#axiomatics

[geopolicraticus]

The Auxiliary Sciences of Formal Thought

The Penumbra of Logic.—Outside the boundary of logic proper, inhabiting the penumbra of logic, if you will, are theories of truth, the nature and role of definitions, and the structures of formal systems. Other modes of formal thought, such a mathematics and mereology, also employ or imply these penumbric ideas. Axiomatics, being the paradigmatic structure of a formal system, has been so thoroughly assimilated into mathematics it is often taken to be a part of it, but the axiomatic method—long called the geometric method because of its association with the exposition of geometry since Euclid—is equally applicable to any of the formal sciences, and indeed is employed in the natural sciences as well when rigor appears to demand it. But axiomatics didn’t appear in a work on logic until the Port-Royal Logic in the seventeenth century (La logique, ou l’art de penser, 1662). Nevertheless, these auxiliary sciences of formal thought have a closer relationship to logic than the other formal sciences, as various systems of logic down through history have included them or excluded them as the logic in question was conceived with a narrower or wider scope. Aristotle gives his theory of doctrine of definition in the Posterior Analytics, so if we take this book to be a part of logic (and it is traditionally part of the Organon), then definition is a part of logic. But the Posterior Analytics is widely recognized as a work in the theory of science (at least, science in the Platonic sense of demonstrative knowledge), so that the position of this work within logic itself is the question of the position of definition within logic writ large. Being shared, then, among the formal sciences, serving as the common language of the formal sciences, are these disciplines in the penumbra of logic actually the essence of formal thought? Rather than being in the gray area of formal thought, ought they to stand out as the most sharply defined exemplars of formal thought?

[geopolicraticus]

The Dimensions of Transcendental Logic

The Problem of Formalization.—The bulk of the problem of logic is not the process of deduction, but the formalization of the problem that precedes deduction. It is how the problem is set up for deduction rather than the deduction itself in which the art of logic lies. And here we must take the “art” of logic seriously, for that part of logic not yet reduced to mechanical derivation remains an art and not a science and cannot be mechanized in the manner of fully formalized proof procedures. How exactly we shoehorn ordinary thought into the strictures of logic so that anything at all can be derived is the real problem of logic, and this is the problem of formalization. This problem of formalization has many dimensions, all of which bear upon the result, but each of which can be taken up separately. How we break down the problem of formalization into discrete parts is itself a part of logic, or, rather, transcendental logic. Ancient axiomatics distinguished between axioms and postulates: axioms are common to all the sciences, while postulates were specific to a particular body of knowledge. We could still break down formalization in this way, but the distinction between axioms and postulates is now regarded as archaic. While the distinction remains valid, it passes over in silence specifying the choice of logic employed in derivation, unless we consider this a tacit function of the axioms. Twentieth century axiomatics distinguished between formation rules and transformation rules: formation rules govern what can be a well-formed formula (WFF), while transformation rules govern transforming one WFF into another WFF. This conceptualization has the virtue of making reasoning (transformation) explicit, but it does not explicitly thematize the elements that may be employed in a WFF (what we could call the ontology of a calculus), which is taken for granted, as the ancient conception took the univocity of reasoning for granted.   

[superpte]

Pure Logic Is An Illusion. Axiomatics Means Choice, Thus Heresy.

Pure Logic Is An Illusion. Axiomatics Means Choice, Thus Heresy.

The problem with pretending that there is something as pure logic is that any logical system involves axiomatic choices, and those, in turn, are a matter of emotions, feelings, proclivities, inclinations, and, generally sentiment, thus emotion.  This initial axiomatic choice is even more basic than Second Order Logic Incompleteness theorems (which say that, given an axiomatic and basic…

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[animatitration]

Axiom of Existence

There is a set

[fractal-ontology]

Obliquity

Nils Guadagnin, Hanging stone, 2014 Mousse polyuréthane, bois, électroaimant, 28 x 28 x 125 cm

Demodulation. There is always a monadic resonance to which a repetition is coupled in order to form a motor or compose an operational line, assembling at the limit a free phylum of machine interconnectivity. Every machine an operator or operand of another functional aggregate, assigned to an eternal…

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[feelingpolitical]

Capitalism is indeed an axiomatic, because it has no laws but immanent ones. It would like for us to believe that it confronts the limits of the Universe, the extreme limit of resources and energy. But all it confronts are its own limits (the periodic depreciation of existing capital); all it repels or displaces are its own limits (the formation of new capital, in new industries with a high profit rate). [...] And it does both at once: capitalism confronts its own limits and simultaneously displaces them, setting them down again farther along.

Deleuze and Guattari, A Thousand Plateaus, “7000 B.C.: Apparatus of Capture,” 463

[departedsouls-rp]

Can I reserve Caroline Forbes please?

Of course! Caroline is now on reserve for 24 hours. 

[charrlut]

So for my Theory and Criticism class, I have to analyze a text (which can be anything) and explain how it demonstrates axioms from what we've been studying. I like to keep things interesting, so I'm going to explain how tumblr is a great example of Barthes's ideas as expressed in "Death of the Author."

I really, really like being an English major.