C.S. Peirce is one who recognized the constitutional independence of mathematical inquiry, finding at its core a mode of operation tantamount to observation and more primitive than logic itself. Here is one place where he expressed that idea.
Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.
— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)
When it comes to the relative contributions of phenomenology and mathematics to logic, I always find myself returning to the picture I drew once before from Peirce’s Syllabus, on the relationship of phenomenology and mathematics to the normative sciences and metaphysics.
Normative science rests largely on phenomenology and on mathematics;
metaphysics on phenomenology and on normative science.
— Charles Sanders Peirce, Collected Papers, CP 1.186 (1903)
Syllabus : Classification of Sciences (CP 1.180–202, G-1903-2b)
I find this “two-footed, thrice-braced” stance has many advantages over the “dufflepud” attempt to stand logic on phenomenology alone.
The arc of the semiotic universe is long but it bends towards universal harmony.
Re: Facebook Discussion • What’s at the End of a Chain of Interpretants?
Semiotic manifolds, like physical and mathematical manifolds, may be finite and bounded or infinite and unbounded but they may also be finite and unbounded, having no boundary in the topological sense. So unbounded semiosis does not imply infinite semiosis.
Here are two points in previous discussions where the question of infinite semiosis came up.
☯ TAO ☯
Trials And Outcomes
Expression | Impression
Effectors | Receptors
Exertion | Reaction
Conduct | Bearing
Control | Observe
Effect | Detect
Poke | Peek
Note | Note
Just a few notes to be developed later …
Pragmatism makes thinking to consist in the living inferential metaboly of symbols whose purport lies in conditional general resolutions to act. (Peirce, CP 5.402 n. 3).
Such reasonings and all reasonings turn upon the idea that if one exerts certain kinds of volition, one will undergo in return certain compulsory perceptions. Now this sort of consideration, namely, that certain lines of conduct will entail certain kinds of inevitable experiences is what is called a “practical consideration”. Hence is justified the maxim, belief in which constitutes pragmatism; namely:
In order to ascertain the meaning of an intellectual conception one should consider what practical consequences might conceivably result by necessity from the truth of that conception; and the sum of these consequences will constitute the entire meaning of the conception. (Peirce, CP 5.9, 1905).
I once wrote a “pure empiricist” sequential learning program that took this sort of approach to the data in its input stream.
Here is the manual, that will give some idea —
The program integrated a sequential learning module and a propositional reasoning module that I thought of as The Empiricist and The Rationalist, respectively.
The learning module was influenced by ideas from the psychologists Thorndike and Guthrie and the statisticians Fisher and Tukey. The reasoning module made use of ideas about logical graphs from C.S. Peirce. There is a kind of phase transition as we pass from finite state adaptation covered by the learning module to context-free hypothesis generation covered by the reasoning module, but it happens that some aspects of the latter are already anticipated in the former.
I think Peirce would say that any struggle to pass from the irritation of doubt toward the settlement of belief is a form of inquiry — it’s just that some forms work better than others over the long haul. Instead of a demarcation Peirce describes a spectrum of methods, graded according to their measure of success in achieving the aim of inquiry.
I’m about to be diverted for a couple of weeks but this is an ever-ongoing question so I know I’ll be coming back to it again. The short shrift goes a bit like this —
The gist of the idea that Peirce dubbed the pragmatic maxim is really a mathematical principle that has always been hard to render in ordinary language, largely because of the Procrustean subject-predicate embedding that most of the languages we know and love impose on its core structure. The primal form is more like one of those bistable gestalts — duck-rabbit, Necker cube, old-young woman, etc. One way to get a mental handle on the matter is to mull over the many variations on its underlying theme, such as the ones I quoted and discussed in my blog post —
C.S. Peirce’s pragmatic maxim marks the place where the tire of theory meets the test track of experience — it tells us how general ideas are impacted by practical consequences. If our concept of an object is the sum of its conceivable practical effects then the truth of a concept can be defeated by single outcome outside the sum.
Defining minimal negation operators over a more conventional basis is next in order of logic, if not necessarily in order of every reader’s reading. For what it’s worth and against the day when it may be needed, here is a definition of minimal negations in terms of and
To express the general case of in terms of familiar operations, it helps to introduce an intermediary concept:
Definition. Let the function be defined for each integer in the interval by the following equation:
Then is defined by the following equation:
If we take the boolean product or the logical conjunction to indicate the point in the space then the minimal negation indicates the set of points in that differ from in exactly one coordinate. This makes a discrete functional analogue of a point-omitted neighborhood in ordinary real analysis, more exactly, a point-omitted distance-one neighborhood. In this light, the minimal negation operator can be recognized as a differential construction, an observation that opens a very wide field.
The remainder of this discussion proceeds on the algebraic convention that the plus sign and the summation symbol both refer to addition mod 2. Unless otherwise noted, the boolean domain is interpreted for logic in such a way that and This has the following consequences:
The following properties of the minimal negation operators may be noted:
It will take a few more rounds of stage-setting before I can get to concrete examples of applications but the following should indicate the direction of generalization embodied in minimal negation operators.
To begin, let’s observe two different ways of generalizing the operation of exclusive disjunction (XOR) or symmetric difference.
Let = the boolean domain
Generalizing in that sense would continue the sequence as and so on. These are known as parity sums, returning if there are an even number of ’s in the sum, returning if there are an odd number of ’s in the sum.
The triple sum can be written in terms of 2-place minimal negations as follows:
It is important to note that these expressions are not equivalent to the 3-place minimal negation