1. Introduction
This paper explains the Problem of Induction (the Problem) as it was set up by Hume (Hume, 1989) and will briefly discuss the major groupings of attempts made to refute such; including a particular emphasis on the attempt to dissolve the Problem. To conclude, the author will lay out an independent theory of induction constructed while considering the topic. The ‘Category-Property Theory of Induction’ (C-P Theory), as it is to be called, will be based on an analysis of induction that uses categories and property sets, and is perceived to provide a potential new direction from which to approach a possible resolution to the Problem.
For the purposes of constraining the topic to something manageable within the word limit, this essay will be limited to discussing enumerative induction, or universal inference. That is, inference from a sample to a general hypothesis. It should be noted that contemporary notions of induction extend beyond this; see Carnap’s (1962) taxonomy of induction.
Note also that in this paper, the term ‘object’ is used to refer to any observable component of the universe, irrespective of whether such an object is usually referred to in English by a noun or a verb.
2. Hume on Induction
Induction appears to be critical to everyday actions and choices. Common sense would imply that because the sun has come up every day of our lives thus far and for all of recorded history, it is reasonable to act as if it will do so again tomorrow. Please note the implicit distinction between ‘believe’ and ‘act as if’.
Hume did not use the term ‘induction’. He instead wrote of causal inferences (Vickers, 2016); the process contemporarily called universal inference (Carnap, 1962). Hume was interested in the “habit of the mind”, that we might refer to as behavioural conditioning, that begins with the fact that all past As have been Bs, and leading to an action consistent with the assumption that the next A will also be a B. I phrase the question in this particular manner so as to emphasise what Hume reasoned to be a biological / neurological basis for the prevalence of inductive experience in daily life and in scientific endeavour. That is, he reasoned that causal inferences must be a product of the imagination in some sense, and not of reason.
As regards the proposition that the problem of induction shows that there can be no claim to absolute knowledge (Barseghyan, 2015), the author claims that while correct, it is not because induction is not rationally justifiable. There can be no (verifiable) claim to absolute knowledge because in reality there are no absolute facts. Facts themselves are contingent; they are features which are stable and observable across a period of time, within a particular context. That is, one could say that properties of an object are dependent on their space-time coordinates. Induction is not justifiable when taken across the ‘infinity’ of time as ultimately all linguistic categories will break down at this scale.
The formal sense in which inductive logic is usually written is as follows:
1) All observed As have been Bs
2) O is an A; where O is an as yet unobserved object
Therefore, O is a B
The Problem as postulated by Hume (1989) and summarised by Salmon (1967) is a dilemma with deductive and inductive parts. Firstly, it is clear that inductive logic cannot be rationally justified via deduction as the conclusion does not follow necessarily from the premises because of its future predictive element. Secondly, we are unable to use induction to justify induction, as this is begging the question.
3. Attempts to resolve the Problem
There have been many attempts to show that induction is, in fact, rational, however many of these can be reduced to a case of using induction to justify induction.
None of the attempts at rationalising induction have been successful in my view, and instead of spending time summarising them here, I will move on to what I consider more interesting approaches which seek to dissolve the problem.
Those who seek to dissolve the problem attempt to offer a third solution to the dilemma: showing that the Problem is erroneous in its construction. This is usually aimed at by demonstrating the existence of a flaw in one of the few assumptions underlying the Problem. One of the main thrusts of dissolution attempts has been to show that the Problem (“Is induction rationally justifiable?”) is ill-formed. It is said to be an absurd formulation; as it is claimed, rationality assumes the use of induction (Skyrms, 1966). That is to say, the Problem of Induction is a kind of tautology; equivalent to asking “Is the Pope a Catholic?”
I believe that there is merit to this line of argument and I will come back to this topic of whether induction is a precondition of what is classified as rationality when I discuss the C-P Theory. In the interim, let us turn to a particular attempt to dissolve the Problem as an example of the direction these particular arguments can go in.
The Problem of Induction lies, in part, with how the conclusion of some inductive inference is interpreted. Induction necessarily includes projection into the future, however, this projection is not to be understood as a claim to knowledge, but a hypothesis which will either be shown to be true or false given further observations.
John D. Norton has presented both a Material Theory of Induction (Norton, 2003) and a Material Solution to the Problem of Induction (Norton, 2009). He begins by identifying an assumption of the Problem, which is that inductive inference is subject to universal rules and claims that without this assumption, the Problem cannot be laid out. In his Material Theory, he demonstrates that inductive inferences are “warranted by facts”. Norton notes that facts are contingent and not themselves logical truths. This, in turn, implies that there are no universal supports for any inductive inference and that each inductive system is applicable exclusively in the domain where the supporting facts are true.
Under Norton’s Material Theory of Induction, the Problem as classically set up by Hume collapses to the problem of justifying facts. The answer to the resultant question “What justifies a warranting fact?” is, according to Norton, the exploration back through the interconnected sets of scientific knowledge and back through the history of this knowledge towards the origins of belief. It should be clear that one doesn’t have to move far back into biological (and geological) history when tracing these ideas before we are dealing with the beliefs and perceptions of creatures vastly unlike modern man, and yet eerily familiar in their neuro-anatomy and now we have moved quickly out of the realm of philosophy.
The justification of facts, Norton points out, will also be based on some inductive processes, which in turn will be based on more facts; and so we launch into a multidirectional regress. Rather than appearing as petitio principii, Norton argues that this is a demonstration of the “inductive solidity of science”; the notion that science is a nested and interlocking set of facts and inductive processes with give to each other strength and integrity, like stone blocks in an arch.
4. The Category–Property Theory of Induction
In presenting this (new) theory of induction, the author does not mean to imply that he believes a complete refutation of Hume’s argument will follow. However, it is perceived that the theory presented will go some way in furthering the discussion.
C-P Theory begins with the entirely reasonable assumption that categorisation is the basis of cognition. The potentially infinite amount of sensory information confronting a conscious being needs to be filtered if it is to be processed and a decision to act made as a result. This is the reason for the emphasis of ‘act as if’ over and above ‘believe’ in the opening paragraph of section 3: the driving force is the need to act toward a specific aim. It is beneficial for the success of an organism if the system of categories that are inferred from observations of the world are congruent with reality. Induction, under this view, is the formalisation of the characteristic processes of all conscious human persons. That is, they use neurological structures to map their environment (including its components), themselves, and the relation between the two.
Take for example the inductive argument that every emerald observed has been green. Therefore the next emerald to be observed will also be green. Say that among your prior observations you observed an emerald that was blue. Would you acknowledge this object entry into the category ‘emerald’. I would argue that you would not, as ‘green’ is a part of the pervading definition of emerald. It would be classified as something else, or the category of emerald would need expansion to include these new blue emeralds. It is the claim of the C-P Theory that induction is actually a taxonomic process which forms the substrate of conscious thought. Rationality then is built on top of induction, as those who have attempted to dissolve the Problem have argued.
4.1 Formulating the Theory
The C-P Theory can be formulated as follows. All As are Bs is equivalent to saying that all objects categorised as A have the properties of objects categorised as B. This is equivalent to saying that there is sufficient overlap between property sets such that they can be considered to be within the same category. Or, that all things categorised as A have a particular property bi which is a property of Bs.
More formally, the definition of a category X is the set of properties {x1, x2, … , xn}. If, when observing an object, any one of the properties is not observed, the object cannot be admitted to membership of the category X without modification to the category. The inductive conclusion usually formulated all As are B, is then actually a definitional statement that all As have the property (of being) B, irrespective of whether B itself is a category of objects or a property. If the proposition is all Xs are xi, then any observation that does not have the property xi is categorically not-X.
Let us apply C-P Theory to the classic example of swans. All past swans have been white; therefore the next swan observed will also be white (and similarly, all swans are white). Under C-P Theory, ‘white’ is one property in the property set that defines the category ‘swan’.
The category of swan (S) describes objects with the property set {s1, s2, … , ‘white’, … , sn}. There is, therefore, at this point, by definition, no such thing as a black swan. The inductive inference that ‘All swans are white’ is true by definition. It is a taxonomic question.
Now suppose you observe an object with the property set {s1, s2, … , ‘black’, … , sn}. You then have to make a decision. Either, you create a new category ‘black swan’ (Sblack), or you alter the property set that defines swan (S) such that this new object may be admitted. Now given that there is sufficient overlap in the property sets defining S and Sblack, the original category S is renamed ‘white swan’ (Swhite) and the property set defining ‘swan’ (S) is modified to { s1, s2, … , (‘white V ‘black’), … , sn}. That is, the property ‘white’ is replaced by the property ‘white OR black’.
Note that C-P Theory handles the Grue paradox. One common misconception because of the time component in the definition of the word grue, is that after time t, all things which are green prior to time t will change to being blue thereafter. This is not the case. The grue paradox is resolved, I claim, by the understanding of an expansion in the property set of the category in question as outlined above. That is, the category ‘emerald’ (E), originally defined by the property set {e1, e2, … , green, … , en}, is subsumed by the category ‘Grue emerald’, which in turn is defined by the property set {e1, e2, … , grue, … , en}, where grue is understood to contain a time component. At most, the grue paradox demonstrates that induction is context and language (read category) dependent, an assumption of C-P Theory.
The corresponding grue-like term with regard to the case of swans is ‘whack’ (White-black): that is, white up to time t, then white or black thereafter.
4.2 The C-P Theory and Karl Popper
Let’s say that your category ‘swan’ has thus far stood the test of repeated observations. In the Popperian sense, it is the accepted hypothesis. It has not yet been falsified but is indeed falsifiable. We have to consider what would be a falsification of the definition of swan.
One such falsification could take the form of an observation of an object that has all of the properties (and only the properties) of an object that would regularly be granted membership in the class ‘swan’ except that it does not have the property of being white. It is instead black. The question then arises, has induction failed? Well, no, not if we adjust the category of ‘swan’ to include things that are swans in every way except for the fact that they are black instead of white; or if we define a new category to encompass these new observations. Either way, induction is the process by which categories are continually realigned with reality as it is observed.
‘Scientific induction’ is the formalisation of the process by which we explore and learn about the world systematically. In this sense, even Popper cannot escape induction, for all (falsifiable) hypotheses are themselves the conclusions of some inductive process, which are tested by observation. They are the result of someone making observations about the world (or the current state of scientific literature) and inferring some generalisation (in fact they are forming a category) which they then set about testing.
5. Conclusion
Is induction rationally justifiable? The C-P Theory suggests that the question is unfounded. The question is better formulated: “Does induction lead to rational behaviour?” Even when induction ‘fails’ to make a correct prediction, it has succeeded in its purpose. That purpose being to bring into line the mental categories one uses to navigate the world, with reality itself. If induction provided no negative feedback, it would surely lead to irrational action, as it would allow categories to become misaligned with reality. In that sense, induction is not in itself rational, but nor is it irrational. Induction provides the cognitive basis for rationality (expressed as action in line with categories of objects, that are themselves in line with reality).
References
Barseghyan, H. (2015). HPS100 Lecture 02: Absolute Knowledge. Retrieved from https://www.youtube.com/watch?v=Ni0foAtFois
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