Russell: introduction to mathematical philosophy:
propositional functions
Russell begins here with a definition of ‘proposition’-
he says ‘proposition’ should be limited to symbols – and such symbols as give expression to truth and falsehood –
much would depend here on the definition of symbol – and one’s basic idea of truth and falsehood
by symbol – could we not mean any descriptive act?
of course such would include the propositions of ordinary language – but would it not by definition include other artistic creations – poetic expression – and any act of visual art i.e. painting – sculpture architecture etc.– and perhaps even acts of gesture? etc.
so it depends how much you want to let into ‘symbol’ –
and ‘truth’ – to cut quickly to the chase I see it as assent – and falsehood – as dissent –
really just a jump to the left or a jump to the right
and of course acts of assent and dissent can take on any number of forms – any number of expressions
I favour the idea that a proposition is a proposal – of whatever kind or form
and in the most general sense it is a proposal ‘of a state of affairs’
now any observer of such a proposal can give their assent to the proposal – or can dissent from it
that is they can affirm it – or deny it
so a proposition is a proposal that can be affirmed or denied –
is capable of being affirmed or denied
in normal parlance – it would seem to be of the nature of a proposition (proposal) that it can be affirmed or denied in some manner of speaking
a visitor to an art gallery whose response to a work of abstract art is broadly speaking one of approval – has affirmed the proposal
the same proposition in the shape of abstract expressionism can be ‘denied’ by the very next observer
perhaps if analysed such a response would mean something like ‘I don’t agree with how the world is portrayed in this painting’
anyway
‘propositional function’ is defined by Russell as an expression containing one or more undetermined constituents – such that when the values are assigned – the expression becomes a proposition
it is a function –whose values are propositions
or as he also describes it – ‘a mere schema, a mere shell, an empty receptacle for meaning, not something already significant.’
an example –
‘x is human’ is a propositional function
as long as x remains undetermined it is a propositional function – it is neither true nor false
but when a value is assigned to x it becomes a true or false proposition
I like propositional functions – but I think for reasons quite different to Russell
the beauty of a propositional function in my terms is just that it is a function with undetermined values
‘undetermined values’ here means unknown values
and the point of the propositional function is that it shows that function is not dependent on determination – on knowing
which is to suggest that function is quite independent of knowledge
I think it is even possible that Russell might agree to this view of things – in a limited way
I think that the propositional function really points to the basis of logic in scepticism – and much as Russell was known for his sceptical frame of mind – I doubt that he would have ever conceived of such a notion
the propositional function is a proposal – in the absence of determination – of knowledge –
nevertheless a proposal
Russell wants to distinguish sharply between a propositional function and a proposition
and this is where the definition of – or one’s understanding of - the nature of proposition is relevant
if as I have put – a proposition is any proposal that can be asserted or denied – what then of a propositional function?
Russell as I noted distinguishes proposition and propositional function – in terms of truth function
the proposition can be regarded as true or false – but not the propositional function?
is that so?
that is in the example above ‘x is human’ – while x is left undetermined – as an unknown – a proposal is put –
and it is the proposal that there is something that can be described as human –
and it is a proposal that can be regarded as true or false -
now you might wonder how could it be rationally denied?
under what conditions could such a statement be false?
this matter only depends on one’s definition of ‘human’
i.e. it is conceivable for instance that in the future with developments in genetic engineering and or bio-technology that the classification ‘human’ could be regarded as obsolete
in such a circumstance it could well make sense to regard the statement ‘x is human’ as no straightforward matter – and quite possibly false - either in general or in relation to certain classifications of ‘species’
so in such a case even though x is undefined – ‘human’ is up for grabs
this is not perhaps the best example to take of propositional functions
a more interesting case is one Russell goes on to consider ‘all A is B’
Russell says ‘A and B’ have to be determined as definite classes before such expressions becomes true or false
but is that so?
‘all A is B’ is a proposal for identity
such a principle or a version of such is required for arithmetic – calculation depends on the assumption that the left and right hand sides of the ‘=’ sign are equivalent
however in other contexts it is not so straightforward -
can you i.e. apply it in philosophy of mind?
i.e. are all sensations brain processes?
so the question is really about the appropriate application of such a propositional function –
it is clear that in some contexts such a propositional function – does function – has value
in other contexts – its status is uncertain
the point is – it is a proposal for relating one class to another in a certain manner
even that ‘certain manner’ can be a question – that is the ‘is’ in ‘all A is B’ is not uncontroversial – it can have a number of meanings
the propositional function even though its values are indeterminate – is not a statement without meaning or significance
one needs to accept it as a proposition – for the determined propositions to follow
so it can be regarded as true or false
it is quite extraordinary that in the twentieth century – and I suspect even in this century – logicians have seriously put that the propositions of their activity are not subject to truth conditions –
the absurdity of it is quite staggering
do they seriously suggest that the propositions with truth value are derived from propositions with no value?
a more cynical view might be to suggest that they find security in not subjecting their own propositions to the question of truth value
perhaps it is just that logic – main stream logic has never got past Plato
anyway such a view of logic of propositional functions is deluded nonsense
my overall point is that the propositional function is a proposal – is a proposition
the issue is really all about function
in my view a propositional function – asserts function
and the proposition (in Russell’s terms) – is a function asserted – meaning the values are declared – the ‘variables’ determined
now as I have just argued – the propositional function does not exist in metaphysical empty space – its validity depends on its epistemological context
so it is true or false – but to see this you need to be able to look to its use – and the context of its use
both the propositional function and the functioning proposition are proposals
and in an even more general sense they are propositional acts
to understand an act you need to understand its context – or at least make start in that direction – get an idea of it
so finally in relation to propositional functions -
the variable in a propositional function is an unknown value
the fact of the propositional function shows us quite clearly that we can and do function with unknowns
that is the fact of the unknown value does not prohibit function
the function in a propositional function – is the act proposed – and the value of the act is unknown
it is on this foundation – the unknown – that all ‘determined’ propositions rest – it is their ground and source
if to be is to be the value of a variable
and the variable qua variable is unknown
to be is to be the value of the unknown
© greg. t. charlton. 2008.
