Plausibility, verisimilitude, novelty, nontriviality, versus optima, probabilities, information, n-ary givens
Latest significant edit: January 7, 2016. This post still needs work.
IT has sometimes been noted that deductive conclusions claim nothing unentailed by their premisses, nothing informative or newsy in view of the premisses, yet often bring to their premisses a new aspect or perspective (a newness occasionally called 'psychological novelty'). That opposition between information and new perspective or aspect seems an instance of a noteworthy pattern.
The two series spelt out in this post's title line up nicely in such oppositions.
1. Optima & feasibles VERSUS plausibility as natural simplicity whereby surmise compensates for being wild.
Note: by "surmise" I mean abductive inference, pretty much, but definable (novelly or not, I don't know) as inference where the premisses neither deductively imply, nor are deductively implied by, the conclusion.
2. Probabilities VERSUS verisimilitude a.k.a. likelihood (in C. S. Peirce's sense) whereby induction compensates for being expansive.
Note: by "induction" (except in the case of mathematical induction, which is deductive) I mean inference from a sample or fragment to a whole, pretty much, but definable (not too novelly, I hope) as inference where the premisses do not deductively imply, but are deductively implied by, the conclusion.
3. Information, news, VERSUS new aspect whereby 'forward-only' deduction compensates for being constrictive.
4. Givens, data, facts, as n-ary complexuses, VERSUS nontriviality whereby equipollential deduction compensates for being utterly faithful (to the premisses).
Note that the four modes of inference explicitly or implicitly defined above have no generic overlap and exhaust the basic possibilities of classical entailment relations between premisses and conclusion.
I will discuss the oppositions not individually but collectively, and will focus most of all on the patterns made by each series. I will assume that the reader attributes at least some importance to the mathematics of optimization, probability, etc. As to the importance of perspectives brought by conclusions, I will point out that a conclusion generally needs to offer a more or less promising or fruitful perspective or aspect (be it plausibility, likelihood, new aspect, or nontriviality) as a merit and indeed a rationale, in order to help motivate inference and reasoning; each of those perspectival merits is hard to quantify usefully, yet they form a system reflecting that of the inference modes that they help motivate.
The perspectival merits in inference in Table (B) are one-to-one reminiscent of, but NOT equatable to, the deductive topics in Table (A):
|1. Optima & feasibles.
||3. Information, 'news'.
4. Givens (data, etc.)
as n-ary complexuses.
Roots, bases, as arities,
allied to other relations,
brought by surmise.
2. Verisimilitude (in
C. S. Peirce's sense)
brought by induction
sample to whole).
||3. New aspect
Both tables have the following pattern, with notable oppositions along its diagonals:
|1. Simple, doable, compelling.
2. Apt, consistent, consonant.
|X||3. New, distinctive. |
4. Complex, structured.
Note: I reached the ordering optima, probabilities, information, givens (i.e., logic) as reflecting some broad correlations with levels of concrete phenomena (see Table (D)), such that it seems an order of being or of (decreasing) abstractness, and its reverse seems an order of (decreasing) familiarity.
|1.||Optima & feasibles. |
|4.||Givens, data, facts, base̅s |
(for further conclusions).
The topics in Table (A) 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)).
The perspectival merits of inference listed in Table (B) have familiar forms and some intellectual history (see further below).
Optima & feasibles: of, or as if of, the almost-now, more or less along the surface of the future.
Probabilities: of, or as if of, the more gradually addressable future. (Zadeh's possibility theory seems to pertain to this area too; he calls 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: 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 not exact, universal (across events), or 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.
Fruitful aspects or perspectives given by conclusions to premisses
1. Simplicity of a surmise's conclusion (an explanatory hypothesis) is a familiar idea. One version of it is parsimony or Ockham's Razor; C. S. Peirce held that logical simplicity is secondary and that at its extreme it would add no explanation to a surprising observation; he explored simplicity of explanation as plausibility, facility, naturalness, instinctual attunement, and Galileo's 'natural light of reason' (see the linked passage from "A Neglected Argument," 1908).
2. An inductive conclusion's verisimilitude, in Peirce's sense, is an idea familiar in the form of that which people mean in speaking of an induction as an inductive generalization, expecting the whole to resemble a sample (preferably a fair one).
3. New or novel aspect of conclusion of a 'forward-only' deduction such as a categorical syllogism has been noted by various people including Peirce, and they have typically seen it as pertinent to deduction generally. It is familiar in the sense that it is considered perspectivally redundant, for example, to conclude a deduction merely by restating a premiss in unchanged form.
4. Nontriviality or, in stronger dose, depth of a mathematical conclusion is a familiar idea among mathematicians, and has been an element in the formation of ideas of complexity. Conclusions in pure mathematics tend to be equipollent (propositionally equivalent) to their premisses, as Aristotle noted (Posterior Analytics, Bk. 1, Ch. 12, link to pertinent text).
- Conclusions in a given mode of inference vary in the degree to which they put their premisses into the pertinent aspect (simplicity, verisimilitude, novelty, nontriviality). However, none of those aspects seems generally quantifiable in a useful way. In particular: 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." ("Notes on the Doctrine of Chances," Collected Papers Volume 2, Paragraph 663); and see complexity theorist Cosma Shalizi's note "Complexity Measures."
- Moreover, such a fruitful aspect stands out better when one is unsure of the inference and conclusion that present the aspect. When its premisses and conclusions have become well established, one is at least less likely to call an inference or its conclusion (merely) cogent or 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 say of a well-established conclusion: "that's 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 premisses entail the conclusions and whether the conclusions entail the premisses, while needful, do not suffice to formulate fruitful inference. Specialized forms (for example the traditional syllogistic forms), however, are often defined that do formulate some involvement of fruitful aspects.
Each fruitful aspect counterbalances, quasi-ironically, the essential structure of the entailment relations between premisses 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 fruitful aspects.
1. A surmise's conclusion (with 'surmise' as defined below) is complex in the sense of both adding to, and subtracting from, what the premisses claim, and is of interest when it nonetheless brings a simple perspective.
2. An induction's conclusion is novel in the sense of adding to, but not subtracting from, what the premisses claim, and is of interest when it nonetheless brings a 'conservative' or 'frugal' (verisimilitudinous or 'likely') perspective.
3. A 'forward-only' deduction's conclusion retrenches in the sense of subtracting from, but not adding to, what the premisses claim, and is of interest when it nonetheless brings a novel perspective.
4. An equipollential deduction's conclusion is simple in the sense of neither adding to, nor subtracting from, what the premisses claim, and is of interest when it nonetheless brings a complex or nontrivial perspective.
Note: Induction as actually framed in practice sometimes has a conclusion that does not fully entail its premisses, 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 premisses 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 premisses 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 premisses 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 natural plausibility, and
3. A 'forward-only' deduction's novelty of aspect, and
|Quasi-modal topics of so-called 'applied' but abstract and significant mathematically deductive areas (below).||Aspects with merit in inference (below).
Valuable, fruitful aspects or perspectives into which conclusions put premisses are sorted below by inference mode.
|1.||Optima & feasibles.
||Naturalness, simplicity, plausibility, viability, cogency.
Suggested technical name: viatility.
||Verisimilitude, likelihood, in C. S. Peirce's sense.
Suggested technical name: veteratility.
|3.||Information, 'news'.||New aspect, elucidativeness, significance in a sense.
Suggested technical name: novatility.
|4.||Givens (data, etc.) as nullary, unary, binary, n-ary, complexuses. (See Arity.)
||Nontriviality, depth, complexity, 'lessonfulness'.
Suggested technical name: basatility.
The complexity theorist Cosma Shalizi (in his notes "Complexity" and "Complexity Measures") has said that complexity is ill-defined and its proposed general measures not actually useful. Now, the idea of complexity seems based on the idea of nontriviality in mathematics ("deep" in mathematics means "very nontrivial," I've been told), and the effort at quantifying complexity seems aimed at finding an information-like quantity. Yet, the idea of information as a quantity is rather simple. Now, one might argue that the analogous idea for complexity will be complex because it's about COMPLEXITY, of course. But that is to say that it is not a quantity on a mathematical par with information in the sense that probability is. Complexity, at least in that which seems the idea's core sense of nontriviality and depth as in mathematics, seems akin in various ways, including its importance and tantalizing character, to aspectual novelty, verisimilitude (in C. S. Peirce's sense), and plausibility (natural simplicity). So classing it, one ends up with four kindred aspects in inference, aspects that resist computation-friendly formulations but are vital as forms of value or merit in inference. Data or, more generally, givens, in their character of adicity or arity, with its definabilities and its own share of importance, seem a better "complexity" counterpart to information. For logic, there are, first of all, the structures of alternatives (such as that represented by the variable x) and of compounds conjunctive, conditional, etc., and such is the turn of interest that groups mathematics of logic with those of information, probability, and optimization. Now, when each element of such a compound or relation is itself a relation with the same arity, the arity is like a root or base raised to successive powers, though its sheer size is not the only point. Anyway, logic gains stature as a serious subject, as Quine pointed out, when it comes to the study of relative terms (in polyadic quantification), which have arities of their own; relative terms are a complicating factor in logic. Now, such a relation as the dyadic '__discussing__' is itself not a compound conjunctive, disjunctive, or otherwise; nor are relations such as pure mathematical operations, functions, etc.; yet the forms of such relations are what is applied to represent the forms of alternatives and other such compounds which, for their part, are in a sense (be it literal or figurative) relations among worlds.