In 1879 the German mathematician Gottlob Frege published a little book with the title Begriffsschrift (Concept Notation). Although it was largely ignored at the time (mathematicians found it too philosophical, philosophers too mathematical) it has subsequently been held to be one of the most important works on logic in over 2 000 years. I will look at some of the central themes of Frege, from the Bergriffsschrift and his other writings of the 1880s and 1890s, and ideas that have been developed from them.
The first idea is that of a truth-function. This is best explained by analogy with a mathematical function. If we have a function such as
we can see that the value of this numerical function is determined in a systematic way by the numerical values that are put in the x place: if the value of x is 2 then the value of the function is 9; if x is 12 then the value is 19.
This can be extended to propositions. If we have any proposition we can form a new proposition by prefixing ‘It is not the case that’ to the proposition. For example, if we have the proposition ‘Snow is white’ we can form the new proposition
It is not the case that snow is white.
In this example the original proposition was true and its negation is false. If we take a false proposition, ‘Coal is white’, its negation
It is not the case that coal is white
will be true.
What is interesting here is that when forming the negation of a proposition we are not concerned with the proposition’s meaning but only whether it is true or false: the negation of a true proposition is false and the negation of a false proposition is true.
Like the mathematical case, the value is determined In a systematic way, except that we are here dealing with functions of truth-values (true and false) rather than numerical values.
We can abbreviate this information concerning negation in a truth-table:
if we let ‘p’ stand for any proposition, ‘-‘ for ‘It is not the case that’, and ‘T’ and ‘F’ for ‘true’ and ‘false’ we have
which is a short way of giving all of the facts about negation.
There are several other truth functions. One is conjunction. If we have two propositions p and q, then their conjunction p and q will be true only when both p and q are true, and false otherwise. If we abbreviate ‘and’ by ‘.’
we have the truth-table
p q p . q
T T T
T F F
F T F
F F F
This lists all of the possible combinations of truth and falsity of the two propositions and gives the overall truth-value in the column under the ‘.’
Other truth-functions include ‘or’, ‘if ../ . then’ and ‘if, and only if’. Any textbook on modern logic will give these truth-tables and show how they can be developed into a technique for determining the validity of arguments concerning propositions.
The idea of a function can be extended to propositions themselves. For example, if we have a proposition
Frege had a beard
we can see that this consists of a referring expression ‘Frege’ and a predicate ‘had a beard’. The proposition is true if the object referred to by ‘Frege’ (the person Frege) has the property of being bearded; the proposition is false if the referent of ‘Frege’ does not have a beard.
Again we can think in terms of functions: if we have a propositional-function such as
x had a beard
this will be true for some values of x (Frege, DH Lawrence) and false for others (Bertrand Russell, Virginia Woolf). Note, however, that we only have truth or falsity when some value is supplied for x: ‘x had a beard’ has no truth-value as it stands.
We are now ready for an account of generality We will use the abbreviations
‘~x’ for ‘For all x’, and
‘~x’ for ‘There is at least one x’
so we can write:
‘Every-one is mortal’ as ~x, x is mortal
‘someone is mortal’ as ~x, x is mortal.
and ‘no-one Is mortal’ as -~x, x is mortal.
We can now write the four forms of the categorical propositions (See ‘Introduction to Logic Part One’) as
(A): ~x, If x is S then x is P
(E): ~x, if x is S then x is not P
(I): ~x, x is F and x is G
(0): ~x, x is F and x is not G.
Read these out in English and convince yourself that they do correspond to the categorical propositions.
So how is Frege such an advance on traditional logic? The reason is that Frege can go much further than the categorical propositions and the syllogism. Traditional logic – and traditional grammar – would read the proposition
Brutus killed Caesar
as being of subject / predicate form, with Brutus as the subject. It would also read the same proposition (in passive form)
Caesar was killed by Brutus
as having Caesar as the subject. Frege’s logic, on the other hand, reads it as a relation
x killed y
which is true for some values of x and y (Brutus and Caesar in that order) and false for others (Bertrand Russell and DH Lawrence in any order.)
Frege’s notation also extends to relations with any number of terms, for example
York is between Edinburgh and London
is a three termed relation
x is between y and z
which is true for the values York, Edinburgh and London, in that order (as well as for Northampton, Oxford and Cambridge) and false for other values.
Because Frege’s notation allows the expression of relations it can express the argument
All circles are plane figures
Therefore, whoever draws a circle draws a plane figure
an argument that resists explanation in terms of traditional logic.
A final achievement of the Begriffsschrift that will be mentioned is that Frege gives a complete deductive system for logic. Anyone who has studied geometry from Euclid will be aware that The Elements starts from various definitions concerning points, lines and planes; continues with axioms -statements giving ‘obvious truths’ about points, lines and circles; and then proves theorems about circles, triangles and squares – giving at each stage of the proof the definitions, axioms or previous theorems used. Hence, if we accept the axioms and definitions at the start, The Elements provides a rigorous proof of all of the statements of geometry.
In the same way, Frege intends the Begriffsschrift to provide a rigorous formalisation and proof of all mathematical theorems – he did, after all, start out as a mathematician. More generally, he intended his system to apply to any scientific language. So his achievement is that he was able to formalise a large part of arguments in a natural language such as English (or German) and rigorously prove the validity of those arguments.
The Law of Non-contradiction
The Law of Non-contradiction is sometimes stated in the form ‘Nothing can both be and not be’ but can for our purposes be stated as ‘A proposition cannot be both true and false’. In the above notation, this can be written as – (p . -p), with the usual mathematical convention concerning brackets.
This can be demonstrated to be valid as follows:
|p||-p||p . -p||– (p . -p)|
Column (1) contains the possible truth-values for the proposition p, T and F; from the truth-table for ‘-‘ we have column (2); from columns (1) and (2) and lines 2 and 3 of the truth-table for ‘ . ‘ we have column (3); column (4) comes from column (3) and the truth-table for ‘-‘.
Any truth-table with only Ts in its final column is called a tautology. Showing that something is a tautology, as the above shows, demonstrates that it is valid.