A survey of Euclid's Elements, this text provides an understanding of the classical Greek conception of mathematics and its similarities to modern views as well as its differences. It focuses on philosophical, foundational, and logical questions — rather than strictly historical and mathematical issues — and features several helpful appendixes.

A methodology is proposed as a tool for explanation of form in zoomorphology, in particular its design, diversity, and transformation. An alternate use of descriptive, inductive/comparative, and deductive methods is suggested. The basic concepts required are summarized. Following an extensive anatomical analysis a specific deductive methodology is developed, comprising three major parts: 1) Formal analysis of systems, using optimal design. 2) Transformation of an initial system's model by simulating modifications via maximizing the model for specific functional requirements. 3) (...) Testing by comparison of deduced modifications with real systems having the maximized functional capacity, trends in specific morphoclines, and empirical analogues from technology.The methodology produces a morpho-space of modifications that has the form of branching pathways. That pattern results from innovation, bifurcation, and radiation moments in the deduction. The morpho-space is converted in a mechano-space by including next to constructional characters those of operation and rolefulfilment. If the initial system is verified as the ancestor and the transformation-function is verified as the evolutionary theme of change, the explanation turns -in addition to a function anatomical one- also into an evolutionary one. The method is illustrated by reviewing some recent analyses about avian feeding: maximization of avian pecking for probing and filter feeding capacities. (shrink)

This classic undergraduate treatment examines the deductive method in its first part and explores applications of logic and methodology in constructing mathematical theories in its second part. Exercises appear throughout.

We provide a new link between deductive and probabilistic reasoning fallacies. Illusory inferences from disjunction are a broad class of deductive fallacies traditionally explained by recourse to a matching procedure that looks for content overlap between premises. In two behavioral experiments, we show that this phenomenon is instead sensitive to real-world causal dependencies and not to exact content overlap. A group of participants rated the strength of the causal dependence between pairs of sentences. This measure is a near (...) perfect predictor of fallacious reasoning by an independent group of participants in illusory inference tasks with the same materials. In light of these results, we argue that all extant accounts of these deductive fallacies require non-trivial adjustments. Crucially, these novel _indirect_ illusory inferences from disjunction bear a structural similarity to seemingly unrelated _probabilistic reasoning_ problems, in particular the conjunction fallacy from the heuristics and biases literature. This structural connection was entirely obscure in previous work on these deductive problems, due to the theoretical and empirical focus on content overlap. We argue that this structural parallelism provides arguments against the need for rich descriptions and individuating information in the conjunction fallacy, and we outline a unified theory of deductive illusory inferences from disjunction and the conjunction fallacy, in terms of Bayesian confirmation theory. (shrink)

In [2] Hempel and Oppenheim give a definition of “explanation” for a certain formal language. In [1] Eberle, Kaplan, and Montague prove five theorems demonstrating that the Hempel and Oppenheim definition is not restrictive enough. In [3] Kim proposes two further conditions to supplement the Hempel and Oppenheim definition in order to avoid the objections posed in [1]. In this paper it is shown that the definition of Hempel and Oppenheim supplemented by Kim's conditions is open to a trivialization very (...) analogous to that given in [1]. (shrink)

The aim of the article is to outline the historical background and the present state of the methodology of deductive systems invented by Alfred Tarski in the thirties. Key notions of Tarski's methodology are presented and discussed through, the recent development of the original concepts and ideas.

I express my dissatisfaction with the common ways to treat the semantic paradoxes. Not only do they give rise to revenge paradoxes, they ignore the wisdom contained in the ordinary reaction to paradoxes. I instead propose an account that vindicates the ordinary reaction to paradox by putting the blame on us philosophers. It is the wrong conception of what a valid inference is, one that is central to “the ideal of deductive logic” that gives rise to the problem. The (...) solution outlined gives us a new way to accept defeat in light of the paradoxes: the arguments that lead to them are based on valid forms of reasoning, but their conclusions are nonetheless rationally rejected. (shrink)

There is a broad debate in contemporary philosophy of logic on the informativeness of proofs. In this context, informative proofs are demonstrations whose premises do not include the content of the conclusion. D’Agostino and Floridi (Synthese 167(2):271–315, 2009) claimed that proofs are informative if they use _virtual information_. In their terminology, this is the data carried by _dischargeable hypotheses_, assumptions entertained during proof and eliminated before concluding. Although these authors capture several cases of informative demonstrations, they do not explain the (...) nature of virtual information. Why does the use of dischargeable hypotheses increase the informativeness of a demonstration? Following a Kantian inspiration, D’Agostino and Floridi claim that virtual information adds a synthetic feature to our proofs. However, this appeal to Kantianism is misleading: for Kant, mathematics is synthetic because it requires the construction of concepts in pure intuition. On the other hand, the entertainment of provisional hypotheses does not seem to involve the construction of figures in (transcendental) imagination. Exploring a _dialogical_ characterization of logic, I suggest that virtual information results from the dynamics between a reasoner and her audience. When a reasoner provisionally assumes _P_, she enacts potential interlocutors who commit to _P_. Thus, “virtual information” denotes the content of some assumptions of the possible audience of a demonstration. This process of departing from one’s original premises to embrace the suppositions of other people is informative, but it is not synthetic: in this way, the reasoner does not entertain intuitions but only needs to reason through the conceptual content of other people’s premises. (shrink)

Background As more people are living longer, they become frail and are affected by multi-morbidity, resulting in increased demands from the ambulance service. Being vulnerable, older patients may have reduced decision-making capacity, despite still wanting to be involved in decision-making about their care. Their needs may be complex and difficult to assess, and do not always correspond with ambulance assessment protocols. When needing an ambulance, older patients encounter ambulance clinicians who are under high workloads and primarily consider themselves as emergency (...) medical care providers. This situates them in the struggle between differing expectations, and ethical conflicts may arise. To resolve these, providing ethical care, focussing on interpersonal relationships and using ethical competence is needed. However, it is not known whether ambulance clinicians possess the ethical competence required to provide ethical care. Thus, the aim of this study was to deductively explore their ethical competence when caring for older patients with reduced decision-making ability. Methods A qualitative deductive and exploratory design was used to analyse dyadic interviews with ambulance clinicians. A literature review, defining ethical competence as comprising ethical sensitivity, ethical knowledge, ethical reflection, ethical decision-making, ethical action and ethical behaviour, was used as a structured categorization matrix for the analysis. Results Ambulance clinicians possess ethical competence in terms of their ethical knowledge, highlighting the need for establishing an interpersonal relationship with the older patients. To establish this, they use ethical sensitivity to interpret the patients’ needs. Doing this, they are aware of their ethical behaviour, signifying how they must act respectfully and provide the necessary time for listening and interacting. Conclusions Ambulance clinicians fail to see their gut feeling as a professional ethical competence, which might hinder them from reacting to unethical ways of working. Further, they lack ethical reflection regarding the benefits and disadvantages of paternalism, which reduces their ability to perform ethical decision-making. Moreover, their ethical knowledge is hampered by an ageist approach to older patients, which also has consequences for their ethical action. Finally, ambulance clinicians show deficiencies regarding their ethical reflections, as they reflect merely on their own actions, rather than on their values. (shrink)

In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the lattice-theoretical point of view; this set is represented by the lattice FACSs A of all algebraic closed-set systems C on A such that (A, C) ε FGModS. We relate some properties of these (...) structures with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSs A is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to hold only for deductive systems in more restricted classes like the protoalgebraic or the weakly algebraizable ones. The paper ends with a section on examples and counterexamples. (shrink)

The idea that knowledge can be extended by inference from what is known seems highly plausible. Yet, as shown by familiar preface paradox and lottery-type cases, the possibility of aggregating uncertainty casts doubt on its tenability. We show that these considerations go much further than previously recognized and significantly restrict the kinds of closure ordinary theories of knowledge can endorse. Meeting the challenge of uncertainty aggregation requires either the restriction of knowledge-extending inferences to single premises, or eliminating epistemic uncertainty in (...) known premises. The first strategy, while effective, retains little of the original idea—conclusions even of modus ponens inferences from known premises are not always known. We then look at the second strategy, inspecting the most elaborate and promising attempt to secure the epistemic role of basic inferences, namely Timothy Williamson’s safety theory of knowledge. We argue that while it indeed has the merit of allowing basic inferences such as modus ponens to extend knowledge, Williamson’s theory faces formidable difficulties. These difficulties, moreover, arise from the very feature responsible for its virtue- the infallibilism of knowledge. (shrink)

Logical Form, Probability Interpretations, and the Inductive/Deductive Distinction.

We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, (...) whereas Hilbert dismisses elucidation and consequently treats the primitives as schematic. (shrink)

Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way . Heterogeneous fibring, allowing the original logics to have different presentations , has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus (...) and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semi-decidability is investigated. (shrink)

In this paper we study interpretations and equivalences of propositional deductive systems by using a quantale-theoretic approach introduced by Galatos and Tsinakis. Our aim is to provide a general order-theoretic framework which is able to describe and characterize both strong and weak forms of interpretations among propositional deductive systems also in the cases where the systems have different underlying languages.

T. Govier's description of ‘conductive argument’ and 'A priori analogy' is taken as a start to investigate non-deductive argumentation. It is here argued, that the nature of those types can be better understood when taking up a dynamic view (in addition to the usual structural view). The concepts of ‘frame’ and ‘position’ are constructed in order to establish such a twofold approach.

Almost every formal model of explanation thus far proposed has been demonstrated to be faulty. In this paper, a new model, proposed by Raimo Tuomela, is also demonstrated to be faulty. In particular, one condition of the model is shown to be too restrictive, and another condition of the model is shown to be too permissive.

We are justified in employing the rule of inference Modus Ponens (or one much like it) as basic in our reasoning. By contrast, we are not justified in employing a rule of inference that permits inferring to some difficult mathematical theorem from the relevant axioms in a single step. Such an inferential step is intuitively “too large” to count as justified. What accounts for this difference? In this paper, I canvass several possible explanations. I argue that the most promising approach (...) is to appeal to features like usefulness or indispensability to important or required cognitive projects. On the resulting view, whether an inferential step counts as large or small depends on the importance of the relevant rule of inference in our thought. (shrink)

I address a type of circularity threat that arises for the view that we employ general basic logical principles in deductive reasoning. This type of threat has been used to argue that whatever knowing such principles is, it cannot be a fully cognitive or propositional state, otherwise deductive reasoning would not be possible. I look at two versions of the circularity threat and answer them in a way that both challenges the view that we need to apply general (...) logical principles in deductive reasoning and defuses the threat to a cognitivist account of knowing basic logical principles. (shrink)

The purpose of this paper is to present in a uniform way the commutator theory for k-deductive system of arbitrary positive dimension k. We are interested in the logical perspective of the research — an emphasis is put on an analysis of the interconnections holding between the commutator and logic. This research thus qualifies as belonging to abstract algebraic logic, an area of universal algebra that explores to a large extent the methods provided by the general theory of (...) class='Hi'>deductive systems. In the paper the new term ‘commutator formula’ is introduced. The paper is concerned with the meanings of the above term in the models provided by the commutator theory and clarifies contexts in which these meanings occur. The work is presented in an abstracted form: main ideas are outlined but proofs are deferred to the second part of the paper. (shrink)

The proposed reflection intends to present the problem of judicial adjudication as a substantially-axiologically founded autonomous moment on the practical realization of law, and to explore this understanding in confrontation with external exigencies, mostly teleologically determined—hence, beyond strict deductive application, as a syllogistic reference of facts to norms, and finalistically determined decision, as an option among possible alternatives to achieve specific aims. The main objective is to enter into a discussion on the methodological meaning of “integrity”, “hard cases” and (...) “right answer”, as presented by Ronald Dworkin, and a critical reflection on the criticism of that approach levelled by Neil MacCormick, so as to confront the relevance of principle and policy arguments, in order to bring about a different methodological approach, an alternative jurisprudentialist conception of adjudication, incorporating a practical-normative constitutive dialectics between legal controversy and legal system, such as that presented by Castanheira Neves. The focus will, then, be the legitimacy of the connection of arguments of principle and consequentialist arguments in adjudication, its selection and its justification, stating, therefore, a specifically assumed judicium, a judicative decision, having the legal system as its horizon of normative reference and of substantial and institutional autonomy. (shrink)

Until quite recently, the epistemology of logical laws has not been much discussed and neither has how one can be justified in claiming that a particular inference is valid. The transfer of warrant from premises to conclusion in modus ponens will be examined in the paper through assessing Paul Boghossian's inferentialist proposal of assuming 'blind reasoning'. It will be argued that merely being justified in inferring according to a logical law a priori is worthless unless one can also be justified (...) in claiming that the inference is valid. An alternative to Boghossian's solution will be presented which addresses this need as well as the needs Boghossian had identified previously. (shrink)

Hempel and Oppenheim have stated in Part III of their paper “Studies in the Logic of Explanation” [2] a set of conditions for deductive explanation. However, their analysis has come under damaging systematic criticisms in a recent paper by Eberle, Kaplan and Montague [1], The principal aim of the present paper is to review the Hempel-Oppenheim analysis and propose a strengthened version of it that avoids the recent criticisms.