Few notes on the terms used
The “intentional matter” is transcendent (objective) in that that it transcends (keeps its identity) across different varying properties of the intentional acts, e.g. numerically different acts (i.e. you can think about same thing more then once), qualitatively different acts (you can wish, need, hate, love, or think about the same thing); Also transcends the subjects in that that multiple subjects can think (or have other intentional act) about same intentional matter, which in turn opens possibility for communication about the same intentional matter (if two people can’t think of/about same intentional matter it is hard to see how they will communicate about it). This later transcendence I will call inter-subjective transcendence of intentional matter (transcends the thinking-of of multiple subjects), and the other one intra-subjective transcendence of intentional matter (transcends the different intentional acts of single person). If I also mention objectivity, or objectivity of intentional matter, I would mean same thing as transcendence of intentional matter. Also I might leave out the words “of intentional matter”, and use only transcendence or objectivity to mean transcendence of intentional matter. Both transcendence and objectivity point to the same thing – the disconnectedness between the identity of whatever is intentional matter, and the other varying properties of intentional acts, including the identity of the subjects intending.
Relation between intra-subjective and inter-subjective transcendence
First let me put forward the note that it seems obvious to me that there is no way that inter-subjective transcendence can appear, if we don’t have intra-subjective transcendence, i.e. if I can’t think of same intentional matter multiple times, there is no way that me and somebody else can talk about that certain intentional matter. For even if the person in his answer to my question about X does talk about the same X, if I can’t understand it as being about same X (as it would be a separate intentional act then the one in which I asked about X), there would be no possibility of communication.
Or take a case of learning words in one simple language game – ostensive teaching/training (as in Wittgenstein Philosophical Investigations), where…
teacher’s pointing to the objects, directing the child’s attention to them, and at the same time uttering a word; for instance, the word “slab” as he points to that shape.
And then the teacher can request the child itself to point to the objects when he pronounces some of the words, or alternatively:
…the learner names the objects; that is,he utters the word when the teacher points to the stone
But for this second part to be possible (ignoring the inter-subjective dimension for now), the intra-subjective transcendence has to be there, it must be possible for the child to think of the same intentional matter, be it the stone (reference) or the act of pointing to specific stone (use), when it again hears the teacher utter the same word (of course even the word itself needs to be recognized as the same word).
So to give account of inter-subjective transcendence, we need first to have account of intra-subjective transcendence.
But the reverse doesn’t seem necessary. There is no need to have inter-subjective transcendence, in order to have intra-subjective transcendence. Pythagoras could’ve wondered about the relations of the sides of the right triangles one day, and comprehend the necessity of the relation which we know today as a Pythagorean theorem the next day. And while there being intra-subjective transcendence of intentional matter (namely the relation between the sides of right triangle) he could have failed to communicate it to anyone. In same way I could notice and observe a thing, which I don’t have a word about, nor someone hinted me about existence of such a thing. Of course, in our thought the inter-subjective transcendence is also necessary, but this is not a question if it is or not necessary, but if it is necessary for the intra-subjective transcendence.
Sometimes this asymmetric relation between inter-subjective and intra-subjective transcendence/objectivity is unnecessarily complicated by talking about meanings; and by talking about meanings implicitly the inter-subjective dimension is added, as meaning is something which is meant by something else. Only a sign – word spoken, written, a nod or shaking the head, pointing with the finger, laughing and so on… can mean something; but what those mean is not because of that meaning per se – it doesn’t fall into some realm of meanings. That which is meant (which is signified) is the other thing (other then the sign), which happens to be in relation of meaning. So for example, “tree” can mean tree, but tree is not because of that a meaning per se. It is intentional matter.
Anyhow, it seems to me that this asymmetric relation hints that inter-subjective transcendence is grounded in the intra-subjective transcendence.