Post-Established Harmony: Kant and Analogy Reconsidered

This essay is a response to John Milbank’s comparison of Kant and Aquinas’ theories of analogy in ‘A Critique of the Theology of Right’. A critique of Milbank’s essay forms the point of departure for my reconstruction of Kant’s actual theory of analogy. I show that the usual focus on the Prolegomena for this end is insufficient; in fact, the full extent of Kant’s theory of analogy only becomes clear in the Critique of Judgment. I also consider the significance of the Analogies of Experience in the Critique of Pure Reason. In conclusion, I draw on the work of Michel Guérin to designate Kantian analogy, ‘post-established harmony’.     

Morality andin vitro fertilization

The question of the morality ofin vitro fertilization is examined. One of the central questions to be answered is whether the zygote loss that seems inseparable from the process is morally justified. Even when embryo transfer occurs, many zygotes which have been intentionally created are intentionally destroyed; they are used as means to the alleged benefits that others will attain (the benefit to the infertile couple, to the child produced by the process, and to those who might benefit from the increase of genetic knowledge that allegedly will occur fromin vitro research). The justifications advanced in defense of early abortion are discussed, and it is shown that these justifications must be totally distinct from those advanced in support ofin vitro fertilization. A theory of values is proposed which shows why a set of reasons may justify early abortion, but not an abortion late in the pregnancy. This theory states that the value characteristics of an entity are notidentical to the characteristics which make the entityhuman. It is concluded that if certain key empirical assumptions are correct, then in vitro fertilization is a morally permissible process; however, the falsity of these assumptions, or the unsoundness of the theory of value, might well result in a reversal of this judgment.     

Does Kant'sMetaphysical Foundations of Natural Science fill a gap in theCritique of Pure Reason?

In 1792 and 1798 Kant noticed two basic problems with hisMetaphysical Foundations of Natural Science (MAdN) which opened a crucial gap in the Critical system as a whole. Why is theMAdN so important? I show that the Analogies of Experience form an integrated proof of transeunt causality. This is central to Kant's ‘answer’ to Hume. This proof requires explicating the empirical concept of matter as “the moveable in space”, it requires the specifically metaphysical principle that every physical event has an external cause, and it requires a metaphysical principle regarding the individuation of spatio-temporal things. These three doctrines are not defended in the firstCritique, but only in theMAdN. Kant's transcendental analysis of the conditions of experience thus requires the “special metaphysics” of theMAdN. This marks an important shift in Kant's view of the metaphysical basis of the transcendental philosophy.     

Cut-elimination for Weak Grzegorczyk Logic Go

We present a syntactic proof of cut-elimination for weak Grzegorczyk logic Go. The logic has a syntactically similar axiomatisation to Gödel–Löb logic GL (provability logic) and Grzegorczyk’s logic Grz. Semantically, GL can be viewed as the irreflexive counterpart of Go, and Grz can be viewed as the reflexive counterpart of Go. Although proofs of syntactic cut-elimination for GL and Grz have appeared in the literature, this is the first proof of syntactic cut-elimination for Go. The proof is technically interesting, requiring a deeper analysis of the derivation structures than the proofs for GL and Grz. New transformations generalising the transformations for GL and Grz are developed here.     

Proofs,pictures, and Euclid

Though pictures are often used to present mathematical arguments, they are not typically thought to be an acceptable means for presenting mathematical arguments rigorously. With respect to the proofs in the Elements in particular, the received view is that Euclid’s reliance on geometric diagrams undermines his efforts to develop a gap-free deductive theory. The central difficulty concerns the generality of the theory. How can inferences made from a particular diagrams license general mathematical results? After surveying the history behind the received view, this essay provides a contrary analysis by introducing a formal account of Euclid’s proofs, termed Eu. Eu solves the puzzle of generality surrounding Euclid’s arguments. It specifies what diagrams Euclid’s diagrams are, in a precise formal sense, and defines generality-preserving proof rules in terms of them. After the central principles behind the formalization are laid out, its implications with respect to the question of what does and does not constitute a genuine picture proof are explored.     

Assertion and grounding: a theory of assertion for constructive type theory

Taking Per Martin-Löf’s constructive type theory as a starting-point a theory of assertion is developed, which is able to account for the epistemic aspects of the speech act of assertion, and in which it is shown that assertion is not a wide genus. From a constructivist point of view, one is entitled to assert, for example, that a proposition A is true, only if one has constructed a proof object a for A in an act of demonstration. One thereby has grounded the assertion by an act of demonstration, and a grounding account of assertion therefore suits constructive type theory. Because the act of demonstration in which such a proof object is constructed results in knowledge that A is true, the constructivist account of assertion has to ward off some of the criticism directed against knowledge accounts of assertion. It is especially the internal relation between a judgement being grounded and its being known that makes it possible to do so. The grounding account of assertion can be considered as a justification account of assertion, but it also differs from justification accounts recently proposed, namely in the treatment of selfless assertions, that is, assertions which are grounded, but are not accompanied by belief.     

From the Knowability Paradox to the existence of proofs

The Knowability Paradox purports to show that the controversial but not patently absurd hypothesis that all truths are knowable entails the implausible conclusion that all truths are known. The notoriety of this argument owes to the negative light it appears to cast on the view that there can be no verification-transcendent truths. We argue that it is overly simplistic to formalize the views of contemporary verificationists like Dummett, Prawitz or Martin-Löf using the sort of propositional modal operators which are employed in the original derivation of the Paradox. Instead we propose that the central tenet of verificationism is most accurately formulated as follows: if \({\varphi}\) is true, then there exists a proof of \({\varphi}\). Building on the work of Artemov (Bull Symb Log 7(1): 1–36, 2001), a system of explicit modal logic with proof quantifiers is introduced to reason about such statements. When the original reasoning of the Paradox is developed in this setting, we reach not a contradiction, but rather the conclusion that there must exist non-constructed proofs. This outcome is evaluated relative to the controversy between Dummett and Prawitz about proof existence and bivalence.     

LE^{t}_{ \to } , LR^{ \circ }_{{\widehat{ \sim }}}, LK and Cutfree Proofs

Two consecution calculi are introduced: one for the implicational fragment of the logic of entailment with truth and another one for the disjunction free logic of nondistributive relevant implication. The proof technique—attributable to Gentzen—that uses a double induction on the degree and on the rank of the cut formula is shown to be insufficient to prove admissible various forms of cut and mix in these calculi. The elimination theorem is proven, however, by augmenting the earlier double inductive proof with additional inductions. We also give a new purely inductive proof of the cut theorem for the original single cut rule in Gentzen’s sequent calculus $ LK $ without any use of mix.     

Curry’s Paradox and ω -Inconsistency

In recent years there has been a revitalised interest in non-classical solutions to the semantic paradoxes¹. In this paper I show that a number of logics are susceptible to a strengthened version of Curry’s paradox. This can be adapted to provide a proof theoretic analysis of the ω-inconsistency in ?ukasiewicz’s continuum valued logic, allowing us to better evaluate which logics are suitable for a naïve truth theory. On this basis I identify two natural subsystems of ukasiewicz logic which individually, but not jointly, lack the problematic feature.     

Estimation of the correlation coefficient in the case of a bivariate normal population when one of the variables is dichotomized

It is shown that the problem of estimation of the correlation coefficient of a bivariate normal population when one of the variables is dichotomized may be attacked with “probit analysis” methods. This represents an extension of the work of Gillman and Goode (3), as it was possible to find by this approach an approximation to the large-sample variance of the resulting estimateG ofρ. An empirical investigation was undertaken with the object of obtaining some information about the distribution ofG for large sample size. Methods for determining the “pass-fail” cut-off are considered.     

On the theory of test discrimination

This paper discusses the properties of distributions of test scores and advances the view that the properties of the distribution should depend on the function which the test is intended to perform. A theory of test discrimination is developed which defines discriminatory capacity in terms of the number of relations of difference established by the operation of administering a test ofk items to a sample ofn individuals. A simple proof is presented which indicates that maximum discrimination between individuals is achieved when tests are constructed to yield distributions of the rectangular form. A coefficient of test discrimination is developed. The problem of obtaining in practice distributions approximating to the rectangular form is briefly discussed.This paper was prepared under the auspices of the Defence Research Board, Ottawa, Canada.     

A theory of cheating

The theory of motivated cheating postulates that test takers may cheat when they do not know an answer. With probabilityk, an “observer” is unsure of an answer and will copy from a nearby “target” with probabilityc. The corresponding parameters for the target may be entirely unrelated to those of the observer. Thus, the undesirable feature of bidirectionality of parameters found in correlational techniques is not an inherent feature of this theory of cheating. Predictions are derived, and estimates ofk andc are proposed. Statistically large values of c suggest that an observer was copying from a target. High values ofc for both the observer and the target suggest collusion. The theory is applied to a 40-item five-choice test taken by students in an introductory psychology section. From the full paired comparison matrix of target × observer parameter estimates, the method identifies 2 students who were probably in collusion.     

Understanding Negation Implicationally in the Relevant Logic R

A star-free relational semantics for relevant logic is presented together with a sound and complete sequent proof theory (display calculus). It is an extension of the dualist approach to negation regarded as modality, according to which de Morgan negation in relevant logic is better understood as the confusion of two negative modalities. The present work shows a way to define them in terms of implication and a new connective, co-implication, which is modeled by respective ternary relations. The defined negations are confused by a special constraint on ternary relation, called the generalized star postulate, which implies definability of the Routley star in the frame. The resultant logic is shown to be equivalent to the well-known relevant logic R. Thus it can be seen as a reconstruction of R in the dualist framework.     

Inferentializing Semantics

The entire development of modern logic is characterized by various forms of confrontation of what has come to be called proof theory with what has earned the label of model theory. For a long time the widely accepted view was that while model theory captures directly what logical formalisms are about, proof theory is merely our technical means of getting some incomplete grip on this; but in recent decades the situation has altered. Not only did proof theory expand into new realms, generalizing the concept of proof in various directions; many philosophers also realized that meaning may be seen as primarily consisting in certain rules rather than in language-world links. However, the possibility of construing meaning as an inferential role is often seen as essentially compromised by the limits of proof-theoretical means. The aim of this paper is to sort out the cluster of problems besetting logical inferentialism by disentangling and clarifying one of them, namely determining the power of various inferential frameworks as measured by that of explicitly semantic ones.     

Inertial eigenvalues, rod density, and rod diameter in length perception by dynamic touch

Four experiments addressed the relevance of the eigenvaluesI _k of the inertia tensor for perceiving length by dynamic touch. Experiments 1–2 focused on the consequences of limiting variation in the minimum eigenvalueI ₃. Both revealed that perceived length is a function ofI _k . Whether the contribution ofI ₃ is detected, however, depends on the range of values that characterize a particular object set. Experiments 3–4 considered the relationship between an independent index of a rod’s diameter, which does not affectI _k , and actual manipulation of a rod’s diameter, which does affectI _k . Whereas the former appeared as satisfaction of implicit instructions to alter reports of perceived length, the latter entailed actual differences in perceived length in accordance withI _k . Results are discussed with respect to the links among actual length, perceived length, andI _k , as well as, in particular, how these links guarantee that perceived length is in the range of actual lengths.     

On some proof theoretical properties of the modal logic GL

This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.     

Diagrammatic reasoning in Frege’s Begriffsschrift

In Part III of his 1879 logic Frege proves a theorem in the theory of sequences on the basis of four definitions. He claims in Grundlagen that this proof, despite being strictly deductive, constitutes a real extension of our knowledge, that it is ampliative rather than merely explicative. Frege furthermore connects this idea of ampliative deductive proof to what he thinks of as a fruitful definition, one that draws new lines. My aim is to show that we can make good sense of these claims if we read Frege??s notation diagrammatically, in particular, if we take that notation to have been designed to enable one to exhibit the (inferentially articulated) contents of concepts in a way that allows one to reason deductively on the basis of those contents.     

Unique Best Deserver Theory and Arguments From Misclassification

It is a core commitment of Epistemic Two-Dimensionalism (E2D) that an utterance is 1-necessary iff it is a priori. But Jeff Speaks's Argument from Misclassification proves that, on a natural interpretation, E2D assigns necessary 1-intensions to many utterances that speakers deem a posteriori. Given that 1-intensions are meant to formalize a speaker's own understanding of the words she utters, this proof raises serious difficulties for E2D. In response, Elliott, McQueen, and Weber point out that the Argument from Misclassification presupposes a controversial theory of reference-fixation for proper names and argue that E2Dists ought to reject this theory. They discuss three alternatives to the theory that render E2D immune to the Argument from Misclassification. I demonstrate here that each of these alternatives either (i) replicates the faults of the original theory by assigning necessary 1-intensions to a posteriori utterances, or (ii) makes 1-intensions creatures of darkness. I argue that, although the Argument from Misclassification does indeed rely on a controversial theory of reference-fixation, Elliott, McQueen, and Weber have indicated no promising alternative to this theory.