Russell: introduction to mathematical philosophy:
mathematics and logic
according to Russell mathematics and logic are one –
logic is the youth of mathematics and mathematics the manhood of logic
after a survey of all that has come before in his book Russell asks the question
‘What is this subject, which may be called indifferently either mathematics or logic? Is there any way we can define it?’
to begin Russell says
in this subject we do not deal with particular things or properties –
we deal formally with what can be said to be any thing or any property
logic does not deal with individuals – because they are not relevant or formal
in the syllogism the actual truth of the premises is irrelevant – all that is important is that the premises imply the conclusion
a syllogism is valid in terms of its form – not in virtue of the particular terms occurring in it
and we are therefore faced with the question – what are the constituents of a logical proposition?
if we take a relation between two terms we may represent the general form of such propositions as xRy – i.e. x has the relation R to y
in the assertion ‘xRy is sometimes true’ i.e. there are cases where dual relations hold – there is no mention of particular things or relations
we are left with pure forms as the only possible constituents of logical propositions
the form of a proposition is that which remains unchanged when every constituent of the proposition is replaced by another
logic is concerned only with forms – and stating that they are always or sometimes true
in the proposition ‘Socrates is human’ – the word ‘is’ is not a constituent of the proposition – but merely indicates the subject predicate form
in the proposition ‘Socrates is earlier than Aristotle’ ‘is’ and ‘than ‘ merely indicate form
however form can be the concern of a general proposition even when no symbol or word in that proposition designates form
Russell argues we can arrive at a language in which every formal belonged to syntax and not vocabulary
in such a language we could express all the propositions of mathematics even though we did not know one word of the language
we should have symbols for variables such as ‘x’ an ‘R’ and ‘y’ arranged in various ways – and the way of arrangement would indicate something was being said of all or some of the values of the variables
there are symbols with constant formal meanings – these are ‘logical constants’
‘logical constants’ will always be derivable from each other - by term for term substitution
and that which is in common is ‘form’
all constants that occur in pure mathematics are logical constants
logical propositions are those that can be known a priori – that is without study of the actual world
logical propositions have the characteristic of being tautologous – as well as being expressed in terms of variables and constants
this gives us the definition of logic and pure mathematics
Russell says he does not know how to define tautology
and in a note to this matter says –
‘The importance of “tautology” for a definition of mathematics was pointed out to me by my former pupil Ludwig Wittgenstein, who was working on the problem. I do not know whether he has solved it, or whether he is alive or dead.’
yes – one gets the impression here that Russell was not all that keen on the tautology
or at the time of his writing the above all that keen on the student who introduced the ‘importance’ of it to him
for he doesn’t even bother to give a definition of tautology
we are left wondering – perhaps indeed it is just a bad smell
also one would imagine a query here a word there could have settled the question of whether Wittgenstein was alive or dead –
perhaps though the first world war was the reason for placing Wittgenstein in a disjunction
my view on all this is -
all propositions are actions – we can say propositional actions
their basis is necessity – practical necessity
that is we propose descriptions of the world – so as to be able to operate effectively in the world –
we can therefore say any proposal is a proposal for order
we need order so as to operate effectively
this is a premise for any propositional behaviour
logic is a description of the possibilities of propositional order
that is logic displays the order inherent in propositional behaviour
so my first point is that logic is a descriptive action
the propositions of logic describe what is possible with propositions – that is how they can be ordered – how they can be related
i.e. a proposition can be put – and its opposite can be put
the second proposition is negation of the first - the relation here is negation
two propositions can be conjoined – and in such a case their relation is conjunction
propositions can be disjoined – and their relation is disjunction
implication is a relation where one proposition is said to imply another
describing the relations between propositions (negation conjunction disjunction implication) tells us not only how propositions are used – but also if the question arises – how they can be used - in relation to each other
in this sense logic is the study of propositional relations
and it is an account or description of propositional behaviour that applies to any propositional usage – mathematical or empirical
mathematics is primarily concerned with calculation
that is to say it is a particular or specialized propositional usage –
this is not to suggest that mathematics is in any way limited –
for it is clear that any kind of experience can be made the subject of calculation
logic though is not an activity of calculation – even though there is calculation in logic
it is a description of the possibilities of propositional behaviour – one form of which is mathematical action
and it is in that sense a description or a proposal about what actually occurs
for this reason it makes no sense to speak of it as being a priori
logic as a descriptive activity only exists because propositional behaviour exists –
because that is how the world is in terms of human beings and their actions
the propositions of logic are descriptions of what occurs or can occur when people use propositions
Russell mentions the law of self-contradiction as a logical proposition – and somewhat reluctantly the tautology
‘it is raining and it is not raining’ is a self-contradiction – it is a proposition that contradicts itself – it is a logically false statement
which to my mind means quite simply it has no use
‘it is raining and it is raining’ – is tautologous – it is a proposition which takes the value true for all assignments of truth values to its atomic expressions
again it like the self-contradiction is a propositional form that has no utility – no use
now I make this point to raise the question whether it makes any sense to speak of ‘propositions of logic’
if as Wittgenstein argues and Russell comes along for the ride – the propositions of logic are all tautologous –
then as a set of propositions they are useless
but they are only useless in this sense because they are being treated in an artificial manner
they are being taken out of any context – even out of the world
and then the question is asked – well what is their significance or their meaning?
well the answer of course is that they have none – they’ve been placed in a void –
and the very point – theoretical point of a void is that it has no significance
this bizarre result is a consequence – firstly of regarding propositions as in some sense special entities – when in fact all they are is the expression of the human need to make known – which is I would suggest the most basic of human needs
and they are therefore actions in the unknown – actions of defiance if you like
now to describe these actions – the propositional actions – to get an idea of how they do and can work is just another propositional action designed to shine some light into the darkness
logical activity is just the same action as any other propositional action – it has no specials status
it is a descriptive activity
its subject is propositional behaviour
so it is a propositional account of propositional behaviour
its an ‘in house’ activity – or action within the action
it’s ground if you like is all propositional behaviour
and the ground of all propositional behaviour is simply the unknown
for it is the unknown that is the object of all propositional behaviour
through our propositions we make platforms for action
it is on such platforms that we get about the business of living
logic is simply a way of seeing how we do this
© greg. t. charlton. 2009.
