Part II. On Logic and Mathematics. II. GENERALITY

5. The proposition “The circle is in the square” is in a certain sense independent of the assignment of a particular position. (In a certain sense it is totally unconnected.)

yes – ‘the circle is in the square – does not address the question of particular position – it leaves that question – open –

if you assign position – you have a different and separate proposal

this ‘open’ proposition –

is logically speaking no different to any other proposal – any other proposition –

a proposal – a proposition – is open to question – open to doubt –

the ‘openness’ – we are talking about here – is uncertainty –

a proposition is uncertain – a proposition expresses uncertainty

any claim to the contrary – any claim of certainty – is no more than pretence –

such claims are not logical – they are rhetorical

6. The proposition “This circle is in the square” is not a disjunction of cases.

‘the circle is in the square’ – is not a disjunctive statement

you could well put forward a disjunctive proposal regarding position –

but that proposal would be quite different to ‘the circle is in the square’

on the basis of this proposition ‘the circle is in the square’ – yes the circle has position – and further – the position of the circle is uncertain

the proposition is straightforward –

we are dealing fairly and squarely with uncertainty

and it is not ‘a darkness veiling possible position etc.’

it is uncertainty in the clear light of day – as clear as day

if you come from a philosophic tradition that regards uncertainty as the enemy –

understanding that uncertainty is the reality – hard and fast –

might require a big conceptual and psychological shift –

basically you need to divest yourself of pretence and rhetoric –

get to the clarity of logic

7. The inadequacy of the Frege-Russell notation for generality

Wittgenstein’s question is whether “('x)” has the applicability that is claimed for it – has the generality –

‘The proposition “there are only two things that are circles in this square” (construed on the model of the proposition “there are only two men who have climbed the mountain”) sounds crazy with good reason. That is to say nothing is gained by forcing the proposition “there are two circles in this square into that form; it only helps to conceal that we haven’t cleared up the grammar of the proposition.’

a proposition is a proposal – open to question – open to doubt – is uncertain –

this applies whatever form a proposition takes – be that – as in this case –

notation or ordinary language

and it is clear that we use different propositional forms for different purposes –

and no formulation used – for whatever purpose –

is beyond question

can any one form – i.e. logical notation or ordinary language – be used to apply in all circumstances?

I hate to bring all this crashing down to earth – but it is really just a question of use –

if a particular form is used – by a user in all the propositional contexts that he operates in – then you would be inclined to say – yes –

so generality as an application is really just a matter of the constant use of a propositional form

does it suit our purposes to use the one form constantly?

I would think – that as a matter of fact – the answer is – no –

and obviously so

8. Criticism of my former view of generality

‘My view about general propositions was that ($x).fx is a logical sum and that though

its terms aren’t enumerated here they are capable of being enumerated (from the dictionary and the grammar of language).

For if they can’t be enumerated we don’t have a logical sum. (A rule, perhaps, for the construction of logical sums).

Of course the explanation of ($x).fx as a logical sum and of (x). fx as a logical product is indefensible. It went with an incorrect notion of logical analysis in that I thought that some day the logical product for a particular ($x).fx would be found. –

Of course it is correct that ($x).fx behaves in some ways like a logical sum and (x). fx like a product; indeed for one use of the words “all” and ‘some’ my old explanation is correct. – for instance for “all the primary colours occur in this picture” or “all notes of the C major scale occur in this theme”. But for cases like “all men die before they are 200 years old” my explanation is not correct. …’

the logical sum interpretation of ($x).fx – as Wittgenstein himself notes has limited applicability

the point is that ‘all’ – is open – to interpretation – and one interpretation – or if you like – one use of ‘all’ – does not close off – or exclude – other interpretations and uses

if you think you can define a term a priori – you have it wrong

a word only has meaning in use – and just what that meaning is – will be open to question – to doubt – will be uncertain –

and it is this logical uncertainty – that gives rise to different uses – different interpretations – different meanings –

as Wittgenstein realised the logical sum interpretation of ‘all’ – in not all there is to ‘all’ –

and if you are looking for an account of ‘all’ that covers all possibilities – you misunderstand language –

there is no complete analysis of any term –

any term – and any analysis – is ‘open’ – that is the point – open – to question – to doubt

language is uncertain

‘The generality notation of our ordinary language grasps the logical form even more superficially than I earlier believed. In this respect it is comparable with the subject-predicate form.’?

logical form – here – is a theory of language structure and use –

(by the way – open to question – open to doubt – uncertain)

so the real question is –

is it ordinary language that is wanting – relative to such a theory –

a theory expressed in symbolic logic

or the theory – that is inadequate relative to ordinary language?

and the answer is yes

the reality is that any form – will be found to be inadequate relative to another form –

if you believe one form is authoritative – and the other lacking in authority

and it seems that it is logical form that gets the authoritative guernsey here from Wittgenstein

we operate with different propositional forms in different contexts

and the only real ‘authority’ – is use

this is the reality that has to be grasped – if we are to have an understanding of language –

an understanding of our world

symbolic logic is a stripped down language-game –

and an intriguing one at that

the language of logical notation can serve – does serve – certain purposes –

but the idea that it can or does or should function as a bedrock to all propositional action is at the best – fanciful – and at the worst – rhetorical rubbish – based I would suggest on outmoded authoritarian epistemology

‘Generality is as ambiguous as the subject-predicate form’?

it is not that generality is ambiguous – it is uncertain

‘There are many different “alls” as there are “ones” ‘?

and just what ‘all or ‘one’ – amounts to is – logically speaking – never clear cut –

these terms are – like any other proposal – open to question –

and there is no point at which – logically speaking – the questioning stops

‘So it is no use using the word “all” for clarification unless we know its grammar in this particular case.’?

even when a grammar – a context – a use – is proposed –

questions can still be asked – doubts can always be raised –

so any so called ‘knowledge’ here – will be uncertain

the reality on the ground – is just that we do use the word ‘all’ – without clarification

and I would suggest the reason for that is not ignorance or incompetence –

but rather a native understanding – that there will be no definitive clarification –

that the whole point of ‘all’ – is to leave the matter – open –

open to question

9. The explanation of generality by examples

‘The mental process of understanding is of no interest to us (any more than the mental process of an intuition).

“Still there’s no doubt that someone who understands the examples as arbitrary cases chosen to illustrate the concept doesn’t understand the same as a man who regards them as a definitely bounded enumeration.” Quite right, but what does the first man understand that the second doesn’t? Well, in the things he is shown he sees examples to illustrate certain features; he doesn’t think I am showing him the things for their own sake as well. –

I would like to call the one class “logically bounded” and the others “logically unbounded”.’

‘the mental process of understanding is of no interest to us’?

let’s be clear here – ‘mental process’ – is an account of – an ‘explanation’ – of understanding –

it is a description – a proposal – for explaining ‘understanding’ –

and as with any other proposition put – as an explanation of understanding – it is open to question – open to doubt – it is uncertain

if such an account ‘is of no interest to us’ – so be it –

this does not mean however –

that the argument of such a proposal is not worth considering –

that such a proposal can be of no use in some context –

we approach the unknown with different conceptual schemes – different philosophies

different world view

and no one approach – is beyond question – beyond doubt –

in reality in practise – it is just a question – an empirical question – of what approach is used – where and by whom

what is of interest to us – logically speaking is just what is proposed – whatever that is

and how that proposition works – how it is proposed that it works

with any ‘explanatory’ proposition – we are invited to see the world in a certain way –

this is where we begin – to question – doubt – to deal with the uncertainty – that is at the base of any proposal – of any proposition

and dealing with proposals – dealing with uncertainty is argument

and yes different proposals – different propositions – are the currency of understanding

and different proposals – different descriptions – whether made public or not – will be different understandings

and yes you can classify these understandings – these proposals – as ‘logically bounded’ and ‘logically unbounded’ –

this is just another kind of description of the different proposals –

examples – ‘bounded’ or ‘unbounded’ – are proposals – open to question – open to doubt –

windows into uncertainty

‘Yes, but is it really true that he sees only these features in the things? In a leaf, say, does he see only what is common to all leaves? That would be as if he saw everything else as blank like an uncompleted form with the essential features ready printed. (But the function “f( …)” is just such a form .)

well indeed you could speculate in this way –

once it is understood that the ground of any proposal is uncertainty – then it is clear that speculation – inventive and fascinating as it may be – is all we have

‘But what sort of process is it when someone shows me several different things as examples of a concept to get me to see what is common to them, and when I look for it and then actually see it? He may draw my attention to what is common – But by doing this does he make me see the object differently? Perhaps so; for surely I may take a special look at one of the parts, when otherwise I would have seen the whole with equal clarity. But this seeing is not the understanding of the concept. For what we see isn’t something with an empty argument place’

examples of a concept?

first up a ‘concept’ of – is an explanation of – what? – a word usage –

and then – explaining – the concept – via examples?

the giving of examples is a means of explaining the concept

so the idea is that the concept should capture – or indicate – possible usage

when you strip this down – what it amounts to is a proposal – the proposing of examples – of usage

what I see – ‘when someone shows me several different things’ – is a proposal

how I ‘see’ – interpret the proposal – is logically speaking – an open question

‘But seeing is not the understanding of the concept’ –

understanding of – whatever – is whatever I propose – as understanding –

and any such proposal – will be open to question – open to doubt –

understanding – is uncertain

‘For what we see isn’t something with an empty argument place’

when we propose – we propose the argument place

and with each new proposal – a new argument place

the argument and the argument place are one in the same

language – and therefore the world – is a constant argument

‘So it is the rules governing the example that make it an example’ –

and as for ‘rules’ – what you are talking about here is accepted or proposed propositional practice – in some context –

a rule – once you divest it of pretence – is just a proposal – open to question –

the logical reality is that with or without the ‘rules’ – we are dealing with and in uncertainty at every turn –

‘… and when I hear the word “plant” it isn’t that there comes before my mind a picture I then describe as a plant. No I make the application as it were spontaneously.’

either proposal – the picture – or the ‘spontaneous application’ – is an explanation of what happens

in the absence of any explanation – what happens is – unknown –

the action of proposing – is the action of knowing

any ‘knowledge’ we operate with is uncertain – open to question – to revision – to doubt

when I put forward a different proposal to what you propose – or a different ‘explanation’

can I be sure that I know what it is you mean – and can you be sure that you know what I am saying?

no – the matter is uncertain

when I operate with a similar – or even identical proposal or explanation –– to the one you put – can I be sure that I know what you are proposing – and that you know what I am proposing?

no – the matter is uncertain

yet it is just this uncertainty that is the traffic of our communication –

that I might decide that I know what you are on about – is at best a pragmatic decision

it is a decision to proceed – in order to proceed –

and to proceed in uncertainty

‘The only thing of interest to us is the exact relationship between the example and the behaviour that accords with it”

the relationship between an example and the behaviour that accords with it –

is a matter of perception in the first place – and as for any statement regarding the relationship – any such proposal will be open to question – open to  doubt – will be uncertain

and if so – what sense can you make of ‘exactness’?

surely any claim of ‘exactness’ can only be rhetorical

‘The example is a point of departure for further calculation’ –

the ‘example’ is an invitation – an invitation to propositional inquiry –

it is a devise of usage – a focus for – the adventure that is –

uncertainty

‘There is one thing I always want to say to clarify the distinction between instances that are offered as examples for a concept and instances that make up a definite closed group in the grammar … F(a, b, c, d, e) is the disjunction of all the cases we have actually used, but there is also other cases (we won’t of course mention any) that make true the general proposition “F(a, b, c, d, …)”. And here of course we can’t put the general proposition in place of F(a, b, c, d, e).’

F(a, b, c, d, e) is a disjunction – and therefore – closed –

the disjunction – is a bounded statement – a bounded generality

“F(a, b, c, d, …)” – is an unbounded statement – an unbounded generality

what we are talking about here is restricted generality – and unrestricted generality –

here are two ways in which the ‘concept’ generality is defined – is used

’So this is how it is: “bring me a flower” can never be replaced by an order of the form “bring me a or b or c”, but must always be “bring me a or b or c or some other flower”?

But why does the general sentence behave so indeterminately when every case which actually occurs is something I could have described in advance?

But even that seems to me not to get to the heart of the matter; because what matters I believe, isn’t really the infinity of the possibilities, but a kind of indeterminacy. Indeed if I were asked how many possibilities a circle in the visual field has of being within a particular square, I could neither name a finite number, nor say that there are infinitely many (as in a Euclidean plane). Here, although we don’t ever come to an end, the series isn’t endless in the way in which ½1, x, x + 1 | is.

Rather , no end to which we come is really the end; that is, I could always say: I don’t understand why these should be all the possibilities. – And doesn’t that just mean that it is senseless to speak of “all the possibilities”? So enumeration doesn’t touch the concepts “plant” and “egg” at all.’

so this is how it is – yes a disjunctive statement is quite different – to an unrestricted general statement

‘when every case is something I could have described in advance’?

any proposal for ‘every case’ – will be open to question – to doubt – will be uncertain

one shouldn’t get ahead of oneself – in logic or in life

‘what matters is a kind of indeterminacy’ –

yes – and at the heart of indeterminacy – is uncertainty

‘no end to which we come is really the end’

yes – any proposal is open to question – open to doubt – is uncertain

‘ends’ – are not logical – they are pragmatic points – for moving on

‘senseless to speak of all possibilities’?

it is never senseless to leave a matter open – it is to be logical

‘So enumeration doesn’t touch the concepts “plant” and “egg” at all.’?

well it does touch them – perhaps only just – but touch them it does

the point is any explication of a term – is open to question – to doubt – is uncertain

logically speaking there is no complete analysis of any term –

there is only what occurs –

by way of explication – by way of proposal –

we are fooled by syntax – into thinking that words – are definite –

language – in whatever presentation – is argument –

and it is as ongoing – as you want it to be – need it to be – are prepared for it to be –

we are in the practise of life – limited –

limited by time by space and our very human concerns –

logic – not so

‘I would like to say: in grammar nothing is supplementary, no stipulations come after others, everything is there simultaneously.’

grammar here is a gaming of language – a structuring of language

there are obviously good reasons for this

it enables functionality at a basic level – on a common platform

for ‘grammar’ to have functionality – to enable functionality – however –  there must be adherence to its rules – by language users –

that is the game must have players – and players who play according to rules –

what this comes down to is the pragmatics of language

a proficient game player – plays well according to the rules

a player who doesn’t play proficiently – let us say haphazardly – relative to the rules –

(and this would be most of us – most of the time – I would suggest –)

still uses language –  and who’s to say – ineffectively?

yes the well constructed game – leaves nothing out – really if it did – it would be a failure as a game

whether in fact language actually operates in accordance with such a model – is quite another question

how language is used – what grammar – or grammars people operate with – is open to question – to doubt – is in fact – uncertain

the point is – you can look at language use – and develop a theory from what you observe – all to the good –

really all you have with language – is action – bare and bold – action that works –

the why and the how – are actually irrelevant – to the brute fact of language use

all you can say of a language users who don’t operate with your grammar – is that they don’t –

and perhaps the consequence is that you don’t get what they are on about – that you don’t understand them –

perhaps they say the same of you?

we are indeed used to everyone communicating with each other – or at least using language at each other –

be a big job to find out if everyone is following the rules – and who’s rules they are

and if they weren’t being followed what would be the difference?

you are still faced with the logical reality that whatever is proposed –

is open to question to doubt – is uncertain –

‘What is said about enumeration of individual cases cannot ever be a roundabout explanation of generality.’

enumeration might be a way of defining generality – of dealing with generality –

is that how it is used – as an illustration of generality in a particular context?

and look if generality is to have some use for us – mustn’t it be made usable?

otherwise you could say – no better case for Oakum’s razor

and really isn’t it the case that generality is always ‘cut down’ – placed in a domain –

given definition – given context – i.e. ‘all plants’ – is not  meant to be a reference to ‘all trees’?

my point is I suppose – we ‘explain’ generality – just by how we use the concept –

that is the reality

the concept – really has no significance – outside of use – it’s not in fact there

and how the ‘concept’ – is used – as with any other concept – is open to question – to doubt – is always – on the ground – uncertain

finally –

you can play this conceptual game – with all its complexity – i.e. rules – but as presented here by Wittgenstein – it is essentially a priori –

the question is whether this explanation of usage – is useful?

if in some context – this conceptual – a priori  proposal bears fruit – then I suppose the answer will be – yes

my point really is that any ‘explanation’ of usage – is usage –

so logically speaking we can drop this notion of ‘explanation’ – as a superfluity –

and as to how ‘examples’ work – that too is a question of context – of usage

we have proposals – and proposals in relation to proposals – that’s the story – that’s language use

as for ‘generality’ – it is a classificatory proposal – that becomes redundant – once you realise that any proposal is – logically speaking – open – ‘unbounded’ –

in practise we may pretend ‘boundedness’ – even ‘definitiveness’ –

this is behaviour – not logic – pragmatic yes – but not logical

a general word – a general proposition – is an invitation to question – to doubt – to uncertainty –

an invitation to the propositional life

10. The law of a series “And so on”

‘The expression “and so on” is nothing but the expression “and so on” (nothing, that is but a sign in a calculus which can’t do more than have meaning via the rules that hold it; which can’t say more than it shows).

That is, the expression “and so on” does not harbour a secret power by which the series is continued without being continued.’

‘The expression “and so on” is nothing but the expression “and so on”’?

yes – and no –

the expression is never used – without definition –  without context – i.e. ‘a sign in the calculus’ –

and yes – in any language game in which it is used – it will have definition in terms of the rules of that game

a sign – any sign – only has significance – in terms of some explication – some proposal for meaning

‘which can’t say more than it shows’?

once an interpretation has been adopted – then yes that is what it shows –

this is a pragmatic point

logically speaking – a sign – any sign – is open to question – open to doubt –

is uncertain

‘That is, the expression “and so on” does not harbour a secret power by which the series is continued without being continued.’

this strikes me as obvious –

however that is not really the point here –

how the expression ‘and so on’ – is interpreted – how it is used –

is logically speaking – an open matter –

‘Of course it doesn’t contain that, you will say, but still it contains the meaning of infinite continuation.’

well perhaps – perhaps not – 

it depends who’s using it and in what context

again it is a question of use – and ‘use’ – is language use –

open to question – open to doubt – uncertain

‘But we might ask: how does it happen that someone who now applies the general rule to a further number is still following this rule? How does it happen that no further rule was necessary to allow him to apply the general rule to this case in spite of the fact that this case was not mentioned in the general rule?

And so we are puzzled that we can’t bridge over the abyss between individual numbers and the general proposition.

“Can one imagine an empty space?” (Surprisingly, this is where this question belongs.)

It is one of the most deep rooted mistakes of philosophy to see possibility as a shadow of reality.

But on the other hand it can’t be an error; not even if one calls the proposition such a shadow.’

‘in spite of the fact that this case was not mentioned in the general rule’?

it really does depend on how you interpret the general rule –

if you regard it as open –  what doesn’t it include?

‘possibility as a shadow of reality’? –

this is poetry –

the logical reality is – uncertainty –

and what that means is that how we interpret any proposal – any proposition – even within an operating interpretation – is open to question – open to doubt –

‘possibility’ – without the poetics –

is propositional uncertainty

‘What troubles me is that the “and so on” apparently has to occur in the rules for the sign “and so on.” For instance, 1 + 1 and so on.  =  . 1 + 1 + 1 and so on,  and so on.”

as I interpret it – ‘and so on’ – is a direction for use –

the sign ‘and so on’ is a rule

any rule – stated – made pubic – is a sign

in ‘1+1 and so on’ – the ‘and so on’ – refers to the action to be taken with ‘1+1’ –

so we have a rule – for ‘1+1’ –

now do you include the rule in the equation?

yes – for clear direction –

does it look strange on either side of an ‘=’ sign?

perhaps – if you think the equals sign only applies to ‘determinate’ propositions  – determinate process – determinate states of affairs

in this case – it doesn’t –

no big adjustment I would think

‘The possibility of introducing further numbers. The difficulty seems to be that the numbers I’ve in fact introduced aren’t a group that is essential and yet there is nothing to indicate that they are an arbitrary collection: Out of all numbers just those numbers that have been written down.’

there is no essence / accident issue here –

there is simply a learned language behaviour – a learned practise

in the same way as there is no grammar to ordinary language use –

yes – we have ‘explanation’ of ordinary language use –

but then the question can always be asked – what is the grammar of this grammar?

in reality we have different languages – different language uses –

how and why this has come about – is in the realm of speculation –

speculation which can be very productive and useful

we learn forms of use – forms of practise

introducing further numbers is no more mysterious than – introducing further words –

it is what we do – when we do it

and in general we do it within established forms – in accordance with accepted practise –

which we can if we wish to – speculate upon –

such speculation has led to the development of logics –

proposals to explain language use – sign usage –

at the heart of the matter is this:

any ‘explanation’ is open to question – is open to doubt – is uncertain

I would suggest anyone who deals in explanation understands this reality – intuitively

as to why we explain – again a matter of speculation –

I would suggest the reason is – for the pure excitement of – facing and exploring – uncertainty

‘and so on’ – is such a pedestrian phrase – and that is as it should be –

for what it expresses –  is nothing startling – it is in fact ordinary everyday behaviour across the board –

philosophy may leave everything as it is –

but not before dressing it up – and dressing it down – in all manner of costume

‘What do we see “1, 1+1, 1 +1 +1….” as?

As an inexact form of expression. The dots are like extra numerals indistinctly visible.

It is as if we stopped writing numerals, because after all we can’t write them all down, but as if they are all there  all right in a kind of box. Again, it is something like when I sing only the first notes of a melody distinctly, and then merely hint at the rest and let it taper off into nothing. (Or when in writing one writes only a few letters of a word distinctly and ends with an unarticulated line.) In all such cases the indistinctly has a distinctly corresponding to it.’

‘what do we see “1, 1+1, 1 +1 +1 ….” as?

the expression ‘1, 1+1, 1 +1 +1 ….’ – has no meaning outside of interpretation –

and the context in which it is used will determine how it is seen – which is to say how it is used

when I look at “1, 1+1, 1 +1 +1 ….”  -

I see a directive – a directive to – ‘keep going’ –

and to keep going in the way already described

(Wittgenstein when asked by Malcolm what it was he liked to eat – said he liked to eat anything – so long as it was the same)

‘We are inclined to believe that the notation that gives a series by writing down a few terms plus the sign “and so on” is essentially inexact, by contrast with the specification of the general term.’

as to exactness –

any proposal – in any form is open to question – open to doubt – is uncertain –

and this includes any rules for its use – and the system in which it is used

any claim of exactness is open to question – and for that matter any claim of inexactness –

the issue is not exactness or inexactness – it is function and utility

and the function and utility of any proposal is open to question – open to doubt –is uncertain

Wittgenstein concludes here –

‘Is this notation inexact? (referring to a line of logical notation). It isn’t supposed by itself to make anything graphic: all that matters are the rules for its use, the system in which it is used. The scruples attaching to it date from a train of thought which concerned itself with the number of primitive signs in the calculus of Principia Mathematica.’

the search for propositional certainty is naive –

it is pre-logical

© greg t. charlton. 2015.