Semantic Criteria of Correct Formalization

This paper compares several models of formalization. It articulates criteria of correct formalization and identifies their problems. All of the discussed criteria are so called “semantic” criteria, which refer to the interpretation of logical formulas. However, as will be shown, different versions of an implicitly applied or explicitly stated criterion of correctness depend on different understandings of “interpretation” in this context

The Principle Of Excluded Middle Then And Now: Aristotle And Principia Mathematica

The prevailing truth-functional logic of the twentieth century, it is argued, is incapable of expressing the subtlety and richness of Aristotle's Principle of Excluded Middle, and hence cannot but misinterpret it. Furthermore, the manner in which truth-functional logic expresses its own Principle of Excluded Middle is less than satisfactory in its application to mathematics. Finally, there are glimpses of the "realism" which is the metaphysics demanded by twentieth century logic, with the remarkable consequent that Classical logic is a particularly inept instrument to analyze those philosophies which stand opposed to the "realism" it demands

Freedom as the basis of truth and reality in Russell's positivism and Stace's mysticism

Thesis (Ph.D.)--Boston UniversityPurpose. The central purpose of the dissertation is to localize the ultimate grounds of philosophic (and particularly metaphysical) differences, with the goal of achieving an eventual synthesis. To do this, the dissertation (1) analyzes the nature of metaphysical systems in general, (2) seeks a perspective (termed a meta-metaphysics) from which any metaphysical system can be derived, and (3) proposes to combine phenomenological and non-phenomenological methodologies. Method. First, the dissertation reviews representative attempts of philosophic synthesis. Those dealing with the general problem of synthesis, without specifying any particular one, are by Schiller, Montague, and Pepper. One specific attempt discussed consists of several syntheses of the existentialist and linguistic-analytic methods. Second, the dissertation presents an extended analysis of two antithetical philosophies: Russell's positivism and Stace's mysticism. By discovering the fundamental assumptions in each of these two conceptions of truth and reality and the values embodied and preserved by them, a comparison can be made with the view to finding the precise area of differences and agreements. Results. Russell is shown to hold that an acceptable definition prescribes and describes. He rejects coherence and warranted assertibility theories of truth, and defines truth as "logical, not epistemological, correspondence." The fundamental assumption in Russell's correspondence definition of truth is the postulation of qualitative metaphysical dualism, while the values embodied in his conception of definition and of truth are those found in the belief in the existence of an external world, including, of course, the existence of other minds. Russell, at times, conceives of reality as a "construction" (a notion first developed in his logic) of sensory perspectives, and at times as an inference from data. Fourteen assumptions are involved in the conceptions of reality. Among them we find "the universe is knowable through reason," "the given is a clue to a reality beyond," "the transcendent (in Kant's sense) use of the laws of inference is valid," and "the simple is more likely to correspond with events beyond sensations than the complex." Stace's position on mysticism is first examined as an interpretation of the mystical experience, and then as a theory of reality. The mystical experience may be characterized as either an experience of value, truth, or reality In mysticism as a theory of reality, value, truth, and reality each has a special ontological status. To understand mysticism is to know the interrelations of these terms. Since "assumption" is a logical term and mysticism is alogical, the phenomenological analysis of the mystical experience involves "priorities" (genetic, logical, phenomenological, axiological) in addition to assumptions. Among the twelve assumptions we find "naturalism and mysticism are two orders of one reality," "the mystic can legitimately choose his own meaning criterion," "extra-logical cognition is possible," "reality itself is extra-logical," and "subject-object bifurcations are not necessary." Conclusions. The ultimate ground for accepting these assumptions, priorities, and values is the possibility of choosing them freely, spontaneously, and autonomously. Russell must justify the possibility of correspondence (truth) and his metaphysical dualism (reality) on the grounds that they embody values which he has, in fact, freely chosen to accept. Stace's assumptions and priorities are likewise true and real because the values they embody are freely chosen. The dissertation constructs a meta-metaphysics that discloses positivism and mysticism (and any other empirically coherent metaphysics) as equality adequate world-views, differing only in the free commitment (Urentschluss) to these grounding assumptions and values. The exposition of this meta-metaphysics is the detailed description of its fundamental categories (the given, freedom, value, the logical and empirical refractory, and organizations of this given). Each of the above assumptions, priorities, and values is then interpreted in terms of this meta-metaphysics

There is no standard model of ZFC

Main results are:(i) Let M_st be standard model of ZFC. Then ~Con(ZFC+∃M_st), (ii) let k be an inaccessible cardinal then ~Con(ZFC+∃k),[10],11]