Part II. II

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 –

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.’

‘($x) has generality in that it is non-specific as to what it applies to –

what it applies to is unknown

‘there are only two circles in this square’ –

and ‘there are only two men who have climbed the mountain’ –

these statements have the same form

they are specific – two men who have climbed the mountain – and two circles in the square –

and they have similar formal scope – all men – and all things in the square

‘clearing up the grammar of the proposition’ –

is first off understanding that a syntactical identity between words or phrases in different uses – does not entail a grammatical identity –

how ‘there is’ is understood will vary with the different propositional contexts –and usages

yes there may be formal similarities between certain propositional usages  –

but there will always be relevant differences

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 –

it is just pretentious to hold that one model of language – applies to all propositional usage

and the same argument applies to the subject-predicate model –

these theories and models have their uses – have their functions – but as with any account of propositional action and behaviour – their applicability – their value – is open to question –

and we should at all times avoid the trap of seeking – through one language use (a theory or model) – an explanation of all language use

generality in ordinary language – is in all its uses – open to question – open to doubt – is uncertain

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

the subject-predicate form – is simply a model of limited applicability

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).

yes – the point of a proposition is that it is put –

this ‘mental process of understanding’ is an account of how the proposition comes about –

such an account might be of interest to us – but it is a back story – a back story to the proposition put –

the immediate issue is the assessment and use of the proposition

“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”.’

yes different understandings – different propositional constructions

what does the first man understand that the second doesn’t?

he understands a different proposal – a different proposal in relation to the original concept / proposal –

yes ‘logically bounded’ and logically unbounded –

fair enough

the point is we have – in different propositional contexts – different understandings –

and always the possibility of different descriptions

‘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.)’

yes – the function “f( …)” can be seen as such form –

which is to say the function “f( …)” – is a way of seeing

‘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’

yes – rules – if you like –

but better termed ‘propositional practices’

‘… 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 an 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 of all cases actually used – and therefore – closed –

this 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 – as you need it to be –  as you 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

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 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 following

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 regard this notion of ‘explanation’ – as superfluous –

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”’?

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

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

the series as proposed is regarded as on-going –

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

continuation – in terms of the rule governed action already established

infinite?

‘infinite’ here is the proposal of – a logically open propositional action

‘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 –

also – the general rule – within certain parameters may be a working proposition

and there is no rule to say you can’t change a rule

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

there is no abyss

the is a proposal (general proposition) for the use of signs (numbers) – for a propositional action

‘Can one imagine an empty space?’ – ‘possibility as a shadow of reality’? –

‘empty space’ and ‘shadows of reality’ –

this is poetry –

the propositional reality is – successive rule governed propositional action –

‘possibility’ here – without the poetics –

is the next step in a propositional action

‘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.”                                                                                                                             

‘and so on’ – is a direction for propositional action –

it is if you like a proposal within a proposition

the sign ‘and so on’ – is a rule

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

so we have a rule for the use of ‘+1’ –

do we need the additional ‘and so on’ – as in ‘1 + 1 and so on.  =  . 1 + 1 + 1 and so on,  and so on.’?

in so far as repetition is the point of ‘and so on’ – the second ‘and so on’ – strikes me as missing the point

‘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

(This reminds me of a story regarding Wittgenstein –

when asked by Malcolm’s wife what it was he liked to eat – he 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