical, as well as a systematic point of view.

For an arbitrary similarity type of Boolean Algebras with Operators we define a class ofSahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities arecanonical, that is, their validity is preserved under taking canonical embedding algebras. (...) Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V.We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity calledHenkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the first-order correspondents of these two equations are equivalent over the class of cylindric atom structures. (shrink)

For an arbitrary similarity type of Boolean Algebras with Operators we define a class of Sahlqvist identities. Sahlqvist identities have two important properties. First, a Sahlqvist identity is valid in a complex algebra if and only if the underlying relational atom structure satisfies a first-order condition which can be effectively read off from the syntactic form of the identity. Second, and as a consequence of the first property, Sahlqvist identities are canonical, that is, their validity is preserved under taking canonical (...) embedding algebras. Taken together, these properties imply that results about a Sahlqvist variety V van be obtained by reasoning in the elementary class of canonical structures of algebras in V. We give an example of this strategy in the variety of Cylindric Algebras: we show that an important identity called Henkin's equation is equivalent to a simpler identity that uses only one variable. We give a conceptually simple proof by showing that the firstorder correspondents of these two equations are equivalent over the class of cylindric atom structures. (shrink)

This is an extensively revised edition of Mr. Quine's introduction to abstract set theory and to various axiomatic systematizations of the subject. The treatment of ordinal numbers has been strengthened and much simplified, especially in the theory of transfinite recursions, by adding an axiom and reworking the proofs. Infinite cardinals are treated anew in clearer and fuller terms than before. Improvements have been made all through the book; in various instances a proof has been shortened, a theorem strengthened, a (...) space-saving lemma inserted, an obscurity clarified, an error corrected, a historical omission supplied, or a new event noted. (shrink)

We propose an extension of Aczel's constructive set theory CZF by an axiom for inductive types and a choice principle, and show that this extension has the following properties: it is interpretable in Martin-Löf's type theory. In addition, it is strong enough to prove the Set Compactness theorem and the results in formal topology which make use of this theorem. Moreover, it is stable under the standard constructions from algebraic set theory, namely exact completion, realizability models, forcing as (...) well as more general sheaf extensions. As a result, methods from our earlier work can be applied to show that this extension satisfies various derived rules, such as a derived compactness rule for Cantor space and a derived continuity rule for Baire space. Finally, we show that this extension is robust in the sense that it is also reflected by the model constructions from algebraic set theory just mentioned. (shrink)

Philosophical issues often turn into logic. That is certainly true of Moore’s Paradox, which tends to appear and reappear in many philosophical contexts. There is no doubt that its study belongs to pragmatics rather than semantics or syntax. But it is also true that issues in pragmatics can often be studied fruitfully by attending to their projection, so to speak, onto the levels of semantics or syntax — just in the way that problems in spherical geometry are often illuminated by (...) the study of projections onto a plane. To begin I will describe a potentially vast landscape of logics of a certain form, with some illustrations of how they appear naturally in response to some problems in philosophical logic. Then I will turn Moore’s Paradox into logic, within that landscape, and show how far it can be illuminated therein. (shrink)

Kurt Gödel’s incompleteness theorems and the limits of knowledge In this paper a presentation is given of Kurt Gödel’s pathbreaking results on the incompleteness of formal arithmetic. Some biographical details are provided but the main focus is on the analysis of the theorems themselves. An intermediate level between informal and formal has been sought that allows the reader to get a sufficient taste of the technicalities involved and not lose sight of the philosophical importance of the results. Connections are established (...) with the work of Alan Turing and Hao Wang to show the present-day relevance of Gödel’s research and how it relates to the limitations of human knowledge, mathematical knowledge in particular. (shrink)

On the intended interpretation of intuitionistic logic, Heyting's Proof Interpretation, a proof of a proposition of the form p -> q consists in a construction method that transforms any possible proof of p into a proof of q. This involves the notion of the totality of all proofs in an essential way, and this interpretation has therefore been objected to on grounds of impredicativity (e.g. Gödel 1933). In fact this hardly ever leads to problems as in proofs of implications usually (...) nothing more is assumed about a proof of the antecedent than that it indeed is one, and this assumption does not require a further grasp of the totality of proofs. The prime example of an intuitionistic theorem that goes beyond that assumption is Brouwer's proof of the 'bar theorem': For every tree x, if x contains a decidable subset of nodes such that every path through the tree meets it (a 'bar'), then there is a well-ordered subtree of x that contains a bar for the whole of x. Instantiated with an arbitrary tree t, this proposition takes the form P(t) -> Q(t). Brouwer's proof of the bar theorem mainly consists in an analysis of the inner structure that a proof of P(t) must have, where proofs are taken to be primarily mental objects. So here Brouwer engages in phenomenological reflection by considering the acts in which we think about bars. From that analysis he obtains the information from which to construct a proof of Q(t). In this talk I will argue that Brouwer circumvents the problem of impredicativity by resorting to a transcendental argument based on phenomenological description, and defend this application by showing how common objections to transcendental arguments do not apply here. Finally, I will indulge in some historical speculation by relating the foregoing considerations to the remarkable change that Gödel's view on the Proof Interpretation underwent between his Yale Lecture (1941) and the Dialectica paper (1958). (shrink)

I started out as a student of physics, hard-working, interested, but alas, not ‘in love’ with my subject. Then logic struck, and having become interested in this subject for various reasons – including the fascinating personality of my first teacher –, I switched after my candidate’s program, to take two master’s degrees, in mathematics and in philosophy. The beauty of mathematics was clear to me at once, with the amazing power, surprising twists, and indeed the music, of abstract arguments. As (...) our professor of Analysis wrote at the time in our study guide “Mathematics is about the delight in the purity of trains of thought”, and oldfashioned though this phrasing sounded in the revolutionary 1960s, it did resonate with me. Then I had the privilege of being taught set-theoretic topology by a group of brilliant students around De Groot, our leading expert around the time, who worked with Moore’s method of discovering a subject for oneself. Topology unfolded from a few definitions and examples to real theorems that we had to prove ourselves – and the take-home exam took sleepless nights, as it included proving some results from scratch which came from a recent dissertation (as it turned out later). Only at the very end did De Groot appear, to give one lecture on Tychonoff’s Theorem where an application was made of the Axiom of Choice, a sacral act only to be performed by tenured full professors. (shrink)

We investigate cardinal invariants related to the structure of dense sets of rationals modulo the nowhere dense sets. We prove that , thus dualizing the already known [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. Math. 183 59–80, Theorem 3.6]. We also show the consistency of each of and . Our results answer four questions of Balcar, Hernández and Hrušák [B. Balcar, F. Hernández-Hernández, M. Hrušák, Combinatorics of dense subsets of the rationals, Fund. (...) Math. 183 59–80, Questions 3.11]. (shrink)

This chapter contains sections titled: Abstract Introduction Logic and Confronting the Truth Avoiding Contradiction in Discourse Relativism Inside Logic: Ways of Avoiding Contradictions Avoiding Contradiction in the Setting of Agency Conclusion References.

In “Models and Reality”, H. Putnam formulated his model-theoretic argument against “metaphysical realism”. The article proposes a meta-reconstruction of Putnam’s model-theoretic argument in the light of two mutually compatible interpretations of it–elaborated by Manuel Garcia-Carpintero and Igor van Douven. A critical reflection on these interpretations and their adequacy for Putnam’s argument allows us to expose new theses coherent with Putnam’s reasoning and indicate new paths to improve this argument for our reconstruction task. In particular, we show that Putnam’s position may (...) be coherent with van Douven’s versions of Global Descriptivism under some conditions, but Putnam cannot reject realism as quickly as Carpintero suggests. We show that Suszko’s canonic axiomatic system and Sneed’s concept of theory may provide valuable support for Putnam’s argument. Finally, we critically evaluate Carpintero’s theses about the genesis of unintended interpretations of our languages, adopting the machinery of the Upward Skolem–Loewenheim Theorem and Knight’s Theorem. (shrink)