Theory building: from Peirce to Popper

the philosophy of science and epistemology. He, in turn, saw logic as a branch of semiotics, of which he is a founder. In 1886, he saw that logical operations could be carried out by electrical switching circuits, thus anticipating the digital computer […]

His approach concerning scientific reasoning is summarized hereafter:

- active process of theory generation, with no prior assurance of truth;

- subsequent application of the contingent theory, aimed toward developing its logical and practical consequences;

- evaluation of the provisional theory's utility for the anticipation of future experience, and that in dual senses of the word: prediction and control”.

In [Peirce, 1958], three processes are identified which operate in a cyclic fashion:

i) abduction is what we use to generate a likely hypothesis or an initial diagnosis in response to some questioning situation;

ii) deduction is used to clarify, to derive, and to explicate the relevant consequences of the selected hypothesis;

iii) induction is used to test the sum of the predictions against the sum of the data.

In Figure 8, we start from Peirce’s approach and further illustrate the distinction between

“logical” and “practical” consequences by separating “logical implication in the green world of linguistic expressions” from “experimentation in the orange world of causality”:

1. “likely hypotheses” are the fruit of some abduction; in the shape of some linguistic expressions;

2. we suppose that the corresponding language contains derivation rules allowing deduction to operate as a checkable derivation ; this leads to predicted results under the shape of new linguistic expressions;

3. in parallel, we experiment with observable facts (in the orange world, events are causality related; therefore resulting facts come as a causal consequence of initial facts);

4. induction upon predictions and observations leads to diagnostic and further abduction.

Figure 8: Peirce's approach of Theory building

Peirce’s viewpoint is mostly the viewpoint of one scientist having some ‘subject of enquiry’; i.e. that the truth or falsity of expressions has to be decided by a single mind by comparing predicted results with facts based on real events. The bottom of the picture illustrating the arbitration of the truth or falsity of expressions by induction/abduction within a single mind upon experiments in the real world is what [Shapiro, 1991]

formalizes as an ‘oracle’ (see [8.3]); taking inspiration from him, we may adopt the following definition of a ‘partial theory’:

partial theory (semi-formal definition):

Let ‘L’ be a formal language based on ‘expressions’ describing facts, hypotheses and predictions, where predictions are constrained by a set of rewriting rules. Let us suppose a local ‘oracle’ testing ‘expressions’ and answering true or false on the basis of individual understanding. A partial theory is a set of true expressions such that their rewriting produces only true expressions with respect to the local oracle.

When Peirce does not explicitly address the collaboration, Karl R. Popper will later introduce the scientific community and its controversies as a major component in theory building: as emphasized in [Popper, 1959], each time a likely hypothesis is emitted by one scientist, the others are in charge of attempting to falsify or to refute it. Successive cycles of hypotheses and refutations will eventually lead to a temporary consensus called

likely hypotheses predicted results

initial facts b

l u e

abduction

&

induction

l i n g u i s t i c e x p r e s s i o n s

deduction

r e a l w o r l d

causality i n d i v i d u a l

m i n d

A A II

II

II resulting facts

g r e e n

o r a n g e

truth. The scientific status theory is therefore directly related to its capacity to be ascertained or falsified through genuine tests.

Taking into account the collaboration, we may therefore adopt the following:

Theory (semi-formal definition):

Let ‘L’ be a formal language based on ‘expressions’ describing facts, hypotheses and predictions, where predictions are constrained by a set of rewriting rules. ‘L’ is built and shared by a group of people, each with their own understanding, who attempt to falsify partial theories emitted by the others and collectively constitute an ‘oracle’ deciding the truth or falsity of ‘expressions’. A theory is a set of true expressions such that their rewriting produces only true expressions with respect to the collective oracle.

The ‘collective oracle’ is an abstraction for a set of ‘local oracles’, each associated to an individual mind. According to [2.1.1] such local oracles are based on ‘selectionist thinking’; our purpose has therefore become to rule on a logical basis the interaction between selectionist thinking entities; it is interesting to take note at that point from a conjecture formulated by Edelman in the last and more philosophical part of [Edelman &

Tononi, 2000]:

“After Homo sapiens and higher-order consciousness appeared, it became possible to create syntactically rich symbol systems, to create codes, and even to create logic… We have taken it as established that after the brain arose in evolution by natural selection, which set up value constraints and major structures, each individual brain operates by process of somatic selection. Instead of being guided by effective procedures, it is governed by effective structures, the dynamics of which allow its correlated activities to arise by selection, rather than by the rules of logic … Clearly, if the brain evolves in such a fashion, and this evolution provided the basis for the eventual discovery and refinement of logical systems in human culture, then we may conclude that, in the generative sense, evolution is more powerful than logic… In any case, the interesting conjecture is that there appear to be only two deeply fundamental ways of patterning thought: selectionism and logic.”

Relying on Edelman’s conjecture, we may therefore rephrase [Lukasiewicz, 1970]³⁷ :

“reasoning which starts from theories and looks for facts is patterned by logical deduction and can be entrusted to machines, that which starts from facts and looks for theories is patterned by selectionist thinking.”

37 [Lukasiewicz, 1970] “reasoning which starts from reasons and looks for consequences is called deduction; that which starts from consequences and looks for reasons is called reduction”