Richard E. Grandy

Psychology and epistemology: match or mismatch when applied to science education?

Cognitive psychologys descriptions of an individuals knowledge resemble those philosophers’ offer of scientific theory. Both offer resources for conceptual change teaching. Yet the similarities mask tensions ‐philosophers stress rationality and psychologists focus on causal structure. Both domains distinguish two kinds of change in knowledge structures‐one common and cumulative, the other rare and non‐cumulative. The structures facilitate incremental development but resist major revisions. Unless instruction actively induces restructuring, students’ knowledge will be confused and incomplete. Knowledge is largely organized by schemata, representing the significant concepts and relations in a domain. But using knowledge also requires procedures for recalling, applying and revising schemata. Questions discussed include: When should we present a theory in the context of justification‐‐where knowledge claims are systematically but a historically delineated; and when in the context of development‐‐where knowl...

Constructivisms and Objectivity: Disentangling Metaphysics from Pedagogy.

We can distinguish the claims of cognitive constructivism from those of metaphysical constructivism, which is almost entirely irrelevant to science education. Cognitive constructivism has strong empirical support and indicates important directions for changing science instruction. It implies that teachers need to be cognizant of representational, motivational and epistemic dimensions which can restrict or promote student learning. The resulting set of tasks for a science teacher are considerably larger and more complex than on the older more traditional conception, but the resources of cognitive sciences and the history of science can provide important parts of the teachers intellectual tool kit. A critical part of this conception of science education is that students must develop the skills to participate in epistemic interchanges. They must be provided opportunities and materials to develop those skills and the classroom community must have the appropriate features of an objective epistemic community.

Conditional Assertions and "Biscuit" Conditionals

1. “Biscuit” Conditionals~1! There are biscuits on the sideboard if you want some~2! If you’re interested, there’s a good documentary on PBS tonightand~3! Oswald shot Kennedy, if that’s what you’re asking me,as they’d typically be used, are examples of what are often called “biscuit” con-ditionals, after J.L. Austin’s example ~~1!, above; see Austin, 1970, p. 212!.Amark of such a conditional is that, after it has been uttered, it can only be a certainkind of joke to ask what is the case if the antecedent is false—“And where are thebiscuits if I don’t want any?”, “And what’s on PBS if I’m not interested?”, “Andwho shot Kennedy if that’s not what I’m asking?”. With normal indicative con-ditionals like,~4! There are biscuits on the sideboard if Bill hasn’t moved them~5! If the TV Guide is accurate, there’s a good documentary on PBS tonightand~6! Oswald shot Kennedy, if there hasn’t been an enormous conspiracy,

Stuff and Things

Metaphysical problems often concern the relation between various types of entities or putative entities. Philosophers at least since Aristotle have puzzled over the relation between particulars and universals, substances and things, movements and actions. Debate over the first of these issues still continues, but I think it is fair to say that there was an increase in clarity when, around the turn of the century, the notion of a set was made clearer and more explicit. For the purposes of logic, at least, the use of a set of objects as the analogue of a universal has been quite fruitful. Interpreting a predicate by means of a set of ordered n-tuples has provided a precise characterization of the semantics of the language which has led to interesting metatheoretical and philosophical consequences. There are still some problems which are not resolved by this approach. For example, it seems intuitively plausible that there could be two distinct universals with the same instances. Nevertheless, the approach at least gives an unproblematic theory of truth for sentences involving only particulars and universals. The issue about distinct universals with the same instances is a disputed one and in any case the set theoretic view enables one to give a sharp statement of the issue in dispute.

Advanced logic for applications

I. Henkin Sets and the Fundamental Theorem.- II. Derivation Rules and Completeness.- III. Gentzen Systems and Constructive Completeness Proofs.- IV. Quantification Theory with Identity and Functional Constants.- V. First Order Theories with Equality.- VI. Godels Incompleteness Theorems: Preliminary Discussion.- VII. Undecidability and Incompleteness.- VIII. Godels Second Incompleteness Theorem.- IX. Tarskis Theorems and the Definition of Truth.- X. Some Recursive Function Theory.- XI. Intuitionistic Logic.- XII. Second Order Logic.- XIII. Algebraic Logic.- XIV. Anadic Logic.- Selected Bibliography.- Index of Names.- Index of Subjects.- Index of Symbols.

Undecidability and Incompleteness

Our main objectives are to show (I) Q(N) is incomplete, i.e. there is a closed A such that neither ⊢ Q A nor ⊢ Q -A. (II) Q(N) is undecidable, i.e. there is no effective way of deciding whether ⊢ Q A. Using (II) for Q we can establish (III) Functional calculus is undecidable. (I) and (II) would only show that Q and N are inadequate formalizations of our intuitive concepts if we could not also show (IV) Any consistent effective extension of Q is incomplete. (V) Any consistent effective extension of Q is undecidable.

Suppes’ Contribution to Logic and Linguistics

Patrick Suppes’ contributions in the areas of logic, linguistics and philosophy of language are marked by a characteristic methodological approach. The end is always a clear formulation of a theory that makes it accessible to empirical test; the means is a formulation within a well-understood mathematical theory. The product of his inquiries is not always, however, a definitive set of observational predictions derived from the more precisely formulated theory; often the inquiry leads to conceptual or mathematical problems that must be resolved before any testable consequences can be derived. Thus the work is best seen as a program directed toward making the theories in question more precise and testable. I have divided Suppes’ work into three areas: the logic of empirical theories, theories of syntax, theories of semantics. In each case I have found it necessary to select from among his considerable writings those which seem most significant and representative. Each area is treated in a section below.

Quantification Theory with Identity and Functional Constants

We wish now to extend our theory to include identity and functions. The system will be an extension of HPC which we will call HPC=. The primitive symbols are -, ⊃, ∀, ), (, and =; an infinite list of individual variables x0, x1, x2,..., an infinite list of predicate letters F 0 n , F 1 n , F 2 n ,... for each n > 0, an infinite list of constants, c0, c1,... and for each n > 0 an infinite list of function letters f 0 n , f 1 n ,...

Henkin Sets and the Fundamental Theorem

We will begin by proving a fundamental result which will be used repeatedly in the proofs of our major theorems. We will prove it for the full language of quantification theory even though some of our systems will have a restricted vocabulary. No change in the proof is required for the restricted vocabularies.

Some Recursive Function Theory

We defined a function f to be computable just in case the relation j(x 1,… x n) = z is n.r. in Q. This implies that for some number e