Plausibility, likelihood, novelty, nontriviality, versus optima, probabilities, information, givens

Latest substantive edits: August 2023. This post still needs work.

Quite a few have remarked over the years that a deductive conclusion can give to its premises a new aspect or appearance (occasionally called "psychological novelty"), even though, being deductive, the conclusion claims nothing unentailed by its premises, nothing decidedly, indeed quantifiably, informative or newsy in full light of the premises. That fraternal opposition — between quantifiable information and new aspect or perspective — is just one among four such oppositions in a noteworthy pattern. I don't know whether anybody else has noticed that.

The two series spelt out in this post's title line up nicely in four such fraternal oppositions, four correlations across a fissure. The pattern reinforces the fissure-crossing correlations' significance and opens a way to check consistency through lockstep thinking about the relationally patterned items:

The above assumes:

• SURMISE as inference with mutual non-entailment between premises and conclusion.
• INDUCTION to a collective whole as inference in which the premises are entailed by, but do not entail, the conclusion.
• FORWARD-ONLY DEDUCTION as inference in which the premises entail, but are not entailed by, the conclusion.
• REVERSIBLE DEDUCTION as inference in which the premises entail, and are entailed by, the conclusion.

Note:  The four modes of inference as defined above:
• lack overlap and
• together exhaust the elementary possibilities of classical entailment relations between premises and conclusions.

So, the mathematically much-studied topics — in Table (A) below — resemble (one-to-one) but are NOT the same as the perspectives generally sought in inference — in Table (B) below:

Both tables have the following pattern, with notable oppositions along its diagonals:

By their central roles in the deductive topics (in Column I in Table (D)), I reached familiar arithmetical terms neatly assembled, in this order:
1. Difference by subtraction.
2. Quotient (or ratio) by division.
3. Logarithm: x = log_y z.
4. Root (or base): y = ^x√z.
Example: 8 = 2³. So, 3 = log₂ 8 and 2 = ³√8.
Fixed base (or root) ♥ varying logarithms (as in fixed-base number systems, e.g., base-10).
Fixed log (or exponent or, to be precise, root's index) ♥ varying roots.
The four kinds of numerical results come from four inverses (or whatever they ought to be called) of the normal three basic arithmetical operations that give:
• additive sum (versus subtractive difference),
• multiplicative product (versus divisional quotient or ratio),
• exponentiative power (versus logarithm
 and
 versus root or base).
Yet, I haven't seen why, from a purely ARITHMETICAL viewpoint, logarithm would precede root (as I have it doing) rather than follow it, rank alongside it, sit unordered with it, or bobble ambiguously ordered with it.

0. Zeration, −1., ?
1. Addition
2. Multiplication
3. Exponentiation
4. Tetration
5. , 6., 7., … → ∞

The four topics in Table (A) above (optima, probabilities, information, givens) are the main topics of abstract and significant mathematically deductive areas concerned with structures of alternatives in terms of timelike or (quasi-)modal cases (see Table (E) below).

The four perspectival merits of inference listed in Table (B) above pertain to systematically defined inference modes, and have familiar forms and some intellectual history (see further below).

(E) Temporal, modal (or at least timelike, quasi-modal) cases, combined into an encompassing picture

Optima & feasibles: of (or as if of) the almost-now, more or less along the surface of the future. Not generically concerned with repeated trials.

Probabilities: of (or as if of) the more gradually addressable future. (Zadeh's possibility theory seems to pertain to this area too; he somewhere called it an alternative to probability theory.)

Information, news: of (or as if of) the just-now, more or less along the surface of the past.

Givens, data, facts, wrinkles, complexity, complication: of (or as if of) the more settled, layered past.

The lightcone-like structure is evoked by assuming, even if only vaguely, that not only is motion relative but also there is a finite speed limit of communication and causation — e.g., the lightspeed constant in the known physical universe. But I guess that the present purpose could be adequately served by a more Galilean, less constrained kind of relativity picture if it were at least to exclude infinite speeds. Anyway, one will also want to encompass situations in which the top practical speed of some things is much lower than a universal signal-speed limit (light speed) and is neither exact, nor universal (across events), nor invariant (across various inertial reference frames); hence, there is often, for example, to allow some vagueness as to whether information arrives at lightspeed or somewhat more slowly, say via electrons or a turtle.

Promising or fruitful perspectives given by conclusions to premises

• Conclusions in a given mode of inference vary in the degree to which they put their premises into the pertinent aspect (simplicity, verisimilitude, novelty, nontriviality). However, none of those aspects seems generally and usefully quantifiable.
  Peirce wrote on the quantifying of verisimilitude ('likelihood') in 1910:

  I hope my writings may, at any rate, awaken a few to the enormous waste of effort it would save. But any numerical determination of likelihood is more than I can expect.
  —C.S. Peirce, "Notes on the Doctrine of Chances," Collected Papers Volume 2, Paragraph 663, reprinted in Buchler's collection, see link.

  On complexity, the complexity theorist Cosma Shalizi wrote in 2012 or earlier:

  C'est magnifique, mais ce n'est pas de la science. (Lots of 'em ain't that splendid, either.) This is, in the word of the estimable Dave Feldman (who taught me most of what I know about it, but has rather less jaundiced views), a "micro-field" within the soi-disant study of complexity. Every few months seems to produce another paper proposing yet another measure of complexity, generally a quantity which can't be computed for anything you'd actually care to know about, if at all. These quantities are almost never related to any other variable, so they form no part of any theory telling us when or how things get complex, and are usually just quantification for quantification's own sweet sake.
  —Cosma Shalizi, "Complexity Measures" (and earliest via Wayback Machine).

  See also Shalizi's note "Complexity".
• Moreover, such a promising or fruitful aspect stands out better when one is unsure of the inference and conclusion that present the aspect. When its premises and conclusions have become well established, one is at least less likely to call an inference or its conclusion (merely) plausible, or (merely) likely (in the sense of verisimilitude), or (rather) novel, or (rather) nontrivial. Peirce saw the question of likelihood or verisimilitude as applying to theories for which evidence, but not enough evidence, has been gathered. It is a standing vein of humor among some mathematicians to call a deductively established conclusion "trivial," simply because it is deductively established. (In that spirit someone may likewise deny novelty in an established conclusion, and deny natural simplicity and verisimilitude in an excluded conclusion.)
• All the same, no mind would bother with reasoning — explicit, consciously weighed inference — in the general absence of these aspects; little would remain of inference in general, mainly activities such as remembering and free-associative supposing. The aspects have value needful in order to motivate reasoning at all and to help choose among various reasonings in a given mode to the extent that other reasons for such choice do not suffice or override. Definitions of inference modes in respect of whether the premises entail the conclusions and whether the conclusions entail the premises, while needful, do not suffice to formulate promising or fruitful inference. Specialized forms (for example the traditional syllogistic forms), however, are often defined that do formulate some involvement of promising or fruitful aspects.

Each promising or aspect counterbalances, quasi-ironically, the essential structure of the entailment relations between premises and conclusions that defines the inference mode to which the aspect pertains. So, the collectively systematic character of the definitive entailment relations is reflected in the collectively systematic character of the promising or fruitful aspects.

Note: Induction as actually framed in practice sometimes has a conclusion that does not fully entail its premises, even though we think of induction as inferring from a part (a sample or fragment) to a whole including the part. See "Deductive vs. ampliative; also, repletive vs. attenuative".)

Deduction is so defined that its conclusions are true if its premises are true, while other modes of inference lack that character; this has been seen as a problem for the other modes. In fact any inference depends on being correctable in its premises and procedure through larger inquiry and the testability of conclusions. Certainly surmise to an explanation is the least secure mode of inference, but also the most expeditious, as Peirce pointed out. Now a valid deduction secures its conclusion's truth if its premises are true, while other inference modes only suggest, with more or less strength, their conclusions' truth, and do so by the perspective they bring and other considerations (methods of sampling, weakness of alternate explanations, etc.) rather than only by general definitive entailment structure. But a deduction's conditional assurance of a true conclusion is counterbalanced by the perspective of novelty or nontriviality that a worthwhile deduction brings; dubitability is natural, and right in a way, in deduction. The deduction's conclusional perspectives (novelty, nontriviality) that, with varying strength, incline one to check one's reasonings and claims for falsity, inconsistency, etc., are the same ones that make deduction worthwhile.

1. A surmise's conclusion's natural simplicity, and
2. an inductive conclusion's Peircean verisimilitude,
suggest that a conclusion might be true although non-deductive, and offer the hope of helping inquiry toward eventually finding (respectively) the real natural simplicities and the real reliabilities in things.

3. A forward-only–deductive conclusion's novelty of aspect, and
4. a reversible deduction's nontriviality,
suggest that a conclusion might be false despite being or seeming deductive (involving an error perhaps in the reasoning or perhaps in the premises — indeed, the main purpose of some deductions is to expose the premises to testing), and offer the hope of helping inquiry toward eventually finding (respectively) the real surprising ramifications and the real deep structures in things.

More on the correlation between (A) a fourfold of deductive quasi-modal topics and (B) a fourfold of inference modes' valuable aspects (plausibility, verisimilitude (a.k.a. likelihood), novelty, nontriviality)

A. Quasi-modal topics of so-called "applied" but abstract and significant mathematically deductive areas (below).

B. Meritorious aspects of conclusions, sorted by inference mode below.

 

A1. Optima & feasibles.
• Differences (distances) with directions.
• Mathematics of optimization (longer known as linear & nonlinear programming) applies extremization, calculus of variations, Morse theory.

B1. Naturalness, simplicity, plausibility.
Suggested technical name: avioidity.

• Pertains to surmise, explanatory hypothesis, that which Peirce called "abductive inference." I propose a (non-Peircean) definition of surmise as inference that automatically preserves neither truth nor falsity (the premises don't deductively imply the conclusions, and the conclusions don't deductively imply the premises). Even if one excludes cases where the premise set is inconsistent with the conclusion set (e.g., "there is a horse, so there isn't a horse"), natural simplicity, etc., remain an important consideration in the surmises that remain. The inference "There is a horse here, so there is a squid here" is a surmise without natural simplicity. Now, one could restate a plausible surmise to make it seem, technically, an induction, its premises implied by but not implying its full conclusions: "Normally after rain the lawn is wet and this morning the lawn is wet. So, normally after rain the lawn is wet, this morning the lawn is wet, and last night it rained." But this is like turning a traditional syllogism into a reversible deduction by restating all the premises in the conclusion. Such representation of the inference does not faithfully enough reflect the guiding inquiry interest of the inference and instead counterproductively obscures it with redundancies.
  
• Studies of concrete phenomena (that is, sciences and studies physical, material, biological, and human / social) are for drawing conclusions that typically are (plausible) surmises (but they do apply the other modes of inference along the way). As Peirce points out in discussing verisimilitude in induction (Collected Papers Vol. 2, in Paragraph 633):

  Strictly speaking, matters of fact never can be demonstrably proved, since it will always remain conceivable that there should be some mistake about it. [....] Indeed, I cannot specify any date on which any certain person informed me I had been born there; and it certainly would have been easy to deceive me in the matter had there been any serious reason for doing so; and how can I be so sure as I surely am that no such reason did exist? It would be a theory without plausibility; that is all.

  Hence, even a verisimilar and high-confidence induction, if it arises from concrete individual cases, rests in the end on premises which are affirmed because their falsity is implausible, or so taken, and which amount in the end to plausible surmises. Hence the ultimate standing of conclusions sought by studies of concrete phenomena is that of plausible surmise. Now, I've cited Peirce saying something that I believe helps imply my conclusion; but in or around the same year (1910), he held that all determination of probabilities of actual objects (such as actual dice) rests on verisimilitudes (Collected Papers Vol. 8, in Paragraph 224) and did not mention a dependence of those verisimilitudes on plausibilities.

 

A2. Probabilities.
• Ratios (such that 0 ≤ the ratio ≤ 1).
• Mathematics of probability applies general measure theory, enumerative combinatorics.

B2. Verisimi­litude, likelihood, in C. S. Peirce's sense.
Suggested technical name: veteroidity.

• Pertains to induction, inference that automatically preserves falsity but not truth (the premises don't deductively imply the conclusions, but the conclusions deductively imply the premises).
    Verisimilitude a.k.a. likelihood in Peirce's sense consists in that, if pertinent further data were to continue, until complete, to have the same character as the data supporting the conclusion, the conclusion would be proven true. The phrase "inductive generalization" is an approximate way of saying "induction with some verisimilitude." The inference "there is a horse here, so there are a horse and a squid here" is an induction without verisimilitude.
• Studies of positive phenomena in general (studies such as inverse optimization, statistics, information theory's induction-oriented areas, and maybe philosophy) are for drawing conclusions typically inductive.

 

A3. Information, news.
• Logarithms.
• "...for every unconstrained information inequality," i.e., every law of information, "there is a corresponding group inequality, and vice versa." — Yeung (2012), p. 381. (Application of group theory reveals non-Shannon-type laws of information, see 2003 tutorial by Yeung.)

B3. New aspect, elucidative­ness, signifi­cance in a sense.
Suggested technical name: novelloidity.

• Pertains to 'forward-only' deduction, inference that automatically preserves truth but not falsity (the premises deductively imply the conclusions, but the conclusions do not deductively imply the premises). The inference "There are a horse and a squid here, so there is a horse here" is a 'forward-only' deduction without novelty of aspect. The inference "All A is B, and all B is C, so all A is C" is a 'forward-only' deduction with some novelty of aspect.
    Novelty of aspect of the kind sought in 'forward-only' deduction involves some tightening of focus, which is lacking in such a 'forward-only' deduction as "Socrates is yonder, so Socrates is here or yonder." (Still, it is important sometimes to remember how "surprisingly" broad alternatives are deductively implied — especially on those occasions when one is surprised by a subtle role of such implications in one's reasoning.)
• 'Applied' but abstract and significant mathematically deductive areas — mathematics of optimization, of probability, of information, and of logic — are for drawing conclusions that typically are 'forward-only' deductions. I don't say that equivalences, e.g., that between p and T→p, are unimportant in those fields. But their general aim at 'forward-only' deduction is why they are correlated with areas of inductive research as their "inverses" or "reverses." On the other hand, reverse mathematics (as it is called) is, if not pure mathematics per se, still mathematical logic about pure mathematics. Still, if it becomes part of pure mathematics, it would not be the first research program to originate in mathematical logic and then grow into a part of pure mathematics, if some old accounts that I recall about non-standard analysis are correct. My understanding, of which I'm doubtful, is that probability theory is also often applied in pure mathematics, and not always trivially. These so-called 'applied' areas need another generic name.

 

A4. Givens, facts, data as n-ary complexuses.
• Roots, bases, as numbers of elements repeatably referencible in tuples in a Cartesian product or in a relation defined in it.
• Mathematical logic applies order theory.

B4. Nontriviality, depth, complexity, 'lessonful­ness'.
Suggested technical name: basaloidity.

• Pertains to reversible deduction, inference that automatically preserves truth and falsity alike (the premises deductively imply the conclusions and vice versa). The inference "There are a horse and squid here, so there are a horse and a squid here" is an equative deduction without nontriviality. The inference "3 × 5 = 15, ERGO 15 ∕ 3 = 5" is a reversible deduction, with at least a jot of nontriviality, from one equality to another that is equivalent to it. When the well-orderedness of the pertinent set is a standing given, then the mathematical-induction step, from the ancestral case conjoined with the hereditary case, to the conclusion, is a reversible deduction.
• 'Pure' mathematics is for drawing conclusions typically deductive through equivalences. 'Forward-only' deduction sometimes occurs, especially when greater-than or less-than statements are involved. I've read that, because of this, in a mathematical induction a premise (such as the ancestral case) is not always to be proven by simply reversing the order of one's scratch work; sometimes instead one must find some other way back.

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