1.. HARD CASES 5. Legal Rights A. Legislation . . . We might therefore do well to consider how a philosophical judge might develop, in appropriate cases, theories of what legislative purpose and legal principles require. We shall find that he would construct these theories in the same manner as a philosophical referee would construct the character of a game. I have invented, for this purpose, a lawyer of superhuman skill, learning, patience and acumen, whom I shall call Hercules. I suppose that Hercules is a judge in some representative American jurisdiction. I assume that he accepts the main uncontroversial constitutive and regulative rules of the law in his jurisdiction. He accepts, that is, that statutes have the general power to create and extinguish legal rights, and that judges have the general duty to follow earlier decisions of their court or higher courts whose rationale, as l

Taking Rights Seriously

In Software Abstractions Daniel Jackson introduces an approach to software design that draws on traditional formal methods but exploits automated tools to find flaws as early as possible. This approach--which Jackson calls "lightweight formal methods" or "agile modeling"--takes from formal specification the idea of a precise and expressive notation based on a tiny core of simple and robust concepts but replaces conventional analysis based on theorem proving with a fully automated analysis that gives designers immediate feedback. Jackson has developed Alloy, a language that captures the essence of software abstractions simply and succinctly, using a minimal toolkit of mathematical notions. This revised edition updates the text, examples, and appendixes to be fully compatible with the latest version of Alloy (Alloy 4). The designer can use automated analysis not only to correct errors but also to make models that are more precise and elegant. This approach, Jackson says, can rescue designers from "the tarpit of implementation technologies" and return them to thinking deeply about underlying concepts. Software Abstractions introduces the key elements: a logic, which provides the building blocks of the language; a language, which adds a small amount of syntax to the logic for structuring descriptions; and an analysis, a form of constraint solving that offers both simulation (generating sample states and executions) and checking (finding counterexamples to claimed properties).

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Software Abstractions: Logic, Language, and Analysis

Argumentation in artificial intelligence

Consider the function defined for $\alpha _{1},\ \alpha _{2} \in \mathbb{R}$ (α 1 2 +α 2 2 ≠ 0) and $\beta _{1},\beta _{2} \in \mathbb{C}$ by the series 

$$\displaystyle{ E_{\alpha _{1},\beta _{1};\alpha _{2},\beta _{2}}(z) \equiv \sum _{k=0}^{\infty } \frac{z^{k}} {\varGamma (\alpha _{1}k +\beta _{1})\varGamma (\alpha _{2}k +\beta _{2})}\ \ (z \in \mathbb{C}). }$$

(6.1.1)

Such a function with positive α 1 > 0, α 2 > 0 and real $\beta _{1},\beta _{2} \in \mathbb{R}$ was introduced by Dzherbashian [Dzh60].

Multi-index Mittag-Leffler Functions

Wittgenstein's work remains, undeniably, now, that off one of those few philosophers who will be read by all future generations.

Remarks on the Foundations of Mathematics. By L. Wittgenstein. Edited by G. H. Von Wright, R. Rhees and G. E. M. Anscombe. Translated by G. E. M. Anscombe. Pp. 400. 37s. 6d. 1956. (Basil Blackwell, Oxford)

Argument: Critical Thinking, Logic and the Fallacies

Russell’s paradox is the most famous of the logical or set-theoretical paradoxes. Also known as the Russell-Zermelo paradox, the paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox. Some sets, such as the set of all teacups, are not members of themselves. Other sets, such as the set of all nonteacups, are members of themselves. Call the set of all sets that are not members of themselves “R.” If R is a member of itself, then by definition it must not be a member of itself. Similarly, if R is not a member of itself, then by definition it must be a member of itself. Although also noticed by Ernst Zermelo, the contradiction was not thought to be important until it was discovered independently by Bertrand Russell in the spring of 1901. Since then, the paradox has prompted a great deal of work in logic, set theory and the philosophy and foundations of mathematics. 1. The Paradox 2. History of the Paradox 3. Early Responses to the Paradox 4. Russell’s Paradox in Contemporary Logic Bibliography Academic Tools Other Internet Resources Related Entries

Russell’s Paradox

1. Epistemology & Nominalism.- 2. What Is Abstraction & What Is It Good For?.- 3. Beliefs About Mathematical Objects.- 5. Field & Fregean Platonism.- 5. ? in The Sky.- 6. Nominalism.- 7. The Logic of Physical Theory.- 8. Knowledge of Mathematical Objects.- 9. Physicalism, Reductionism & Hilbert.- 10. Physicalistic Platonism.- 11. Sets are Universals.- 12. Modal-Structural Mathematics.- 13. Logical & Philosophical Foundations for Arithmetical Logic.- 14. Criticisms of the Usual Rationale for Validity in Mathematics.- Contributors.- Index of Proper Names.

Physicalism in Mathematics

Aristotle is generally recognized as the founder of systematic logic or of what he called “analytics.” He has been the first to systematize universally valid logical laws. Aristotle has been also responsible for the remarkable accomplishment of developing logic in at least two distinct ways, including his almost complete theory of the syllogism and his complex and sophisticated theory of modal logic. In addition, he is noted for his work in axiomatic, and there is some evidence that he also began investigating what is now called “propositional logic,” although he did not develop these investigations systematically. It is widely accepted that Aristotle's main contributions to logic begin—and some would say end—with the prior analytics. Aristotle holds what modern logicians call a classical conception of validity and, correspondingly, a classical conception of logical implication or entailment. This proposal is made in the face of the fact that Aristotle gives no account of these things anywhere in his writings.

Aristotle's early logic

In Politics III.10 and IV.4, Aristotle discusses the difference between governments that are regulated by the rule of law and those that are not. Although he concludes that the rule of law helps guard against arbitrary and injudicious government action, Aristotle is also sensitive to the fact that in a democracy it is essential for the people to remain sovereign over the law. His discussion is helpful for understanding, not only the tension between the ‘rule of law’ and the ‘rule of men’, but also the complex role the rule of law plays in any modern democracy.

A. D. Irvine

Papers

Argument: Critical Thinking, Logic and the Fallacies

Russell’s Paradox

Physicalism in Mathematics

Aristotle's early logic

Aristotle on the Rule of Law