Russell: introduction to mathematical philosophy:
incompatibility and the theory of deduction
by ‘incompatibility’ Russell means that if one proposition is true the other is false
this is obviously a form of inference
it is the incompatibility of truth values
he notes that it is common to regard ‘implication’ as the primitive fundamental relation that must hold between p and q if we are to infer the truth of q from the truth of p – but says for technical reasons this is not the best primitive idea to choose
before coming to a view on the primitive idea behind inference he considers various functions of propositions
in this connection he mentions five: negation, disjunction, conjunction, incompatibility and implication
first he puts forward negation – ‘~p’
this is the function of p which is true when p is false and false when p is true
the truth of a proposition or its falsehood is its truth value
next he considers disjunction – ‘p or q’
this is a function whose truth value is true when p is true and when q is true – false when both p and q are false
conjunction – ‘p and q’ – its value is true when both propositions are true – otherwise it is false
incompatibility – i.e. when p and q are not both true – this is the negation of conjunction
it is also the disjunction of the negations of p and q i.e. ~p or ~q
its truth value is true when p is false and when q is false – it is false when p and q are true
implication i.e. ‘p implies q’ or ‘if p then q’ – that is we can infer the truth of q if we know the truth of p
all five have this in common – their truth value depends upon that of the propositions which are their arguments
a function that has this property is a truth function
he says it is clear that the above five truth functions are not independent – that we can define some in terms of others
Russell chooses incompatibility as the indefinable
incompatibility is denoted by p/q
the next step is to define negation as the incompatibility of a proposition with itself – i.e. ~p is defined as p/p
he then goes on to define disjunction implication and disjunction in this manner
but the first step needs to be looked at carefully
that is negation as p/p
now Russell has put the idea that negation is the incompatibility of a proposition with itself
clearly what this presumes is that incompatibility is a relation –
and clearly this is so
the point though is that a relation here holds between propositions – it is propositions that are incompatible
and this is what is put forward in connection with disjunction implication and conjunction
for clearly disjunction implication and conjunction – are relations between propositions
so the idea of incompatibility on the face of it can be applied to these relations – just because they are relations
but negation?
negation is not a relation between propositions
and more to the point – negation is not a relation
secondly propositions have relations with other propositions
that is the only way in which a relation can exist – between propositions
a proposition does not have a relation with itself
it is not possible for a proposition to ‘have a relation’ with ‘itself’
for there is no ‘itself’ to a proposition
a proposition does not have a self – that it can relate to
God knows what the ‘self’ of a proposition is supposed to be
this idea of a proposition having a relation with itself is just nonsense
a proposition in the broadest sense of the term is a proposal
and as to proposal – in the broadest sense of the term again – it is an action
to negate a proposition is to deny it
that it is to say ‘it is not the case that p’
it is to determine the proposition negatively
if you begin in an argument with p
and then assert ~p
the assertion of p is one action
the assertion of the negation of p is another
yes these two propositions can be related
but the second one – the negated proposition
does not have a relation with itself
it is in every sense a separate proposition
the assertion of a proposition and the negation of a proposition are two different logical acts
the upshot of this that Russel’s theory of incompatibility collapses
incompatibility cannot be applied in the manner he wishes to apply it
and for this reason his account of incompatibility as the primitive idea of inference cannot go forward
Russell says of incompatibility it will be denoted by p/q
negation is p/p – disjunction is the incompatibility of ~p and ~q i.e. (p/p) / (q/q)
implication is the incompatibility of p and ~q i.e. p / (q/q)
and conjunction the negation of incompatibility i.e. (p/q) / (p/q)
so in all but conjunction propositions are rendered incompatible with themselves
and in the case of conjunction what you effectively have is the incompatibility of incompatibility
i.e. – incompatibility is incompatible with itself
this rendering of the various types of inference in terms of incompatibility makes the notion of inference incomprehensible
it brings inference to a dead halt
why incompatibility?
Russell italicizes ‘truth’ in his statement ‘…..it seems natural to take “implication’ as the primitive fundamental relation, since this is the relation that must hold between p and q, if we are able to infer the truth of q from the truth of p.’
now he rejects implication as the primitive
is this because he thinks that implication only applies when the truth value is true?
that is he rejects it on the grounds that it does not apply when the value is false?
it does seems clear that he regards implication proper as only applying in the case of where the issue is truth
and yet at the same time he calls for the ‘widest sense’ of the term.
now the problem with this view is that it ties implication – it ties inference - to truth value
it says only given these truth conditions does this inference occur – or can occur
this to my mind confuses and conflates truth conditions and inference
or to put it another way an inference is a logical act – that is made or can be made regardless of the truth conditions of the propositions involved
and so I would put that we can use implication just as well when the subject is falsity as when it is veracity
there is not a problem with if p is false q is true or if q is true p is false
the general point is that inference – the logical act of inference – is independent of the question of truth value
Russell’s mistake with implication was to limit it to inferences where the only value is truth
to account for falsity in implication he came up with incompatibility
now as I have argued the idea of a proposition being incompatible with itself makes no sense
and furthermore it is not necessary to entertain this concept if truth value is not tied to inference
this is not to say the two cannot be formally related – for this is the issue of validity or invalidity
there is also a more general point to be made about Russell’s incompatibility thesis –
the idea is to find a primitive truth function in terms of which the other truth functions can be derived
the fact of it is though that incompatibility is not on the same logical level as conjunction disjunction and implication
it is clearly a derived truth function
the use of negation in its formulation indicates it is a secondary construction
now straight up – a secondary construction by definition will not serve as a primitive
that is it will always be shown to be reducible – and for that reason fail as a primitive
the question then - is implication the primitive that Russell was seeking?
now that we have removed incompatibility from the equation does implication do the job?
that is can we translate conjunction – disjunction and incompatibility into implication?
we can indeed –
conjunction - if p and q are true the inference is true – if either p or q is false the inference is false
disjunction – if p or q is true then the inference is true – but if p or q are false – the inference is false
incompatibility – if p or q is false the inference is true – if p and q are true – it is false
and the great advantage of the form of implication is just that it really does make clear the separation of inference and truth value
that is it quite literally leaves the question of truth value up in the air
and there is a real intellectual honesty built into implication – the issue of truth and falsity is in the inference left undecided
that is we can make the inference without necessarily knowing the values
it is beautiful in the sense that we can infer without hesitation in a state of uncertainty
in fact the state of uncertainty becomes and is the ground of inference
in logical terms this action demands that it is performed without prejudice
on this view – what is primitive to inference is uncertainty
that is once you make the move to implication as the general form of inference – uncertainty is revealed as the ground of inference
this I think injects health into logic – puts life into it
certainty is a corpse
however it must be remarked that such a view is at odds with standard or given view of deductive inference
Russell says:
‘In order to be able validly to infer the truth of a proposition, we must know that some other proposition is true, and that there is a between the two a relation of the sort called “implication”, i.e. that (we say) the premise “implies” the conclusion.’
my argument is that in implication the truth values of the propositions are conditional and are conditional in relation to each other
and the real point of this is that in implication per se nothing is decided in terms of truth value
when we imply we are effectively leaving open the question of truth
Russell’s argument above is that to infer the truth of a proposition we must know that some other proposition is true
but this I think is wrong
it is not that we must know – it is rather if p is true – then q is true
here the truth of p is an open question
now if you accept the view that deductive inference is implication – and that all forms of deductive inference can be seen as instances of implication
then deductive inference does not depend at all on the truth value of the propositions
rather it only depends on the possibility of truth value
now on such a view of deductive inference – it would seem that validity is never at issue
or to put it another way a conditional argument is neither valid or invalid
what I am getting at in general is that what logic does is not provide us with knowledge – what is does is spell out the conditions for knowledge
and the basis of conditional arguments is uncertainty
© greg.t. charlton. 2008.
