The logic of scientific discovery, Part 3

And back to hanging out with Karl...

Chapter 3, Theories

This chapter heads the second section of the book, where Popper builds up the structural components of his Logic of Science. Here he discusses what a scientific theory is and how it connects to events in the knowable world.

12. Causality, Explanation, and the Deduction of Predictions

Popper defines both scientific theories and the events that they depend on as "statements" without privileging events as more concrete. His theories are universal statements, applying to all possible situations. His events are singular statements, applying to specific objects, times and places. Causality is then explained as a combination of one or more universal statements with one or more singular statements (the initial conditions) that lead to a final condition (the effect).

Popper doesn't assert that every effect has a cause, i.e. that it can be deductively explained (the principle of causality). Depending on the interpretation of the word "can" this is either an empty tautological statement (if we try hard enough we can always find a combination of initial conditions and theories that lead to the observed outcome), or an empty metaphysical one (an unprovable and unfalsifiable assumption about the nature of reality). Either way we don't gain much. Instead Popper again creates a methodological rule to deal with this along the lines that the pursuit of science can never give up the attempt to causally explain the events that we observe (this is essentially "the game never ends").

13. Strict and Numerical Universality

Popper takes a brief detour to distinguish between strictly universal statements - "such and such is true at all places and all times", and numerically universal statements which are only contingently true, e.g. "at the present time there are no persons taller than three metres alive on the Earth". The latter can be verified, in principle, by enumerating all examples and so can be replaced by a finite number of singular statements. The former however are universal assertions about an unlimited number of cases, what Popper calls an all-statement.

We take scientific laws to be comprised of these strictly universal all-statements, again as a methodological principle.

14. Universal Concepts and Individual Concepts

Here Popper closely parallels the previous section in terms of "concepts" rather than "statements". It is obvious from the footnotes that this is part of a lively dialog with other authors relevant at the time but now seems to be largely a semantic exercise. Individual concepts are those localised in some sense that can be defined by proper names or co-ordinates in time and space ("Napoleon", "the Earth", or "the Atlantic"). Universal concepts are those which have no limits to the applicability of their definition ("dictator", "planet", or "H2O"). There are some borderline cases that may be defined one way or the other such "Pasteurization": "the sterilization process invented by Pasteur" or "heated to 80 degrees C and kept there for ten minutes" but this is just use of a single label for two different things. Popper's main point is that a clear distinction exists and statements can be classified one way or another.

15. Strictly Universal and Existential Statements

The previous two sections have built up to this key argument. Popper distinguishes two types of statement using only universal concepts - strictly universal statements and existential statements.

A existential statement has the form of "such and such exists", e.g. "there are non-black ravens".

A strictly universal statement has the form of "such and such does not exist", e.g. "there are no non-black ravens".

These two types of statement are symmetric in the sense that the negation of a strictly universal statement is an existential statement, and vice-versa. Importantly, existential statements are verifiable, while strictly universal statements are falsifiable.

In section 6 we saw that according to Popper's demarcation criteria science is made up of falsifiable statements. Therefore the laws of science are phrased in terms of strictly universal statements prohibiting certain occurrences or states of affairs. While we can observe individual singular statements that can agree with existential statements we don't have the luxury of the infinite series of experiments that it would take to verify them so they don't have status as scientific theories.

16. Theoretical Systems

Popper comments on axiomised systems - this is a mature position for a branch of science where enough work has been done to codify the set of universal statements that determine its behaviour. If this can be done it prevents new assumptions sneaking in without explicitly falsifying some number of the axiom statements. Popper lists the properties these axioms should have in preparation for later sections about falsifiability: consistent with each other; independent of each other; and both sufficient and necessary for deducing all the statements contained within the field.

17. Some Possibilities of interpreting a System of Axioms

What are the nature of axioms though, are they merely well corroborated strictly universal statements? That is, are they scientific hypotheses like any other, or do they have some special status?

Popper takes several pages to basically say that they must simply be hypotheses, expressed as statements like any other. If they are not falsifiable: because they are assumed to be true as "self-evident"; because they are conventions that themselves define whether terms that they introduce are 'allowable'; or because they contain a mixed definition of strictly universal concepts and existential elements, then they lose their empirical basis and no longer have a part to play in a scientific logic.

18. Levels of Universality. The Modus Tollens

The Modus Tollens can be summarised as "If A then B. Not B, therefore not A". Popper applies this construction to structures of hypotheses in a branch of science.

Seeing that our axioms are hypotheses Popper discusses the consequences of deriving secondary hypotheses from primary hypotheses as a kind of tree semi-lattice structure. In principle falsification of these secondary hypotheses cascades upwards, falsifying higher level hypotheses, with the exception of independent hypotheses on another branch of the semi-lattice. In practice it is a bit more complicated than that, which is examined in later chapters.

Chapter 3 Summary

In this chapter Popper has assembled a great deal of structure. He's characterised the elements that make up scientific theories and those that make up the evidence base that they rest upon. Causality has been slotted in as the combination of theories and evidence, though not exactly covered in depth.

We've restated a previous methodological assertions and picked up another one: scientific laws are universal. This seems appropriate - a discovery that scientific laws were different in different places or under different conditions would touch off a frenzy of investigation aimed at discovering the hidden variables that controlled the change. The working assumption would be that we were missing some higher order theory to explain the difference in behaviour - exactly Popper's point.

Scientific theories are firmly pinned to the mechanism of falsifiability - it's only a small exaggeration to say that the two words are synonymous in Popper's work. There has also been a brief look at some of the mechanics of falsifiability of our theories, and we still have the open question of how to tie our theories to the real world. Further exploration of these two questions occupies the next three chapters of the book.

All in all this is a good gutsy chapter that builds many basic components of Popper's swamp hut.