*“The mathematics we have to construct are the mathematics of the human intellect”* [1]

**George Boole (1815-1864)**

Aristotelian logic was considered to be comprehensive until the 19th Century. Kant even believed that nothing significant needed to be added to Aristotelian logic. However the 19th and 20th century witnessed a huge development in logic- a development in which Boole was one of the most notable pioneers. Though it had become evident that Aristotelian logic could be empowered, it seems that the modern development in logic was more of a continuation (and improvement) of Aristotle’s legacy rather than a total repudiation of his work.

Boole was an empiricist- not a surprise considering the English epistemological tradition. He believed that his algebra was not only a sound logical system, but also a rigorous way of reasoning that could be applied in other fields of knowledge. Boole was very much acquainted with (he knew both Latin and Greek) the literature of ancient philosophy, indeed, in his writings, he often gives credit to Aristotle for his work on logic*

*[There is an interesting discussion by Aristotle on the limits of logic: In the Posterior Analytic (Book 1, part 3), Aristotle talks about two schools critical of scientific knowledge: one claims that absolute certainty is impossible because if there are always new premises on top of the ones we know this leads to infinite regress, whereas if there is no infinite regress then some premises are non-demonstrable. The other school, instead, believes that all logic is circular reasoning, self-referential.]

Boole’s insight in logic is to use the power of algebra and to simplify reasoning with two states of affairs (1,0). He wants to express thoughts into a mathematical form. Needless to say Boolean algebra is applied today in a multitude of fields (with electrical engineering standing out).

**Mathematical Analysis of Logic (1847) [1]**

PREFACE

Boole makes clear that his work in logic is not just an intellectual curiosity, but also an attempt to have logic applied into practical matters.

INTRODUCTION

What renders Logic possible is the “existence in our minds of general notions”- the ability of our minds to conceive an abstract class which contains individual members. Hence the theory of logic is “intimately connected with that of language”.

Boole claims Aristotelian syllogistic logic lacks rigorous symbols as the ones of mathematics.

Alike geometry, logic needs to be founded upon axioms.

BODY OF THE BOOK

[this book anticipates the themes of ‘The Laws of thought’. Much of the book is about how to manipulate the symbols of Boolean algebra in accordance with the elementary rules of logic]

Universe represented by 1. X,Y,X are the members of the class 1. The product xy represents the class whose members are both Xs and Ys.

A proposition can either affirm or deny. A proposition has the three-fold structure *subject*–*copula* (is, is not)-*predicate*. The subject of the proposition may be Universal or Particular. The combinations are:

Universal affirmative: All Xs are Ys [xy = x]

Universal-negative: No Xs are Ys [xy = 0]

Particular affirmative: Some Xs are Ys [v=xy] with ‘v’ being a separate class

Particular negative: Some Xs are not Ys [v=x(1 – y)]

A proposition in logic can be converted from affirmative to negative and vice versa such as:

‘Every poet is a man of genius’ can become ‘He who is not a man of genius is not a poet’

**THE CALCULUS OF OF LOGIC (1848) **[3]

Boole believes that his logic can express the “operations of the mind in reasoning”. Logic is about relating different ‘classes’ together. According to Boole our intellect works according to a few basic laws through which we can develop new interpretations by deductive reasoning. Boole seems to suggest these laws are the law of identity, of non-contradiction, of equality and the basic rules of algebra like the commutative, distributive laws.

Boole believes that his logic of algebric operations represents ‘mental processes’.

“the connexion of two symbols xy as in multiplication, represents the mental operation of selecting from a class Y those members

which belong also to another class X, and so on.”

**The Laws of Thought (1854)** [2]

CH.1

-The aim is to give to logic a symbolic language. The goal of logic is “to deduce

correct inferences from given premises”; “translate our data into the rigorous language of symbols” (ch.4)

-Empirical inquiry is probabilistic since any proposition requires the ‘confirmation of experience’, while logical propositions are true independently of empirical tests.

-Boole claims that he will show why he think Aristotelian logic is incomplete.

-The language of logic should be mathematical ( a ‘Calculus’).

-Boole on the difficulty of finding trustworthy correlations in natural sciences:* ““Fickle as the wind,” is a proverbial expression”… “the seeds of general truths which lie buried amid the mass of figures”*

-Studying logic implies studying the nature of our mind as well (for instance is our mind abiding to certain laws like the natural phenomena?)

CH.2

-Boole begins with a reflection on linguistic. A sign is defined as ” an arbitrary mark, having a fixed interpretation”.

-If we need a language as an instrument of reasoning, then we need the following signs for the operations: 1) literal symbols to represent things as subjects or adjectives (x,y,z) [e.g let x be the class of ‘men’ and let y stand for ‘good’- by writing ‘xy’ we mean ‘good men’]; 2) signs of operations (+,-); 3) signs of identity (=) [e.g xy = yx].

Another example: let ‘x’ be ‘men’, let ‘y’ be ‘women’ and let ‘z’ be ‘British’. It follows that we can write the expression ‘British men and British women’ in the following algebraic form:

z(x + y)= zx + zy

If we want to express negation by meaning “All the Brits except men”, we could write this in the following way:

y – x = -x + y

-Boole wants to create the ‘Algebra of Logic’ by converting the components of our ordinary language (subjects, adjectives, adverbs) into algebric signs.

In modern terminology we could say:

**Empty set:** 0

**Negation:** 1 – x

**Law of contradiction:** x(1 − x) = 0

**Law of excluded middle:** x + (1 − x) = 1

**Logical ‘or’:** x + y = 0 (either one of the two elements is true)

**Logical ‘xor’:** x(1 − y) + y(1 − x)

**Intersection:** xy (both elements are true)

(Boole’s axiom is that a proposition cannot at the same time be true and false.)

CH.3

-Introduction of the concept of the ‘Universe of Discourse’- it is a limited field in which all of the possible subjects involved in an operation are present.

-Two basic classes: ‘Nothing’ (empty class) and ‘Universe’ (all the possible subjects of the operation). Boole also describes these two classes as the two ‘limits’ of algebric logic.

-‘0’ represents ‘Nothing’, ‘1’ represents ‘Universe’

So another way to express the previous equation (y – x = -x + y) is the following: let ‘1’ be the human beings ‘Universe’ and let ‘x’ be the ‘men’ class’. To get ‘non-men’ (women) as a result we can do a simple subtraction: 1 – x.

CH.4

Boole makes a distinction between primary and secondary propositions such as “the sun shines” (primary proposition) “the earth is warmed” (secondary proposition- because of the logical implication deriving from the first proposition). Primary proposition need to be independent, while secondary proposition are dependent on a proposition. Propositions can be linked together with connectives such as ‘if’, ‘and’, ‘or’, ‘either’…

-+ is used for ‘either’, ‘and’, ‘or’. When x and y are exclusive (or) this can be represented as** x(1 − y) + y(1 − x)**, when they are not exclusive (and) by **x + y(1 – x).**

– the ‘-‘ is used for ‘except’, ‘not’

-‘v’ used as indefinite class

-Primary propositions have subject and predicate. Such propositions can be expressed with the use of the equality symbol such as ‘all **fixed stars** (the subject, x) are **suns** (the predicate, y)’– x=y

An example of a mutually exclusive ‘or’. Let ‘x’ be ‘space bodies’, ‘y’ be ‘planets’ and let ‘z’ be ‘stars’

x= y (1 − z) + z (1 − y)

‘v’ can be used to express what we now call a ‘subset’.

y= vx (it is like saying y is a subset of x- e.g y stands for ‘men’ and x for ‘mortal beings’).

Another alternative for expressing the subset identity is to use an inequality:

x(1 − y) = 0

Boole makes us of the ‘predicate’, ‘copula’, ‘subject’ terminology (showing how much he is still linked to the Aristotelian tradition).

Some men are not wise (where ‘y’ stands for ‘men, ‘x’ for ‘wise’): vy = v (1 − x)

CH.5

Boole acknowledges that some may critic whether a logical interpretation of an algebric proposition can be extended in other areas as well *“It might be argued, that as the laws or axioms which govern the use of symbols are established upon an investigation of those cases only in which interpretation is possible, we have no right to extend their application to other cases in which interpretation is impossible or doubtful”*.

When using symbols, these are the requirements to have a logical language:

-symbols have a fixed interpretation

-the laws of combination of the symbols are clearly pre-determined

-the formal process that leads from one proposition to another follows the pre-determined laws.

-the final interpretation must be in accordance with the laws

Boole applies his algebra in functions as well.

[The next chapters are in dense in algebra and logical intuition, so the next chapters will tend to have briefer summaries. The methods of development, elimination, and

reduction are explained]

CH.6

0/0 is analysed as an indefinite class symbol- “[0/0] indicates that a perfectly indefinite portion of the class, i.e. some, none, or all of its members are to be taken”

CH.8

The 1/0 coefficient that accompanies a term expresses its inexistence just like 1/0 is ‘undefined’ in mathematical notation.

CH. 11

Boolean algebra can infer logical conclusion from a given set of premises. However Boole admits (by using Aristotelian ethics as an example) that this type of reasoning can only describe, not explain the premises. It can also describe the relationships between such premises.

Despite his skepticism towards ethical philosophy and metaphysics, Boole asserts that he believes in the existence of universal principles. It seems that his skpeticism stems from his belief that our opinions are dependent on our limited experiences and conditions. In addition Boole is aware of the epistemological problems concerning the law of causation.

Boole realises that notions of time like ‘sometimes’, ‘eternally’ should be incorporated into logic.

**1** can be used to represent the whole duration of time, and **x** the portion of that time in which proposition **X** is true (this can have useful applications with probability). For instance, **1 – x** denotes the portion of time in which proposition **X** is false.

**0** instead will represent no time whatsoever.

**xy** is the portion of time in which both X and Y are true.

**x(1 – y)** is the time when x is true, but y false;

**(1 – x)(1 – y)** is when X and Y are simultaneously false in a given time;

**x(1 – y) + (1 – x)** is when only one of the two (X,Y) is right, not both

**xy + (1 – x)(1 – y)** is when X and Y are either both true or both false.

**(1 − x)(1 − y)(1 − z)** is when xyz are simultaneously true.

Boole seems to agree with Plato and Aristotle that there are truths that are valid only in a portion of time (such as laws trying to describe the transitory phenomena of nature) and others that are ‘eternal’ (such as geometrical truths).

To express conditional propositions like “if the proposition Y is true, then the proposition X is strue”, the algebra is the following:

y=vx (where v is an indefinite portion of time, vx together means an indefinite portion of time in which x is included).

On the lack of precision of ordinary language, Boole describes the English language as ” that imperfect yet noble instrument of thought”.

CH.12

Boole uses his algebra to analyse an argument made in Plato’s republic on the nature of God.

CH.13

Boole uses his algebra to confute Spinoza and Clarke’s theories on God.

Boole concludes that it is futile to prove the existence of God (as an infinite being) with a priori reasoning.

CH.15

Boole applies his Algebra with the Aristotelian tradition of the syllogism.

CH.16

Boole provides a sharp description of the theory of probability: *“Probability*

*is expectation founded upon partial knowledge. A perfect acquaintance*

*with all the circumstances affecting the occurrence of an event would change*

*expectation into certainty, and leave neither room nor demand for a theory of*

*probabilities… The rules which we employ in life-assurance, and in the other statistical applications of the theory of probabilities, are altogether independent of the mental phænomena of expectation. They are founded upon the assumption that the future will bear a resemblance to the past; that under the same circumstances the same event will tend to recur with a definite numerical frequency; not upon any attempt to submit to calculation the strength of human hopes and fears”*

probability of the occurrence of an event: p

probability of the non-occurrence of an event: 1 – p

CH.17

Concurrence of x and y: xy

Occurrence of x without y: x(1 – y)

Occurrence of y without x: y(1 – x)

Conjoint failure of x and y (1 – x)(1 – y)

CH.20

*“So to apprehend in all particular instances the relation of cause and effect, as to connect the two extremes in thought according to the order in which they are connected in nature (for the modus operandi is, and must ever be, unknown to us), is the final object of science. “*

CH.22

On the induction problem: “the larger number of the generalizations of physical science possess but a probable or approximate truth.”

Boole on the metaphysics of the Alexandrian platonists: “That kindly influence of human affections, that homely intercourse with the common things of life, which form so large a part of the true, because intended, discipline of our nature, would be ill replaced by the contemplation even of the highest object of thought, viewed by **an excessive abstraction as something concerning which not a single intelligible proposition could either be affirmed or denied”**

Boole believes that one source of metaphysical speculation is the (wrong) belief that there is a correspondence between the forms of our thoughts and the actual constitution of Nature.

Boole is not surprised that mathematics is the language of science, since human thought can reveal itself in mathematical forms-that is making an inference logically.

Boolean algebra has been used by McCulloch and Pitts to produce a model of the ‘artificial neurons’ with the idea of using logical gates just like electrical engineering. Both authors were part of the cybernetics movement of the 1940s, 1950s. Both of the believed the nervous system could be described like a computational device.

In the model, the neurons can have a set of inputs (I 1, I 2, I 3….) and one output (y). W1, W2, W3… are weighted values that can take the following numbers (0,1) (-1, 1). The output, just like Boolean logic, can only be binary.

In the function (Sum) stands for the weighted sum.

SOURCES

[1] http://www.gutenberg.org/files/36884/36884-pdf.pdf?session_id=4ac94f265dd27f7ac5babc474ca49c551390f49b

[2] https://www.gutenberg.org/files/15114/15114-pdf.pdf

[3] http://www.maths.tcd.ie/pub/HistMath/People/Boole/CalcLogic/CalcLogic.pdf