This paper presents a model-theoretic semantics for discourse "about" natural numbers, one that captures what I call "the mathematical-object picture", but avoids what I can "the mathematical-object theory".

This paper considers two reasons that might support Russell’s choice of a ramified-type theory over a simple-type theory. The first reason is the existence of purported paradoxes that can be formulated in any simple-type language, including an argument that Russell considered in 1903. These arguments depend on certain converse-compositional principles. When we take account of Russell’s doctrine that a propositional function is not a constituent of its values, these principles turn out to be too implausible to make these arguments troubling. (...) The second reason is conditional on a substitutional interpretation of quantification over types other than that of individuals. This reason stands up to investigation: a simple-type language will not sustain such an interpretation, but a ramified-type language will. And there is evidence that Russell was tacitly inclined towards such an interpretation. A strong construal of that interpretation opens a way to make sense of Russell’s simultaneous repudiation of propositions and his willingness to quantify over them. But that way runs into trouble with Russell’s commitment to the finitude of human understanding. (shrink)

Discourse carries thin commitment to objects of a certain sort iff it says or implies that there are such objects. It carries a thick commitment to such objects iff an account of what determines truth-values for its sentences say or implies that there are such objects. This paper presents two model-theoretic semantics for mathematical discourse, one reflecting thick commitment to mathematical objects, the other reflecting only a thin commitment to them. According to the latter view, for example, the semantic role (...) of number-words is not designation but rather the encoding of cardinality-quantifiers. I also present some reasons for preferring this view. (shrink)

This paper and its sequel “look under the hood” of the usual sorts of proof-theoretic systems for certain well-known intuitionistic and classical propositional modal logics. Section 1 is preliminary. Of most importance: a marked formula will be the result of prefixing a formula in a propositional modal language with a step-marker, for this paper either 0 or 1. Think of 1 as indicating the taking of “one step away from 0.” Deductions will be constructed using marked formulas. Section 2 presents (...) the model-theoretic concepts, based on those in [7], that guide the rest of this paper. Section 3 presents Natural Deduction systems IK and CK, formalizations of intuitionistic and classical one-step versions of K. In these systems, occurrences of step-markers allow deductions to display deductive structure that is covered over in familiar “no step” proof-theoretic systems for such logics. Box and Diamond are governed by Introduction and Elimination rules; the familiar K rule and Necessitation are derived (i.e. admissible) rules. CK will be the result of adding the 0-version of the Rule of Excluded Middle to the rules which generate IK. Note: IK is the result of merely dropping that rule from those generating CK, without addition of further rules or axioms (as was needed in [7]). These proof-theoretic systems yield intuitionistic and classical consequence relations by the obvious definition. Section 4 provides some examples of what can be deduced in IK. Section 5 defines some proof-theoretic concepts that are used in Section 6 to prove the soundness of the consequence relation for IK (relative to the class of models defined in Section 2.) Section 7 proves its completeness (relative to that class). Section 8 extends these results to the consequence relation for CK. (Looking ahead: Part 2 will investigate one-step proof-theoretic systems formalizing intuitionistic and classical one-step versions of some familiar logics stronger than K.). (shrink)

A model-theoretic approach to the semantics of set-theoretic discourse.

Where $\underline{a}$ is a Turing degree and ξ is an ordinal $ , the result of performing ξ jumps on $\underline{a},\underline{a}^{(\xi)}$ , is defined set-theoretically, using Jensen's fine-structure results. This operation appears to be the natural extension through $(\aleph_1)^{L^\underline{a}}$ of the ordinary jump operations. We describe this operation in more degree-theoretic terms, examine how much of it could be defined in degree-theoretic terms and compare it to the single jump operation.

An Exact Pair for the Arithmetic Degrees whose join is not a Weak Uniform Upper Bound, in the Recursive Function Theory-Newsletters, No. 28, August-September 1982.

I prove that the Boolean Prime Ideal Theorem is equivalent, under some weak set-theoretic assumptions, to what I will call the Cut-for-Formulas to Cut-for-Sets Theorem: for a set F and a binary relation |- on Power(F), if |- is finitary, monotonic, and satisfies cut for formulas, then it also satisfies cut for sets. I deduce the CF/CS Theorem from the Ultrafilter Theorem twice; each proof uses a different order-theoretic variant of the Tukey- Teichmüller Lemma. I then discuss relationships between various (...) cut-conditions in the absence of finitariness or of monotonicity. (shrink)

Part 1 [Hodes, 2021] “looked under the hood” of the familiar versions of the classical propositional modal logic K and its intuitionistic counterpart. This paper continues that project, addressing some familiar classical strengthenings of K and GL), and their intuitionistic counterparts. Section 9 associates two intuitionistic one-step proof-theoretic systems to each of the just mentioned intuitionistic logics, this by adding for each a new rule to those which generated IK in Part 1. For the systems associated with the intuitionistic (...) counterparts of D and T, these rules are “pure one-step”: their schematic formulations does not use □ or ♢. For the systems associated with the intuitionistic counterparts of K4, etc., these rules meet these conditions: neither □ nor ♢ is iterated; none use both □ and ♢. The join of the two systems associated with each of these familiar logics is the full one-step system for that intuitionistic logic. And further “blended” intuitionistic systems arise from joining these systems in various ways. Adding the 0-version of Excluded Middle to their intuitionistic counterparts yields the one-step systems corresponding to the familiar classical logics. Each proof-theoretic system defines a consequence relation in the obvious way. Section 10 examines inclusions between these consequence relations. Section 11 associates each of the above consequence relations with an appropriate class of models, and proves them sound with respect to their appropriate class. This allows proofs of some failures of inclusion between consequence relations. Section 12 proves that the each consequence relation is complete or weakly complete, that relative to its appropriate class of models. The Appendix presents three further results about some of the intuitionistic consequence relations discussed in the body of the paper. For Keywords, see Part 1. (shrink)

Proof uses forcing on perfect trees for 2-quantifier sentences in the language of arithmetic. The result extends to exact pairs for the hyperarithmetic degrees.

Relative to any reasonable frame, satisfiability of modal quantificational formulae in which “= ” is the sole predicate is undecidable; but if we restrict attention to satisfiability in structures with the expanding domain property, satisfiability relative to the familiar frames (K, K4, T, S4, B, S5) is decidable. Furthermore, relative to any reasonable frame, satisfiability for modal quantificational formulae with a single monadic predicate is undecidable ; this improves the result of Kripke concerning formulae with two monadic predicates.

Let I be a countable jump ideal in $\mathscr{D} = \langle \text{The Turing degrees}, \leq\rangle$ . The central theorem of this paper is: a is a uniform upper bound on I iff a computes the join of an I-exact pair whose double jump a (1) computes. We may replace "the join of an I-exact pair" in the above theorem by "a weak uniform upper bound on I". We also answer two minimality questions: the class of uniform upper bounds on I (...) never has a minimal member; if ∪ I = L α [ A] ∩ ω ω for α admissible or a limit of admissibles, the same holds for nice uniform upper bounds. The central technique used in proving these theorems consists in this: by trial and error construct a generic sequence approximating the desired object; simultaneously settle definitely on finite pieces of that object; make sure that the guessing settles down to the object determined by the limit of these finite pieces. (shrink)

Where AR is the set of arithmetic Turing degrees, 0 (ω ) is the least member of { $\mathbf{\alpha}^{(2)}|\mathbf{a}$ is an upper bound on AR}. This situation is quite different if we examine HYP, the set of hyperarithmetic degrees. We shall prove (Corollary 1) that there is an a, an upper bound on HYP, whose hyperjump is the degree of Kleene's O. This paper generalizes this example, using an iteration of the jump operation into the transfinite which is based on (...) results of Jensen and is detailed in [3] and [4]. In $\S1$ we review the basic definitions from [3] which are needed to state the general results. (shrink)

A uniform upper bound on a class of Turing degrees is the Turing degree of a function which parametrizes the collection of all functions whose degree is in the given class. I prove that if a is a uniform upper bound on an ideal of degrees then a is the jump of a degree c with this additional property: there is a uniform bound b<a so that b V c < a.

I catalog several concepts associated with finite cardinals, and then invoke them to interpret and evaluate several passages in Rips et al.'s target article. Like the literature it discusses, the article seems overly quick to ascribe the possession of certain concepts to children (and of set-theoretic concepts to non-mathematicians).

It is generally not recognized that Whewell's conception of necessary truth evolved only gradually; his early statements are misleading. For this reason, and because of certain peculiarities in his expository style over his publishing history, he is commonly thought to have used the term "necessary" in the sense of "absolutely necessary". I argue that, on the contrary, the term is essentially relational in his mature view. This conclusion leads, in turn, to a re-interpretation of his doctrine of "fundamental ideas". Here (...) I argue that the crux of his position is that cognitive principles which are initially regulative become, through invention, repeated employment, and stipulative definition, constitutive ones. (shrink)

Much of the Mill-Whewell dispute was purely verbal, but much was not. Mill did not understand Whewell; the true character of the non-verbal aspect of the controversy emerges only upon adequate analysis of Whewell's actual position. Such analysis shows that Mill's objections to Whewell were misdirected, although suggestive of other which might, if prosecuted, carry. Ultimately, the dispute has to do with the given; neither man gives an adequate account of it. For this reason, the controversy cannot be resolved definitively (...) in favor of either of them. (shrink)

PHILOSOPHY and MODERN SCIENCE By PROFESSOR HAROLD T. DAVIS Indiana University THE PRINCIPIA PRESS Bloomington 1931 Indiana Tho FoiKjiult pnmliiliirn experiment..

Manuscript based on address delivered February, 1998 at the Annual Meeting of the Association of Practical and Professional Ethics, Dallas, Texas.

: This paper discusses the role of ethical considerations in the formulation of public policies aimed at shaping the scientific agenda. Specifically, new and controversial public policy issues will confront us in the twenty-first century as the result of developments on the frontiers of biomedical science. Some of the anxieties in the ethical arena generated by the rapid pace of these developments are likely to result in efforts to place constraints on the shape of the scientific agenda and the application (...) of new knowledge. Our moral propositions, therefore, will be tested and re-tested in their application to evolving social, cultural, and historical circumstances and the ever-changing technological context. (shrink)

In lieu of an abstract, here is a brief excerpt of the content:Kennedy Institute of Ethics Journal 12.1 (2002) 95-102 [Access article in PDF] Bioethics Inside the Beltway Some Initial Reflections on NBAC Eric M. Meslin and Harold T. Shapiro On 3 October 2001, Executive Order 12975 expired, and with it so too did the National Bioethics Advisory Commission (NBAC). Established by President Bill Clinton in 1995, NBAC was the fifth national committee since 1974 created to advise the U.S. (...) government on bioethics policy. In the long political tradition in this country that national bioethics policy advice should use temporary rather than permanent bodies, NBAC followed in the footsteps of the National Commission for the Protection of Human Subjects of Biomedical and Behavioral Research (1974-78); the President's Commission for the Study of Ethical Problems in Medicine and Biomedical and Behavioral Research (1980-83); the Ethics Advisory Board (1978-80); and the Biomedical Ethics Advisory Committee (1988-89). Arguably, other distinguished panels could lay claim to recognition as a "national bioethics committee" including the Advisory Committee on Human Radiation Experiments (ACHRE) (1994-95); the Fetal Tissue Transplantation Research Panel (1988); the Human Embryo Research Panel (1994); and the National Institutes of Health (NIH) Recombinant DNA Advisory Committee (RAC), which continues to function.These committees and commissions form a kind of a continuum, albeit occasionally interrupted by periods during which no such bodies existed. Each arose from a particular confluence of events and circumstances. NBAC was no different; its history can be traced both to a series of discussions within the White House Office of Science and Technology Policy beginning in 1993 (NBAC 1998a, p. 3), and to a specific recommendation from ACHRE that called for a national committee to address ethical issues in human subjects research (ACHRE 1995, p. 817). Nbac's Impact In his useful comparison of the accomplishments of the National Commission and the President's Commission, Brad Gray concluded that seven lessons were learned, the sixth of which was that "nothing substitutes for having a perceptible impact on either policy or practice" (Gray 1995, p. 304). Assessing NBAC's [End Page 95] impact on "policy or practice" is difficult, particularly given the fact that it so recently completed its work. More time may be needed before a full assessment can be undertaken of the impact of its six reports (and their total of 120 recommendations). Instead, in this essay we offer some reflections on NBAC's initial impact, using some specific examples of how bioethics gets done Inside the Beltway. Eliciting Reaction If immediate reaction by the White House is any measure of impact, then Cloning Human Beings (NBAC 1997) had a significant one. The day after he received this report in a Rose Garden ceremony, President Clinton submitted a bill to Congress that would have extended the moratorium on federal funding to create a child using somatic cell nuclear transfer. This bill never passed, nor has any other bill emerged that has enjoyed the support of the House and the Senate. Five states have enacted legislation to date, but these vary in their permissibility from bans on all forms of cloning to bans similar to the one proposed by NBAC, which recommended a time-limited prohibition only on somatic cell transfer to create a child (reproductive cloning).If the Cloning report elicited speedy action from the White House, this paled in comparison to the reaction that led to and, in turn, was created by Ethical Issues in Human Stem Cell Research (NBAC 1999a). Those who believe that government moves slowly need only be reminded of the events of 8-14 November 1998 (and, as we shall mention below, the further events of 14 July 1999). Some will recall that early in the second week of November 1998, James Thomson and his colleagues published in Nature and John Gearhart and his colleagues published in the Proceedings of the National Academy of Sciences that they had isolated and cultured human primordial stem cells and germ cells respectively. Within days of these remarkable reports, Advanced Cell Technology (ACT) announced via an article written by Nicholas Wade in the... (shrink)