In this handout, we will discuss the reduction of theories and / or sciences. That is, given two theories ${T}_{1}$ and ${T}_{2}$, we will discuss what it means to reduce ${T}_{2}$ to ${T}_{1}$ and then examine two different reductions: (i) the reduction of logic to psychology and (ii) the reduction of mathematics (arithmetic) to logic (set theory).

To obtain this handout as a PDF, visit Reductionism in the philosophy of science (www.davidagler.com)

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 ${T}_{2}$ to ${T}_{1}$ occurs when the key propositions, theorems, sentences, or laws of ${T}_{2}$ are derivable (logical consequences) of the laws of ${T}_{1}$. If this is the case, then the basic truths of ${T}_{2}$ are seen to depend upon the more basic principles of ${T}_{1}$.

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 ${P}_{1},{P}_{2},\dots {P}_{n}$, 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 ${P}_{i},{P}_{i+1}\dots {P}_{i+n}$ 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”
${T}_{2}$ to
${T}_{1}$
(or one science to another science), it is necessary that both
${T}_{1}$ and
${T}_{2}$ 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 ${T}_{2}$
is reducible to ${T}_{1}$
if and only if the following two conditions are met:

- every term of ${T}_{2}$ is capable of being translated into a term in ${T}_{1}$
- every theorem of ${T}_{2}$ is a logical consequence of the axioms of ${T}_{1}$

Given the above, Figure 1 can construct the following method for reducing ${T}_{2}$ to ${T}_{1}$

The idea here is that anything that ${T}_{2}$ can prove (or explain) is provable (or explainable) using only ${T}_{1}$.

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 $T$. The theory $T$ would

- translate the terms of both terrestrial and celestial mechanics into the terms of $T$
- show how the various theorems of terrestrial and celestial mechanics are deducible from the axioms of $T$

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,

- Its object of study is generally not considered a subjective matter
concerning the ideas or the minds of human beings but an
**objective**matter concerning the real relations between terms, predicates, propositions, and formulas. - If a proposition $Z$ is a logical consequence from a set of propositions $A,B,C,\dots Y$, then $Z$ is said to follow from this set regardless of whether you or I or any finite mind believes it to follow.
- Logical consequence is said to be a fact and so (i) is independent of what people think about it and (ii) is discovered rather than created

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:

- Articulate the axioms of logic and the axioms of psychology
- Translate the key terms of logic into the language of psychology
- Derive the axioms of logic using the axioms of psychology.

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, $P\vee \neg P$ | Generalization of the psychological fact that some mental states exclude other mental states, namely that certain positive beliefs (a belief that $P$) involves an exclusion of another belief (exclusion of a belief that $not-P$) |

Principle of non-contradiction: P or not P, $P\vee \neg P$ | 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:

- mathematical objects are, on closer inspection, logical objects (the terms of
arithmetic can be expressed entirely in logical terms)
^{2} - mathematical axioms (and theorems) and their derivations from mathematical axioms are derivable from logical axioms

In order for a successful reduction of arithmetic to logic, we would need the following:

- axiomatization of arithmetic and logic
- a way of translating all of the terms (concepts) from arithmetic to logic
- a way of proving every arithmetical truth using the laws of logic

**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))

- The axiom schema of comprehension: informally, for every predicate (property)
$P$
there will exist a set $S$
whose elements (members) are exactly those elements that are $P$.
In simpler terms, when we talk about an object $x$
being
*red*, this talk is interchangeable with talking about $x$ being in the set of*red things*. - The axiom of extensionality: informally, if sets ${S}_{1}$ and ${S}_{2}$ have the same elements, then they are the same set.
- The axiom of infinity: informally, there is a set $S$ that contains the empty set and that for any element $x$ that is a element of $S$, the set ${S}_{i}$ that is constructed by taking the element $x$ and the set of $x$ (viz., $\left\{x\right\}$) is in the set $X$. This axiom guarantees at least one infinite set.

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:

**Zero:**$0=\varnothing $, Zero is the set containing a single element, the empty set.**Successor:**the successor of a set $S$ is the set ${S}_{i}$ that contains every set which contains an element $x$ such that if $x$ were removed from $S$, the result is a set that is an element of ${S}_{i}$.*Example:*the successor of 0 is the set that contains every set that has the following property: it is the set that contains an element such that when that element is removed, the remaining set is an element of 0. In other words, the successor of 0 is a set that contains every set which contains an element which, when removed from the set, results in the empty set. In other words, the successor of 0 is the set of all sets that contain one element. That is, $0=\varnothing ,1=\left\{\varnothing \right\},2=\left\{\varnothing ,\left\{\varnothing \right\}\right\}$, etc.**Natural Number:**a natural number is the collection of all sets with $n$ elements

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.

- Irvine, Andrew David and Deutsch, Harry, ”Russell’s Paradox”, The Stanford Encyclopedia of Philosophy (Winter 2016 Edition), Edward N. Zalta (ed.), URL = https://plato.stanford.edu/archives/win2016/entries/russell-paradox/.
- Speaks, Jeff. 2004. Russell’s reduction of mathematics to logic. URL = https://www3.nd.edu/~jspeaks/courses/mcgill/370/logicism.html
- Klement, Kevin, C. Russell’s Logicism. Forthcoming in The Bloomsbury Companion to Russell, ed. Russell Wahl.
- Musgrave, Alan. 1977. Logicism Revisited.
*The British Journal for the Philosophy of Science*, Vol. 28, No. 2 (Jun., 1977), pp.99-127. - Russell, Bertrand. 1920.
*Introduction to Mathematical Philosophy*London: George Allen & Unwin, Ltd. - Pelletier, Francis J., Elio, Renée, and Philip Hanson. 2008. Is Logic
All in Our Heads? From Naturalism to Psychologism.
*Studia Logica: An International Journal for Symbolic Logic*, Vol. 88, No. 1, pp. 3-66.