Teleology and Realism in Leibniz's Philosophy of Science

This paper argues for an interpretation of Leibniz’s claim that physics requires both mechanical and teleological principles as a view regarding the interpretation of physical theories. Granting that Leibniz’s fundamental ontology remains non-physical, or mentalistic, it argues that teleological principles nevertheless ground a realist commitment about mechanical descriptions of phenomena. The empirical results of the new sciences, according to Leibniz, have genuine truth conditions: there is a fact of the matter about the regularities observed in experience. Taking this stance, however, requires bringing non-empirical reasons to bear upon mechanical causal claims. This paper first evaluates extant interpretations of Leibniz’s thesis that there are two realms in physics as describing parallel, self-sufficient sets of laws. It then examines Leibniz’s use of teleological principles to interpret scientific results in the context of his interventions in debates in seventeenth-century kinematic theory, and in the teaching of Copernicanism. Leibniz’s use of the principle of continuity and the principle of simplicity, for instance, reveal an underlying commitment to the truth-aptness, or approximate truth-aptness, of the new natural sciences. The paper concludes with a brief remark on the relation between metaphysics, theology, and physics in Leibniz

The Automation of Syllogistic II. Optimization and Complexity Issues

In the first paper of this series it was shown that any unquantified formula p in the collection MLSSF (multilevel syllogistic extended with the singleton operator and the predicate Finite) can be decomposed as a disjunction of set-theoretic formulae called syllogistic schemes. The syllogistic schemes are satisfiable and no two of them have a model in common, therefore the previous result already implied the decidability of the class MLSSF by simply checking if the set of syllogistic schemes associated with the given formula is empty. In the first section of this paper a new and improved searching algorithm for syllogistic schemes is introduced, based on a proof of existence of a 'minimum effort' scheme for any given satisfiable formula in MLSF. The algorithm addressed above can be piloted quite effectively even though it involves backtracking. In the second part of the paper, complexity issues are studied by showing that the class of ( 00)o1-simple prenex formulae (an extension of MLS) has a decision problem which is NP-complete. The decision algorithm that proves the membership of this decision problem to NP can be seen as a different decision algorithm for ML

A Brief History of Singlefold Diophantine Definitions

Consider an (m + 1)-ary relation R over the set N of natural numbers. Does there exist an arithmetical formula ZΘ(a0, . . . , am, x1, . . . , xK), not involving universal quantifiers, negation, or implication, such that the representation and univocity conditions, viz., (Formula Presented) are met by each tuple (Formula Presented). A priori, the answer may depend on the richness of the language of arithmetic: Even if solely addition and multiplication operators (along with the equality relator and with positive integer constants) are adopted as primitive symbols of the arithmetical signature, the graph R of any primitive recursive function is representable; but can representability be reconciled with univocity without calling into play one extra operator designating either the dyadic operation [b, n]↠ b n or just the monadic function n ↠ b n associated with a fixed integer b > 1? As a preparatory step toward a hoped-for positive answer to this question, one may consider replacing the exponentiation operator by a dyadic relator designating an exponential-growth relation (a notion made explicit by Julia Bowman Robinson in 1952). We will discuss the said univocity, aka ‘single-fold-ness’, issue-first raised by Yuri V. Matiyasevich in 1974-, framing it in historical context. © 2023 Copyright for this paper by its authors

On generalised Ackermann encodings – the basis issue

In this paper, a generalised version Aβ of the celebrated Ackermann encoding of the hereditarily finite sets, aimed at assigning a real number also to each hereditarily finite hyperset and multiset, is studied. Such a mapping establishes a significant link between real numbers and the theories of such generalised notions of set, so that performing set-theoretic operations can be translated into their number-theoretic equivalent. By appropriately choosing a parameter β, both the Ackermann encoding and the less known map RA arise as special cases; a bijective encoding of a subuniverse of hereditarily finite multisets occurs whenever this parameter is chosen among natural numbers, while if it is taken transcendental and within a peculiar interval of the real positive line, then the function is surmised to ensure an injective mapping of both the aforementioned universes

Continued Hereditarily Finite Set-Approximations

We study an encoding RA that assigns a real number to each hereditarily finite set, in a broad sense. In particular, we investigate whether the map RA can be used to produce codes that approximate any positive real number to arbitrary precision, in a way that is related to continued fractions. This is an interesting question because it connects the theory of hereditarily finite sets to the theory of real numbers and continued fractions, which have important applications in number theory, analysis, and other fields

A graphical approach to relational reasoning

Relational reasoning is concerned with relations over an unspecified domain of discourse. Two limitations to which it is customarily subject are: only dyadic relations are taken into account; all formulas are equations, having the same expressive power as first-order sentences in three variables. The relational formalism inherits from the Peirce-Schröder tradition, through contributions of Tarski and many others. Algebraic manipulation of relational expressions (equations in particular) is much less natural than developing inferences in first-order logic; it may in fact appear to be overly machine-oriented for direct hand-based exploitation. The situation radically changes when one resorts to a convenient representation of relations based on labeled graphs. The paper provides details of this representation, which abstracts w.r.t. inessential features of expressions. Formal techniques illustrating three uses of the graph representation of relations are discussed: one technique deals with translating first-order specifications into the calculus of relations; another one, with inferring equalities within this calculus with the aid of convenient diagram-rewriting rules; a third one with checking, in the specialized framework of set theory, the definability of particular set operations. Examples of use of these techniques are produced; moreover, a promising approach to mechanization of graphical relational reasoning is outlined

Some decidability issues concerning C^n real functions

This paper adapts preexisting decision algorithms to a family RDF = {RDFn | n ∈ N} of languages regarding one-argument real functions; each RDFn is a quantifier-free theory about the differentiability class C^n, embodying a fragment of Tarskian elementary algebra. The limits of decidability are also highlighted, by pointing out that certain extensions of RDFn are undecidable. The possibility of extending RDFn into a language RDF∞ regarding the class C^∞, without disrupting decidability, is briefly discussed. Two sorts of individual variables, namely real variables and function variables, are available in each RDFn. The former are used to construct terms and formulas that involve basic arithmetic operations and comparison relators between real terms, respectively. In contrast, terms designating functions involve function variables, constructs for addition of functions and scalar multiplication, and—outermost—i-th order differentiation D^i[ ] with i ⩽ n. An array of predicate symbols designate various relationships between functions, as well as function properties, that may hold over intervals of the real line; those are: function comparisons, strict and non-strict monotonicity / convexity / concavity, comparisons between a function (or one of its derivatives) and a real term. The decidability of RDFn relies, on the one hand, on Tarski’s celebrated decision algorithm for the algebra of real numbers, and, on the other hand, on reduction and interpolation techniques. An interpolation method, specifically designed for the case n = 1, has been previously presented; another method, due to Carla Manni, can be used when n = 2. For larger values of n, further research on interpolation is envisaged

4CPS-390 Dispensing of anticancer investigational drugs during lockdown for the SARS-CoV-2 pandemic: experience in an oncological centre

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