Some of the
intuitive characterisations given, in the last post, of the notion of internality
of truth-value – such as 'internal meaning determines truth-value'
- sound a lot like a common post-Kantian way of characterising or
defining analyticity, namely as 'truth in virtue of meaning'. This
raises the question of whether the class of

*a priori*truths is the class of analytic truths, and the question of whether there are, or should be, distinct notions here at all. My answers to these questions will be No and Yes respectively.
The aim here will
be to try to clarify an interesting notion of analyticity which is
conceptually and extensionally distinct from all the notions of truth

*a priori*identified in the last post (internality of truth, non-Twin-Earthability of truth, Chalmers' epistemic two-dimensionalist account, and traditional conceptions). It is distinct from, but builds on, our internality conception of the*a priori*.
The account I will
give of this notion is inspired by Kant's account of the
analytic-synthetic distinction in the

*Critique of Pure Reason*, as well as Wittgenstein's remarks on the synthetic*a priori*and concept-formation in the*Remarks on the Foundations of Mathematics*.
It is well known
that Kant's definition, or principal explication, of 'analytic' and
'synthetic' is given in terms of subject and predicate:

In all judgments wherein the relation of a subject to the predicate is cogitated (I mention affirmative judgments only here; the application to negative will be very easy), this relation is possible in two different ways. Either the predicate B belongs to the subject A, as somewhat which is contained (though covertly) in the conception A; or the predicate B lies completely out of the conception A, although it stands in connection with it. In the first instance, I term the judgment analytical, in the second, synthetical.

Since modern logic
and philosophy of language has taught us not to regard every
proposition as being composed of a subject and a predicate, this
definition can't be adequate for us. But it is suggestive, and even
moreso are some of the other things Kant says about the
analytic-synthetic distinction. He says of analytic and synthetic
propositions respectively that 'the former may be called

*explicative*, the latter*augmentative*'. And consider this elaborated version he gives of his main question, that of how synthetic*a priori*knowledge is possible: 'If I go out of and beyond the conception A, in order to recognize another B as connected with it, what foundation have I to rest on, whereby to render the synthesis possible?'.
The idea that
synthetical judgments are 'augmentative', that they 'go out and
beyond' 'conceptions', can, I think, be generalized or abstracted
from Kant's discussion in such a way that it does not depend on
construing all propositions as being of the subject-predicate form.
And we get a hint of how to do this from the following passage about
the syntheticity of the proposition '7 + 5 = 12':

We might, indeed, at first suppose that the proposition 7 + 5 = 12 is a merely analytical proposition, following (according to the principle of contradiction) from the conception of a sum of seven and five.But if we regard it more narrowly[my emphasis], we find that our conception of the sum of seven and five contains nothing more than the uniting of both sums into one, whereby it cannot at all be cogitated what this single number is which embraces both. The conception of twelve is by no means obtained by merely cogitating the union of seven and five; and we may analyse our conception of such a possible sum as long as we will, still we shall never discover in it the notion of twelve. We must go beyond these conceptions, […]

This
regarding-more-narrowly will be the key for us. We said above that a
proposition is

*a priori*iff it contains its truth value, i.e. iff its internal meaning determines its truth-value. Our idea now is that a proposition is analytic iff its internal meaning regarded more narrowly in a certain way – or iff a certain sort of part or fragment of its internal meaning – determines its truth-value. And so the next task is to try to clarify what characterises the aspects of internal meaning we are restricting our attention to here.
To do this, we will
use the notion of concept- or conceptual-structure-possession, and
the notion of understanding. We will not need to involve
considerations of knowledge, judgement,
being-in-a-position-to-see-that, or anything like that. (Later, we
will consider how what we say may shed light on accounts which

*do*involve such considerations.)
As a first
approximation, we will say that a proposition is analytic iff the
bits of conceptual structure – the part of its internal meaning -
one must possess in order to understand it, determines its
truth-value. (This involves a terminological departure from the
possibly more common procedure of regarding analyticity as implying
truth – we say that an analytic proposition can be true or false,
just as we say an

*a priori*proposition can be true or false. This has the nice feature of giving us a simple division among propositions in general, not just truths, so that we can say that for propositions in general, being analytic is just*not being synthetic*, and vice versa.)
We can make this
definition easier to handle and more memorable by giving it in two
parts:

The

*meaning-radical*of a meaningful expression consists in the bits of conceptual structure, i.e. the part of its internal meaning, one must possess in order to understand it.
A proposition is
analytic iff its meaning-radical determines its truth-value.

(As an added bonus,
we now have the general concept of a meaning-radical, which we can
apply to sub-propositional expressions as well as propositions, and
perhaps also to super-propositional expressions such as arguments.)

Consider the fact
that we can come to believe false arithmetical propositions - for
example on the basis of miscalculation, or misremembering, or false
testimony - and that we can apply them.

Contrast the case
of a paradigm analytic proposition, such as 'All bachelors are
unmarried'. (To get around the irrelevant problem that in English
'bachelor' very arguably doesn't mean 'unmarried man', let us just
suppose that it does mean exactly that.) To be sure, someone can
assent to the

*sentence*'Not all bachelors are unmarried', and dissent from 'All bachelors are unmarried', but in such a case we would say that they don't understand this latter as we do – they don't understand our*proposition*'All bachelors are unmarried'. So they don't believe – and here we are using words with our meanings kept intact – that not all bachelors are unmarried.
Kant says that we
can become 'more clearly convinced' of the syntheticity of
arithmetical propositions 'by trying large numbers'. Let us now,
therefore, try to illustrate the notion of a meaning-radical, and in
turn that of analyticity, by considering an example of a false
arithmetical proposition involving numbers larger than 7, 5 and 12.
Say '25 x 25 = 600'.

Despite being false

*a priori*, the proposition '25 x 25 = 600' is something we can mistakenly believe and apply while still understanding it correctly (in some suitably minimal, and natural, sense of 'understand'). We have – wrongly – made a connection between our conception of the product of 25 and 25, and our concept of 600.
Why do we say that
we understand the proposition in its ordinary sense and are wrong,
rather than saying that we are operating in a different system, in
which the sentence '25 x 25 = 600' is true, and that we (therefore)
don't understand the proposition in its ordinary sense? It is not
hard to see what sorts of things make it that way. If we worked it
out on a calculator, or calculated it again ourselves, we would
unmake the connection. Such developments would show that we

*did*understand the proposition correctly (i.e. in its ordinary sense).
(Suppose an illegal
move is made in chess, say that someone moves their king into check
(so that it need not be immediately obvious that it is an illegal
move). If the maker of this move can easily be brought to accept that
their move was illegal, we can maintain that they understand how to
play chess and were playing it according to the ordinary rules, but
playing wrongly. If they cannot, then either they simply do not
understand chess, or are insisting on playing according to deviant
rules.)

The fact that in
the case of the false belief that '25 x 25 = 600', there is this
other option here, if I may put it that way, of saying that we are
operating in a different system – an option which we will have to
reject because of many things about how things are, so in that sense
not an option, but still something which makes sense – shows that
there is a possible system, compatible with the meaning-radical of
'25 x 25 = 600', in which that sentence holds. That is, the
meaning-radical of '25 x 25 = 600' – that bit of conceptual
structure – can be incorporated into a larger structure wherein the
concept of the product of 25 and 25 (although we might not want to
call it that anymore) is connected to that of 600 in such a way that
the sentence '25 x 25 = 600' is true. It will have a different
internal meaning from our proposition '25 x 25 = 600', despite the
system it belongs to incorporating the meaning-

*radical*of our proposition. This is what makes '25 x 25 = 600' synthetic.
On this picture,
the full internal meaning of a concept or proposition-meaning may
involve connections which do not have to be made in order to
understand it.

So, a proposition
is analytic iff it has its truth-value in virtue of the bits of
conceptual structure someone has to possess in order to understand
it. That is, iff the bits of conceptual structure one must have in
order to understand it cannot be embedded in a context such that the
proposition-radical of that proposition gets a completion such that
the resulting proposition has a different truth-value from the
proposition in question.

More briefly, a
proposition is analytic iff its meaning-radical determines its
truth-value.

All

*a priori*propositions, then, on the account I am giving here, will be such that their internal meanings determine their truth-values. But analytic propositions have the further property that their*radicals*determine their truth-values, whereas the radicals of synthetic*a priori*propositions can be incorporated into both true propositions and false ones.

**A Complication**
One complication:
perhaps there is a mistaken assumption of uniqueness built into my
talk above of

*the*meaning-radical of a proposition.*The*bit of conceptual structure one must possess in order to understand it. Perhaps one and the same proposition can be understood from more than one angle, as it were, in which case it may be better to talk about multiple meaning-radicals – distinct bits of conceptual structure all of which individually and minimally suffice for understanding.
This gives rise to
a choice: if that's how things are, should we call the analytic
propositions those which are such that

*all*their meaning-radicals determine their truth-values? Or those which have at least one meaning-radical which determines their truth-value?
I do not want to
try to settle the issue of whether we

*should*recognize a possibility of multiple radicals. Furthermore, I have no opinion about which use of terminology is best, in case we should – perhaps it just doesn't matter. If we use 'analytic' for the first, we may say 'weakly analytic' for the second. Or, if we use 'analytic' for the second, we may say 'strongly analytic' for the first. Or we might make 'analytic' mean 'either strongly or weakly analytic'. Or drop it entirely, and always specify 'strong' or 'weak'.