In this handout, we will discuss the reduction of theories and / or sciences. That is, given two theories and , we will discuss what it means to reduce to and then examine two different reductions: (i) the reduction of logic to psychology and (ii) the reduction of mathematics (arithmetic) to logic (set theory).
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In The Structure of Science, Ernst Nagel defines theory reduction as follows
a reduction is effected when the experimental laws of the secondary science (and if it has an adequate theory, its theory as well) are shown to be the logical consequences of the theoretical assumptions (inclusive of the coordinate definitions) of the primary science (p.352).
The basic idea is that a reduction of a theory to occurs when the key propositions, theorems, sentences, or laws of are derivable (logical consequences) of the laws of . If this is the case, then the basic truths of are seen to depend upon the more basic principles of .
To understand this definition of theory reduction, we need to define a theory and a reduction more precisely.
In this section, a basic definition of a theory is provided.
Definition 1 (theory) A theory is a set of propositions , some of these sentences being axioms, about a subject matter.
So, my theory of the world might include some propositions about the types of things that exist and how they interact.
Definition 2 (axioms) An axiom is a basic proposition in a theory.
In calling the axioms of a theory “basic”, what is meant is that the proposition (the axiom) is a foundational or non-derived part of the theory. These are the core propositions upon which all other propositions of the theory depend. The status of these axioms varies from theory to theory, some are said to be known by intuition, others are generalizations from phenomena, some are said to be self-evident.
Propositions of the theory that are logical consequences of the axioms of the theory are the theorems of the theory. These are propositions that are derived from (logical consequences of) the axioms of the theory.
Definition 3 (theorems) Propositions that are the logical consequences of the axioms of the theory.
Example 1 (a theory about the world) I might assert the following axioms: (A1) God exists, (A2) the world exists, and (A3) God is all-knowing, all-powerful, and all-loving. Two theorem of this theory are that (T1) there is no evil in the world and (T2) anything that appears to be evil is only an illusion.
Finally, a theory often involves various definitions and inference rules. The definitions define various terms while the inference rules specify the rules used for reasoning from axioms to theorems.
With the notion of a theory in place, we can define certain conditions for a reduction.
First, in order to reduce one “theory” to (or one science to another science), it is necessary that both and meet our definition of “theory” put forward above. This requires that the science or theory in question has a certain structure. Namely, that it is axiomatized and that other truths of the theory (its theorems) are deducible from the axioms. Thus, if our theory is an unorganized body of propositions, containing no axioms, then the conditions for reduction are not present.
Second, a theory is reducible to if and only if the following two conditions are met:
Given the above, Figure 1 can construct the following method for reducing to
The idea here is that anything that can prove (or explain) is provable (or explainable) using only .
Some reductions are unproblematic in nature. For example, suppose two different theories, one theory for terrestrial mechanics (movement of objects on earth) and one theory for celestial mechanics (movement of the planets). Supposing both theories are axiomatized and both theories involve a number of theorems, we can imagine how both theories might be reduced to a more general physical theory that explains how objects move in general. Let’s call this more general theory . The theory would
One of the benefits of such a reduction is it brings various sciences and theories into unity. That is, rather than having a number of isolated sciences, when one theory is reduced to another, the result is that the reduced theory is often made a species of the more general science. Thus, celestial and terrestrial mechanics would be a species of the more general physical theory.
However, not all reductions are unproblematic. In what follows, we consider two problematic reductions.
The reduction of logic to psychology is sometimes referred to as psychologism in logic. In this section, we briefly outline how one might try to reduce logic to psychology.
To undertake this reduction, we first need to define “logic”
Definition 4 (logic) Logic is a science that investigates various systems of representation with the primary goal to explicate the nature of logical consequence (truth-preserving inference) and identify valid propositions (tautological formulas).
With respect to logic,
Example 2 (Logic as objective) Consider the following argument: All men are mortal. Socrates is a man. Therefore, Socrates is mortal. Whether the conclusion is a logical consequence of the premises is determined by the structure (form, the terms, etc.) of the argument and not whether individuals are inclined to believe it follows from the premises.
With this general idea of logic in mind, we can now define psychologism
Definition 5 (psychologism) Psychologism, in general, is the theory that the statements of one science / theory (call this the target science are reducible to / or merely abbreviations for statements about psychology (the mind, ideas, human beliefs, our cognitive architecture).
As a general theory, psychologism can be applied to a variety of sciences or theories, but we will look at the view that all logical propositions are reducible to psychological propositions. Using our model of reduction, a proponent of psychologism would reduce logic to psychology in the following manner:
We won’t illustrate each of the steps involved in this reduction. Instead, we will simply show how certain logical axioms (tautologies, valid wffs) might be said to have their basis in basic principles having to do with the human mind (see Table 1).
Law of excluded middle: P or not P,
Generalization of the psychological fact that some mental states exclude other mental states, namely that certain positive beliefs (a belief that ) involves an exclusion of another belief (exclusion of a belief that )
Principle of non-contradiction: P or not P,
Belief and disbelief are two different mental states that exclude each other
According to psychologism, the validity (or truth) of logical laws is said to be based upon the validity of highly general psychological laws that govern the human mind. Logic then is not a science that is distinct from psychology, but is, instead, at most a branch of psychology dealing with laws of the mind and inference.
Another attempted reduction is the reduction of mathematics to logic (set theory). This is the theory of logicism.
Definition 6 (logicism) Logicism is the philosophical view that mathematical propositions are reducible to (an abbreviation of) logical propositions.1
Logicism makes two principal claims:
In order for a successful reduction of arithmetic to logic, we would need the following:
First, with respect to the axiomatization of arithmetic, the axioms of arithmetic are as follows (from Russell, Introduction to Mathematical Philosophy:
Definition 7 (axioms of arithmetic)
In addition, we need logical axioms.
Definition 8 (Three logical axioms (set theory))
Remember that logicism contends that these arithmetical axioms are reducible to logical axioms. That is, the logicist claims that the above arithmetical axioms are derivable from logical axioms. Before we can do this, however, another step is required.
Second, we need a way of translating the key terms. This process of translation involves definitions of terms from one language to another language. These definitions are often referred to as bridge translations. In the case of arithmetic, we need to translate terms like “zero”, “successor”, and “number” into logical terms.
We can do this as follows:
Note 1 It is OK if the definitions above means nothing to you. We are simply walking through the general idea of how the reduction is performed.
Finally, the last step in the reduction is to show how each of the arithmetical axioms is derivable from the logical axioms. I won’t provide a proof of this but basically you go through each of the arithmetical axioms and prove them one by one using only the logical axioms and the bridge definitions.