Penrose's New Argument

It has been argued, by Penrose and others, that Gödel's proof of his first incompleteness theorem shows that human mathematics cannot be captured by a formal system F: the Gödel sentence G(F) of F can be proved by a (human) mathematician but is not provable in F. To this argment it has been objected that the mathematician can prove G(F) only if (s)he can prove that F is consistent, which is unlikely if F is complicated. Penrose has invented a new argument intended to avoid this objection. In the paper I try to show that Penrose's new argument is inconclusive.     

Skolem's Discovery of Gödel-Dummett Logic

Attention is drawn to the fact that what is alternatively known as Dummett logic, Gödel logic, or Gödel-Dummett logic, was actually introduced by Skolem already in 1913. A related work of 1919 introduces implicative lattices, or Heyting algebras in today's terminology.     

Menschen, maschinen und g?dels theorem

Mechanism is the thesis that men can be considered as machines, that there is no essential difference between minds and machines.John Lucas has argued that it is a consequence of Gödel's theorem that mechanism is false. Men cannot be considered as machines, because the intellectual capacities of men are superior to that of any machine. Lucas claims that we can do something that no machine can do-namely to produce as true the Gödel-formula of any given machine. But no machine can prove its own Gödel-formula.In order to discuss and evaluate this argument, the author makes a distinction between formal and informal proofs, and between proofs given by men and proofs given by machines. It is argued that the informal proof capacities of machines are possibly greater and the formal proof capacities of men are possibly smaller than the anti-mechanist claims. So the argument from Gödel's theorem against mechanism fails.Though Gödel's theorem does not prove that minds are different from machines, it is not irrelevant to the analysis of thought and to the mind/machine controversy. It points to the importance of informal methods even within formal sciences and to the need for an analysis of the notion of informal thinking in cognitive science.     

I. Psychologism,Functionalism, and the modal status of logical Laws

In a recent article (Inquiry, Vol. 19 [1976]), J. W. Meiland addresses the issue of psychologism in logic, which holds that logic is a branch of psychology and that logical laws (such as the Principle of Non‐Contradiction) are contingent upon the nature of the mind. Meiland examines Husserl's critique of psychologism, argues that Husserl is not convincing, and offers two new objections to the psychologistic thesis. In this paper I attempt to rebut those objections. In question are the acceptable criteria for determining the possibility or impossibility of systems of logic significantly different from our own. I argue that a criteriological application of our accepted laws of logic to this question commits a circular fallacy. I then argue that, even if we accept logical consistency as a criterion for possibility, a plausible argument for the possibility of valid alternative logics can be constructed by using the functionalist analogy between minds and automata. Finally, I attempt to rebut the claim that in logic the only changes possible are conceptual changes that would not permit a proposition to be both true and false.     

The Greatest Extension of S4 into which Intuitionistic Logic is Embeddable

This paper gives a characterization of those quasi-normal extensions of the modal system S4 into which intuitionistic propositional logic Int is embeddable by the Gödel translation. It is shown that, as in the normal case, the set of quasi-normal modal companions of Int contains the greatest logic, M*, for which, however, the analog of the Blok-Esakia theorem does not hold. M* is proved to be decidable and Halldén-complete; it has the disjunction property but does not have the finite model property.     

Reasoning about partial functions with the aid of a computer

Partial functions are ubiquitous in both mathematics and computer science. Therefore, it is imperative that the underlying logical formalism for a general-purpose mechanized mathematics system provide strong support for reasoning about partial functions. Unfortunately, the common logical formalisms — first-order logic, type theory, and set theory — are usually only adequate for reasoning about partial functionsin theory. However, the approach to partial functions traditionally employed by mathematicians is quite adequatein practice. This paper shows how the traditional approach to partial functions can be formalized in a range of formalisms that includes first-order logic, simple type theory, and Von-Neumann—Bernays—Gödel set theory. It argues that these new formalisms allow one to directly reason about partial functions; are based on natural, well-understood, familiar principles; and can be effectively implemented in mechanized mathematics systems.Supported by the MITRE-Sponsored Research program. This paper is a written version (with references) of an address given at the Partial Functions and Programming: Foundational Questions conference held 17 February 1995 at the University ol California at Irvine.     

Platonism and metaphor in the texts of mathematics: G?del and Frege on mathematical knowledge

In this paper, I challenge those interpretations of Frege that reinforce the view that his talk of grasping thoughts about abstract objects is consistent with Russell's notion of acquaintance with universals and with Gödel's contention that we possess a faculty of mathematical perception capable of perceiving the objects of set theory. Here I argue the case that Frege is not an epistemological Platonist in the sense in which Gödel is one. The contention advanced is that Gödel bases his Platonism on a literal comparison between mathematical intuition and physical perception. He concludes that since we accept sense perception as a source of empirical knowledge, then we similarly should posit a faculty of mathematical intuition to serve as the source of mathematical knowledge. Unlike Gödel, Frege does not posit a faculty of mathematical intuition. Frege talks instead about grasping thoughts about abstract objects. However, despite his hostility to metaphor, he uses the notion of grasping as a strategic metaphor to model his notion of thinking, i.e., to underscore that it is only by logically manipulating the cognitive content of mathematical propositions that we can obtain mathematical knowledge. Thus, he construes grasping more as theoretical activity than as a kind of inner mental seeing.     

A Set Theory with Support for Partial Functions

Partial functions can be easily represented in set theory as certain sets of ordered pairs. However, classical set theory provides no special machinery for reasoning about partial functions. For instance, there is no direct way of handling the application of a function to an argument outside its domain as in partial logic. There is also no utilization of lambda-notation and sorts or types as in type theory. This paper introduces a version of von-Neumann-Bernays-Gödel set theory for reasoning about sets, proper classes, and partial functions represented as classes of ordered pairs. The underlying logic of the system is a partial first-order logic, so class-valued terms may be nondenoting. Functions can be specified using lambda-notation, and reasoning about the application of functions to arguments is facilitated using sorts similar to those employed in the logic of the IMPS Interactive Mathematical Proof System. The set theory is intended to serve as a foundation for mechanized mathematics systems.     

On Logics with Coimplication

This paper investigates (modal) extensions of Heyting–Brouwer logic, i.e., the logic which results when the dual of implication (alias coimplication) is added to the language of intuitionistic logic. We first develop matrix as well as Kripke style semantics for those logics. Then, by extending the Gödel-embedding of intuitionistic logic into S4 , it is shown that all (modal) extensions of Heyting–Brouwer logic can be embedded into tense logics (with additional modal operators). An extension of the Blok–Esakia-Theorem is proved for this embedding.     

Time in philosophy and in physics: From Kant and Einstein to Gödel

The essay centers on Gödel's views on the place of our intuitive concept of time in philosophy and in physics. It presents my interpretation of his work on the theory of relativity, his observations on the relationship between Einstein's theory and Kantian philosophy, as well as some of the scattered remarks in his conversations with me in the seventies — namely, those on the philosophies of Leibniz, Hegel and Husserl — as a successor of Kant — in relation to their conceptions of time.For the physical world, the four dimensions are natural. But for the mind, there is no such natural coordinate system; time is the only natural frame of reference.Gödel, conversation on 15.3.72     

Exhibiting interpretational and representational validity

A natural language argument may be valid in at least two nonequivalent senses: it may be interpretationally or representationally valid (Etchemendy in The concept of logical consequence. Harvard University Press, Cambridge, 1990). Interpretational and representational validity can both be formally exhibited by classical first-order logic. However, as these two notions of informal validity differ extensionally and first-order logic fixes one determinate extension for the notion of formal validity (or consequence), some arguments must be formalized by unrelated nonequivalent formalizations in order to formally account for their interpretational or representational validity, respectively. As a consequence, arguments must be formalized subject to different criteria of adequate formalization depending on which variant of informal validity is to be revealed. This paper develops different criteria that formalizations of an argument have to satisfy in order to exhibit the latter’s interpretational or representational validity.     

Children's understanding of human and super-natural mind

Barrett, Richert, and Driesenga [Barrett, J. L., Richert, R. A., & Driesenga, A. (2001). God's beliefs versus mother's: The development of nonhuman agents concepts. Child Development, 72(1), 50–65] have suggested that children are able to conceptualize the representational properties held by certain super-natural entities, such as God, before they achieve representational understanding of the human mind. The two experimental conditions of the present study aimed at cross-checking the above suggestion. One hundred and twenty children aged from 3 to 7 years were involved in both conditions. In the first, a modified perspective-taking and appearance-reality task, similar to that adopted in Barrett et al.'s study, was used. The task in the second addressed another aspect of representational understanding of the human mind, that is, the early emerging of the rule that knowledge is constrained by perception. The results of the study showed that younger children systematically treat God as a human protagonist regarding the representational properties they possess. Moreover, it was found that children are able to reason, accurately, about God's representational properties, only upon reaching their 5th year of age, when their representational understanding of the human mind becomes stable and robust.     

On detachment-substitutional formalization in normal modal logics

The aim of this paper is to propose a criterion of finite detachment-substitutional formalization for normal modal systems. The criterion will comprise only those normal modal systems which are finitely axiomatizable by means of the substitution, detachment for material implication and Gödel rules.Some results of this paper were announced in the abstract [2].Allatum est die 10 Junii 1976     

Kripke on functionalism and automata

Saul Kripke has proposed an argument to show that there is a serious problem with many computational accounts of physical systems and with functionalist theories in the philosophy of mind. The problem with computational accounts is roughly that they provide no noncircular way to maintain that any particular function with an infinite domain is realized by any physical system, and functionalism has the similar problem because of the character of the functional systems that are supposed to be realized by organisms. This paper shows that the standard account of what it is for a physical system to compute a function can avoid Kripke's criticisms without being reduced to circularity; a very minor and natural elaboration of the standard account suffices to save both functionalist theories and computational accounts generally.I am indebted to Saul Kripke for several helpful discussions of this material. I also benefitted from the discussions following the presentations of earlier versions of this paper at the University of Pennsylvania (February, 1984), UCLA (June, 1984), and Rutgers University (December, 1984), and particularly from my discussions with Elizabeth Spelke and Scott Weinstein.     

Embedding Logics into Product Logic

We construct a faithful interpretation of ukasiewicz's logic in product logic (both propositional and predicate). Using known facts it follows that the product predicate logic is not recursively axiomatizable.We prove a completeness theorem for product logic extended by a unary connective of Baaz [1]. We show that Gödel's logic is a sublogic of this extended product logic.We also prove NP-completeness of the set of propositional formulas satisfiable in product logic (resp. in Gödel's logic).     

On the Meaning of Hilbert's Consistency Problem (Paris, 1900)

The theory that ``consistency implies existence' was put forward by Hilbert on various occasions around the start of the last century, and it was strongly and explicitly emphasized in his correspondence with Frege. Since (Gödel's) completeness theorem, abstractly speaking, forms the basis of this theory, it has become common practice to assume that Hilbert took for granted the semantic completeness of second order logic. In this paper I maintain that this widely held view is untrue to the facts, and that the clue to explain what Hilbert meant by linking together consistency and existence is to be found in the role played by the completeness axiom within both geometrical and arithmetical axiom systems.     

Nature,nurture, and capital punishment: How evidence of a genetic–environment interaction,future dangerousness,and deliberation affect sentencing decisions

Research has shown that the low‐activity MAOA genotype in conjunction with a history of childhood maltreatment increases the likelihood of violent behaviors. This genetic–environment (G × E) interaction has been introduced as mitigation during the sentencing phase of capital trials, yet there is scant data on its effectiveness. This study addressed that issue. In a factorial design that varied mitigating evidence offered by the defense [environmental (i.e., childhood maltreatment), genetic, G × E, or none] and the likelihood of the defendant's future dangerousness (low or high), 600 mock jurors read sentencing phase evidence in a capital murder trial, rendered individual verdicts, and half deliberated as members of a jury to decide a sentence of death or life imprisonment. The G × E evidence had little mitigating effect on sentencing preferences: participants who received the G × E evidence were no less likely to sentence the defendant to death than those who received evidence of childhood maltreatment or a control group that received neither genetic nor maltreatment evidence. Participants with evidence of a G × E interaction were more likely to sentence the defendant to death when there was a high risk of future dangerousness than when there was a low risk. Sentencing preferences were more lenient after deliberation than before. We discuss limitations and future directions.     

The Definability of the set of natural numbers in the 1925 Principia Mathematica

In his new introduction to the 1925 second edition of Principia Mathematica, Russell maintained that by adopting Wittgenstein's idea that a logically perfect language should be extensional mathematical induction could be rectified for finite cardinals without the axiom of reducibility. In an Appendix B, Russell set forth a proof. Gödel caught a defect in the proof at *89.16, so that the matter of rectification remained open. Myhill later arrived at a negative result: Principia with extensionality principles and without reducibility cannot recover mathematical induction. The finite cardinals are indefinable in it. This paper shows that while Gödel and Myhill are correct, Russell was not wrong. The 1925 system employs a different grammar than the original Principia. A new proof for *89.16 is given and induction is recovered.     

Six-year-olds' difficulties handling intensional contexts

Previous research shows that children have difficulties handling intensional contexts even when they can pass a test of false belief (e.g. Cognition 67 (1998) 287; Cognition 25 (1987) 289). Some authors (Perner, J. (1991). Understanding the representational mind. Cambridge, MA: MIT Press; Cognition 25 (1987) 289) place these difficulties in the linguistic and not the mental representational domain. The experiments reported here examined whether 6-year-old children could answer questions in an intensional context that did not require the explicit verbal characterization of a belief. We replicated previous findings and found that children answered according to their own knowledge in an intensional context. This occurred even though they responded by choosing a picture to insert into a protagonist's thought bubble rather than report the belief verbally. Children could correctly answer questions about the knowledge state of the protagonist and pass a test of false belief. Further experiments ruled out methodological explanations. Experiment 2 showed that the difference in answering according to own knowledge between the false belief and intensional stories is not accounted for by procedural factors in the two types of test. Experiment 3 revealed that children did not answer according to their own knowledge by default. Experiment 4 suggested that answering according to own knowledge was not a result of pictorial salience. Results are discussed in relation to the simulation-theory debate.