The Logic of Scientific Discovery

Some Conventionalist Objections

Those opposed to Popper’s use of falsifiability adhere to a process of conventionalism. Conventionalism requires a suspension of disbelief and suggests that certain aspects of the world—such as the simplistic laws of nature—are cheapened by the imposition of falsifiability. Popper views conventionalism as an attempt to escape the application of logic. Although he admires how conventionalism has drawn a distinction between theory and experiment, he rejects the idea that it is a form of science. Conventionalists dismiss evidence that may falsify their claims. Popper argues that the mark of a true scientist is the embrace of new discoveries that may lay waste to the original idea. Falsification leads to new arguments and, therefore, a more refined understanding of the world.

Methodological Rules

Demarcation cannot be applied to systems of statements but only to strict universal statements. One cannot look at a system of statements and determine whether they are empiricist or conventionalist. Popper identifies a singular way of avoiding conventionalist practices: by making a distinct decision to avoid applying those practices. Popper reminds the reader of the four strategies outlined in the previous chapter that are to be used in conventionalist thinking that scientists should avoid and reject if they detect them in the works of others. Auxiliary hypotheses should be constructed to increase the possibility of falsifiability. Hypotheses should be built upon the desire to construct a new system that would advance scientific understanding.

Logical Investigation of Falsifiability

Having set aside the possibility of conventionalism, Popper turns to the task of characterizing falsifiable systems. In this section, the philosopher operates with the assumption that basic statements—the building blocks of strict universal statements—can be falsifiable. He attempts to construct a definition of “empirical” and its relationship to single statements. Empirical theories are not categorized as empirical because they are derived from singular statements nor because they use a series of singular statements to create a foundation. Empirical theories require a falsifiable statement.

Falsifiability and Falsification

In this section, Popper draws a distinction between falsifiability and falsification. Falsifiability represents the conditional quality for statements to be categorized as empirical. Falsification requires specific rules. The basic statements of a theory must contradict one another, and this contradiction must be reproduced. This corroborates the falsification.

Occurrences and Events

Popper dissects the relationship between basic statements and theories to present his ideas in what he describes as a “‘realistic’ mode of speech” (88). He replaces “occurrences” with “basic statements.” Therefore, an experiment may rule out certain theories through possible occurrences. While many philosophers feel that words like “occurrence” or “event” are too slippery to be defined concretely, Popper feels they do justice to the concepts they describe. When two singular statements appear to confirm the same occurrence, this occurrence represents a singular statement. “Event” then represents universal ideas.

Falsifiability and Consistency

One of the most important factors for scientific inquiry into a theory is consistency. Every theoretical system should be driven by consistency.

Perceptual Experiences as Empirical Basis: Psychologism

While empirical science is reliant upon experience, inductive logic renders it ineffective. Empiricism must be paired with deductive reasoning. To avoid dogmatism, scientists must avoid the trappings of “immediate knowledge” that are prevalent in psychologism (94). This means that scientists can gain immediate understandings of truth through sense-experience. Popper argues that humans can never know any scientific statement to be true with infallibility; sense-experience is always based upon a limited point of view.

Concerning the So-Called “Protocol Sentences”

Psychologism has its own version of scientific statements Popper categorizes as “protocol sentences.” These are statements of experience. Popper suggests that basic scientific statements can only be tested under the scrutiny of other statements. He argues that a set of rules must be established to determine whether a statement is discarded or utilized for measurement.

The Objectivity of the Empirical Basis

Popper opens this section by claiming that he will distinguish between his school of scientific thought and that of psychologism. The distinction will be made between objectivity and the personal experience that drives psychologism. While he concedes that knowledge is gained through observation and experience, he rejects any notion that this experience verifies that knowledge. The history of epistemology was historically about theories of the mind; they were untestable via experience because they dealt only with thought.

Empirical science, however, is perceived to be rooted in experience and perception. Popper suggests that it requires the marriage of experience and logical reasoning. However, logic must be broken into smaller pieces and rigorously tested. This method requires critics to formulate tests of their own rather than reject the scientific statement based upon their own feelings. This testing is applied to basic statements, or “singular existential statements” (102). To achieve falsifiability, these statements must have a concrete and observable element. Popper argues that psychologism can never have a wholly observable element because observation requires a materialistic effect.

The Relativity of Basic Statements. Resolution of Fries’ Trilemma

Every scientific test should lead to one of two conclusions: corroboration or falsification. If one of these goals is not achieved, then the test was not effective. If scientists are unable to reach an agreement about the conclusion, then more testing is required. Popper addresses what he calls “Fries’ Trilemma,” a thought experiment that seeks to illustrate which of these three approaches can come closest to achieving a form of truth: dogmatism, infinite regress, and psychologism. Fries determined that psychologism came closest, but Popper argues that one can eliminate the trilemma altogether by taking important elements for each approach.

Theory and Experiment

Those who utilize inductive logic begin their experiments by collecting experience and sense impressions. Deductive science requires the establishment of a theory—a basic statement that can be observed and measured. Without this theory, scientists have no idea what to observe and record. The best theories are based upon their degree of falsification. Rather than choosing the theory that is the simplest, Popper proposes using the one that can be the most rigorously tested.

A Programme and an Illustration

Popper compares the basic statements that form testable theories to the radius of a circle. The circle represents the collection of basic statements of a single system, and the various radii represent the many observable events that can be falsified. He argues that these events must be “incompatible with the theory and ruled out by it” (112). The width of the radius is indicative of its degree of falsifiability. Despite the sheer volume of potential falsifiable statements, a section of the circle must always remain as unfalsifiable. There is always more to learn and no way to determine anything with complete verification. Popper calls this section of the circle “forbidden events.”

How are the Classes of Potential Falsifiers to be Compared?

An infinite number of falsifiable classes exist. To determine which classes of falsifiers are more or less falsifiable, Popper offers three ways of thinking about the concepts of “more” or “less” as it applies to falsifiability. First, the cardinality, or the power, of a class cannot be used to determine the degree because all theories have equal weight. No theory has more power than another. Second, the concept of dimension can be used to determine degree because it measures the number of relations among basic statements. Third, the subclass relation requires that each class have a subclass that is equal in all ways. Therefore, the subclass of one directly matches the other. However, this is not a perfect means to determine degree because the falsifiers of the two classes may intersect.

Degrees of Falsifiability Compared by Means of the Subclass Relation

Popper provides several equations to define and determine the dimension of a theory. For example, if two statements have identical classes, then they will have the same level of falsifiability. If the two statements do not have a relational value in which one statement has a subclass of the other, then the falsifiability of the two statements cannot be compared.

The Structure of the Subclass Relation. Logical Probability

In this section, Popper provides an illustration in Figure I of what subclass relations look like if charted on the circle. The points where the classes intersect are the testable relations that provide the opportunity of falsifiability. This applies only to singular statements, not universal statements.

Empirical Content, Entailment, and Degrees of Falsifiability

The more a basic statement can be falsified, the more information it provides about the world. These classes of statements have two elements: empirical content and logical content. Popper argues that the statements should have equal empirical and logical contents. If there is an imbalance, such as if one statement has greater empirical content than the other, then the amount of the other content—logical—must increase.

Levels of Universality and Degrees of Precision

If a scientist focuses solely on empirical content, then the need for universality and precision becomes greater. Popper provides an example of how universal statements can move to basic statements. The universal statement reads as a natural law: “All orbits of heavenly bodies are circles” (122). This statement becomes more basic as it becomes more refined: “All orbits of planets are circles” (122). Although the first statement has a higher degree of falsifiability, it is more difficult to test. The second statement, if tested and falsified, inherently falsifies the universal statement. If the orbit of a planet is shown to not be a circle, then the first statement is not corroborated.

Logical Ranges. Notes on the Theory of Measurement

The technique a scientist uses to apply measurement further determines how precise an experiment is. Popper’s most important measurement for the effectiveness of a statement is its degree of testability.

Degrees of Testability Compared by Reference to Dimensions

While the methods discussed in this chapter so far are highly effective, they do not always work for every purpose. In the example of conservation of energy—which is a principle with a high degree of universality—reliance on subclasses may not lead to the necessary parameters needed for the experiment. In these instances, scientists may need to look at the dimension of the theory, referring to the number of statements which comprise the field of application. Scientists must determine which method or whether both methods are needed.

The Dimension of a Set of Curves

If one thinks of a field of application as a set of statements within a theory, one can map the theory using a circle. Each point within the circle represents a statement. Therefore, the dimension is equal to the set of curves.

Two Ways of Reducing the Number of Dimensions of a Set of Curves

In this section, Popper provides a graph detailing the various dimensional classes and the points through which the circle should pass. Reducing the dimensions of the scientific approach increases the opportunity for falsifiability.

Elimination of the Aesthetic and the Pragmatic Concepts of Simplicity

Popper rejects traditional understandings of simplicity for their impreciseness and openness to interpretation. He argues that simplicity has no correlation with how an idea is presented or explained. It also is not related to the methodology; just because the performance of an experiment is easier, this does not correspond with Popper’s definition of simplicity.

The Methodological Problem of Simplicity

Now that these definitions have been stripped away, Popper seeks to clarify what is left. He offers that simplicity should “provide a measure of the degree of law-likeness or regularity of events” (138). Positivists assume that a theory that is simple and corroborated represents a law. This fails to acknowledge the infinite number of testable points of any field of application. Although Popper outlines what simplicity is not, he suggests that it is of little importance to provide a clear definition of what it is. Instead, he hopes to respond to the philosophical problem of simplicity.

Simplicity and Degree of Falsifiability

Theories with lower dimension are easier to falsify than those with a high dimension. The universality of the theory increases as it becomes more exacting and has a greater degree of falsifiability. The more universal a statement it is, the simpler it is. These simple statements provide a greater understanding of the natural world.

Geometrical Shape and Functional Form

Although a shape—such as that of a curve—is not a simple concept, the law that represents that shape through a function can be viewed as simple. Popper uses the sine function as an example of this simplicity of function versus complexity of shape. The testability of a geometrical statement requires a physical object upon which the statement can be examined.

Conventionalism and the Concept of Simplicity

Popper warns against utilizing a definition of simplicity that is embraced by conventionalists. Conventionalists do not emphasize falsifiability; therefore, the simplest statements affirm their beliefs.