A natural deduction formulation is given for the intermediate logic called MH by Gabbay in [4]. Proof-theoretic methods are used to show that every deduction can be normalized, that MH is the weakest intermediate logic for which the Glivenko theorem holds, and that the Craig-Lyndon interpolation theorem holds for it.

The usual rule used to obtain natural deduction formulations of classical logic from intuitionistic logic, namely.

In 1939, Curry proposed a philosophy of mathematics he called formalism. He made this proposal in two works originally written then, although one of them was not published until 1951. These are the two philosophical works for which Curry is known, and they have left a false impression of his views. In this article, I propose to clarify Curry’s views by referring to some of his later writings on the subject. I claim that Curry’s philosophy was not what is now (...) usually called formalism, but is really a form of structuralism. (shrink)

Evidence is given that implication (and its special case, negation) carry the logical strength of a system of formal logic. This is done by proving normalization and cut elimination for a system based on combinatory logic or λ-calculus with logical constants for and, or, all, and exists, but with none for either implication or negation. The proof is strictly finitary, showing that this system is very weak. The results can be extended to a "classical" version of the system. They can (...) also be extended to a system with a restricted set of rules for implication: the result is a system of intuitionistic higher-order BCK logic with unrestricted comprehension and without restriction on the rules for disjunction elimination and existential elimination. The result does not extend to the classical version of the BCK logic. (shrink)

Because the main difference between combinatory weak equality and λβ-equality is that the rule \begin{equation*}\tag{\xi} X = Y \vdash \lambda x.X = \lambda x.Y\end{equation*} is valid for the latter but not the former, it is easy to assume that another way of defining combinatory β-equality is to add rule (ξ) to the postulates for weak equality. However, to make this true, one must choose the definition of combinatory abstraction in (ξ) very carefully. If one tries to use one of the (...) more common abstraction algorithms, the result will be an equality, = ξ , that is either equivalent to βη-equality (and so strictly stronger than β-equality) or else strictly weaker than β-equality. This paper will study the relations = ξ for several commonly used abstraction algorithms, distinguish between them, and axiomatize them. (shrink)

We situate the well-trodden debate about defining health and disease within the project of a metaphysics of science and its aim to work with and contribute to science. We make use of Guay and Pradeu’s ‘metaphysical box’ to reframe this debate, showing what is at stake in recent attempts to move beyond it, revealing unforeseen points of agreement and disagreement among new and old positions, and producing new questions that may lead to progress. We then discuss the implications of the (...) two-way benefits between medicine and philosophy when the latter is driven and constrained by medical science. (shrink)

This essay discusses four key criticisms recently leveled against the main attempts to use conceptual analysis to understand health and disease. First, it examines the weaknesses of these attempts and suggests a better way to proceed. Next, it briefly discusses another disease debate concerning pathological mechanisms and suggests that this approach could be more fruitful than that of conceptual analysis. The final section demonstrates how Georges Canguilhem's biological philosophy of disease avoids some of the problems associated with conceptual analysis, and (...) how he can contribute to both debates. Canguilhem's appeal to biological normativity is understood as contributing to a theoretical explanation, and a possible naturalization, of what is specific to healthy and pathological phenomena. (shrink)

In this concluding chapter, we gather the various contributions of this volume and attempt to extract some of the many key insights and challenges raised when it comes to the project of explaining health across the sciences. These insights were distilled down into a selection of the central concepts and issues defended or discussed by the authors, and were organized into a table to see, at a glance, where the attention was given. Reflecting on these insights will go some way (...) towards addressing the question of whether these explanations can be brought together into a coherent or unifying framework or whether we are simply stuck with mutually exclusive views about the multifaceted phenomenon of ‘health’. (shrink)

It would be very desirable to have an account of the relation between mind and world that sustained the integrity of each. In this paper, we will argue that a theory of cognition which is broadly Thomistic can do just that. Many commentators recognize that cognitio is Aquinas’s basic epistemic concept, and that it designates knowledge in the broadest and most basic sense, as distinguished from scientia, or knowledge in the paradigmatic sense. There are several important consequences of this distinction (...) which are relevant to a proper understanding of the way in which a theory of knowledge which is in the Aristotelian-Thomistic tradition approaches the issue of the relation between the mind and the world. First and foremost, knowledge cannot be construed as a univocal concept in the A/t context. There are different senses of knowledge and we should not expect there to be a theory which supplies necessary and sufficient conditions for “S knows that p” that satisfies all the different senses of knowledge. It is perplexing that the contemporary discussion seems to miss this point, for the same sort of systematic ambiguity of the term ‘knowledge’ seems to hold in English. Second, the A/t theory of knowledge is not an epistemological theory in the modern sense, if by “epistemological” we mean one which is clearly demarcated from a metaphysical, psychological, or even semantical theory. The A/t theory is a metaphysics of knowledge. Third, a Thomistic theory of knowledge begins with a theory of cognition which provides the grounds of knowledge across the board, including even that of scientia. (shrink)

Debates in fields studying the biological aspects of aging and longevity, such as biogerontology, are often split between ‘anti-aging’ approaches aimed largely at treating diseases and those focusing more on maintaining, promoting, and even enhancing health. However, it is far from clear what this ‘health’ is that would be maintained, promoted, or enhanced. Interestingly, what few have yet to fully reflect on is that there is still no theory of health within the health or aging sciences that would provide an (...) integrative explanatory framework akin to other scientific theories. After clarifying why such a theory could be useful, I discuss five general features of medical theories that could be used to evaluate the utility of a given proposal. With these features in hand, I suggest that philosophers and scientists work together on analyzing actual medical research (experimental analysis), and the ways in which a theoretical construct of ‘health’ is being progressively identified and measured therein. I conclude by suggesting that research fields studying stress and aging might be particularly helpful in developing candidates for theory construction due to their broad scope, increasing specificity, and potential for providing integrative explanations. (shrink)

Can there be deterministic chance? That is, can there be objective chance values other than 0 or 1, in a deterministic world? I will argue that the answer is no. In a deterministic world, the only function that can play the role of chance is one that outputs just Os and 1s. The role of chance involves connections from chance to credence, possibility, time, intrinsicness, lawhood, and causation. These connections do not allow for deterministic chance.