On the nature of proofs, causality, induction and leaps

Borges

What is induction?

Borges, with his usual mastery, summarizes what inductive reasoning is:

To think is to forget differences, to generalize and to abstract. (in Funes the Memorious).

The popular definition of induction is: to infer the universal or general from the particular. For example (a person, still in bed, looking from his window):

1. (evidence) the sidewalk is wet, 
2. (evidence) the sky is overcast, 
3. (induction)
1. when (the sidewalk is wet) AND (the sky is overcast) 
2. it usually (modal qualifier) indicates (inductive evidence) that it has rained, 
1. To refute (2) at least I should go outside and check my neighbor's lawn - but I'm not in the mood.
4. (conclusion) therefore I believe (with some probability) that it rained last night. 

    Induction is a process of reasoning that moves from the particulars to the general. In other words, given some information about some particular instances of some phenomenon or event, induction is the process to arrive to conclusions about the general principles that explain, as a special case, the particulars at hand. As such, induction entails a movement of reasoning from a state of low information (particulars), to a state of higher information (general). This unbalance of information, in which the general conclusion wouldn't be reached soundly, calls for a leap coming from: probability of opinion, or intuition or any other valid device that facilitates the arrival to the conclusion. Nevertheless, inductive judgment could also have the opposite directionality, i.e. to go from the general to the particular, for cases in which the probatory steps to reach the conclusion are too many, or are uncertain (discontinuous), or just that the premises are necessary but not sufficient to establish the conclusion. Therefore induction involves judgmental leaps more or less endorsed by the premises, or by the reasoning. As such induction must necessarily be accompanied with an appraisal of the uncertainty involved (evidential support). As a footnote, informal fallacies arise in this step.

The framework of Induction - Leibniz's predecessors

Ramon Llull

To understand what induction is we must place it in its proper context, i.e. reasoning. I propose that for this draft we choose Leibniz's model of reason. Leibniz basically proposes that reasoning is an algorithmical/combinatorial operation of the human cognition. Reason combines simple thoughts to form complex thoughts. After all the term "Logic" means to compute, to reckon, count.

It seems that Leibniz got inspired by Ramon Lull and Thomas Hobbes in order to automatize Aristotle formal logic.

• Lull's idea is: it would be possible, by the combination of a set of simple terms, to establish all possible propositions and thus to discover all possible statements and demonstrate all possible truths to which human knowledge can aspire (Pombo 352). 

    Leibniz disagrees with a few technical details from Lull's project, but not with the nucleus; thus Leibniz: (1) accepts a mathematical model of thought, (2) that there are some (few) primitive ideas, and that (3) there are some combinatorial laws to build any complex thought by associating the primitive ideas.

    Likewise Hobbes said in Leviathan:

• reasoning is nothing but reckoning [calculation]...when a man reasons, he does nothing else but conceive a sum total, from addition of parcels; or conceive a remainder, from substraction of one sum from another ... those operations are not incident to numbers [alone], but to all maner of things that can be added together and taken one of another...In sum, in what matter so ever there is place for addition and substraction, there also is place for Reason, and where these have no place, there Reason has nothing at all to do.

Thomas Hobbes

    Both, Lull and Hobbes concurred in the combinatory character of reasoning. Hobbes yet contributed another ingredient to Leibniz theory, namely that reason is a linguistic activity, there are words, not only numbers or images. He adds:

• The use and end of Reason, is not the finding of the sum, and truth of one, or a few consequences, remote from the first definitions ... but to begin at these; and proceed from one consequence to another. 

At this point, we could object that the views of Lull, Hobbes and Leibniz, constitute powerful, yet limited, inferences about reasoning. For instance, talking about combinations and literature, Borges says that the former cannot be reduced to a sort of algebra of combinations like Lull claimed, because there is a connection between the book and the reader, which cannot be grasped by the combinations. (Otras Inquisiciones, 237) And that makes perfect sense, not only in literature. I would say that reasoning sometimes is calculation; or, reasoning sometimes is a simplified modeling of reality with an algorithm. Some types of reasonings can be represented by formal algorithmical rules.

However, I believe that induction lends itself this conceptualization, therefore I will adopt Leibniz's combinatorial idea.

The framework of Induction - Leibniz's model

Leibniz (at Leipzig)

Leibniz advocated a demonstrative character to knowledge from first causes. As Ian Hacking says paraphrasing Leibniz: and "Truth is ultimately demonstration" or "knowledge had always been demonstration from first principles". Given the above, Leibniz proposes two approaches to deal with deductive and inductive reasoning as follows:

1. logically deductive propositions are proven with a finite sequence of sentences
2. contingent propositions are proven with an infinite sequence of sentences: "The reasons given for contingent truths must proceed to infinity.¹

Thus for Leibniz, the proof is an algorithm, a chain o reasoning, an ordered combination of elements, such as words, sentences, numbers, etc. (Belaval 1986); an algebra to combine the epistemological objects. The word combination is used in the same sense as in Ars Combinatoria which is the pursuit of algorithms that allow reason to arrive at new truths, unexplored, unlike the "truths" that are demonstrated by formal logic (Marias ###). Along the same lines, Thomas Reid observed

In reasoning by syllogism from general principles, we descend to a conclusion virtually contained in them [Reid was one of the first to say that the syllogism was circular, i.e. is a petitio principii]. The process of induction is more arduous, being an ascent from particular premises to a general conclusion. The evidence of such general conclusions is probable only, not demonstrative: but when the induction is sufficiently copious, and carried on according to the rules of art, it forces conviction no less than demonstration itself does. Brief Account, 236-237.

My take on induction

1. Inductive reasoning is, by far, the most common type of reasoning in real life. In any common physical situation there are infinite number of steps between two events. (Take for instance the sound emanating from the clapping of hands, there's the macroscopic explanation of 2 surfaces hitting each other, then, the continuum mechanics deformation of elastic bodies, and so on.) Human beings use the inductive reasoning much more frequently than deductive reasoning.
2. Give its frequency, inductive reasoning cannot be infinite, but must be tractable so that can be used in real life.
3. The tractability is achieved by shortening the infinite sequence by inductive "leaps".
4. Each leap (shortening) is a simplification achieved by any of the available inductive actions (association, generalization, analogy), upon all the steps involved between conjoint (separated) instances of thought.
5. Each leap constitutes an inductive reasoning, because it goes from the particulars to the general.

Precautions towards healthy induction reasoning

Leaps, or inductive judgments, are approximations to the true, correct judgment. To avoid fallacies emerging from faulty inductive reasoning, or at least, to keep it transparent and its uncertainty well defined:

1. "Leaps" must be valid, and sound. Hume was concerned with the impossibility of human beings to reach perfect inductive (i.e. deductive) judgments, and that led him to skepticism. However, we shouldn't be skeptical if we know that they're just that, approximations, and we can live with that. 
1. Such approximations must be verified and validated. Their applicability to the problem at hands must be guaranteed.
2. The argument by analogy, is the one that most carefully should be watched carefuly, because it makes many assumptions about the reasoning process like: regularity, equality, etc.
3. The incorrect leaps result in fallacies
4. The fact that the validity of a piece of inductive reasoning lies in the realm of opinion is not necessarily bad. But the degree of belief has to be clearly stated.

Addendum

Collingwood contributes a very nice definition of abstraction in his Speculum Mentis (160ff):

"To abstract is to consider separately things that are inseparable: to think of the universal, for instance, without reflecting that it is merely the universal of its particulars, and to assume that 

one can isolate it in thought and study it in this isolation. This assumption is an error. One cannot abstract without falsifying. To think apart of things that are together is to think of them as they are not, and to plead that this initial severance makes no essential difference to their inner nature is only to erect falsification into a principle.... "

Lucretius in De rerum natura uses vestigia, footsteps, traces of truth.

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^{1. Translated by (c) Lloyd Strickland from Textes inédits tome 1 Gaston Grua (ed) pp 325-326;  Sämtliche schriften und briefe series VI volume 4 Deutsche Akademie der Wissenschaften (ed) pp 1663-1664↩}

References:

Borges, J. (1952) Otras Inquisiciones. Alianza Editorial - 1974.
Belaval, Y. (1986), Leibniz. Encyclopaedia Britannica.
Eco, U. (1993) The search of the perfect language. Wiley - 1997.
Hacking, I. (1975) The Emergence of Probability. Cambridge University Press

Pombo, O. (2010) Three Roots for Leibniz's Contribution to the Computational Conception of Reason. Programs, Proofs, Processes. F. Ferreira et al. (Eds.): CiE 2010, LNCS 6158, pp. 352–361, 2010.