Commenter John and I are having a very productive discussion about intentionality.  I thank him for helping me clarify my thoughts about this fascinating topic.  I begin with some points on which (I think) John and I agree.

a) There is a 'third world' or third realm and it is the realm of abstracta.   (I promise: no jokes about Frege's Third Reich. But I can't promise not to speak of Original Sinn and Original Sinn-ers.) Fregean senses, whether propositional or sub-propositional, are abstracta, but not all abstracta are reference-mediating senses. John and I are operating with a provisional tripartite or tri-categorial ontology comprising the mental, the physical, and the abstract.

b) There are instances of intrinsic intentionality. Neither of us is an eliminativist about intentionality in the manner, say, of Alexander Rosenberg. (See Could Intentionality be an Illusion?)

c) There is no intentionality without intrinsic or original intentionality: it cannot be that all intentionality is derivative or a matter of ascription, pace Daniel Dennett.  (See Original and Derived Intentionality, Circles, and Regresses.)

d) Nothing physical qua physical is intrinsically intentional, although some physical items are derivatively intentional.  (Combine this true proposition with the false proposition that all mental states are physical, and you have an unsound but valid argument for the eliminativist conclusion that there is no intrinsic intentionality.)

Agreement on the foregoing points leaves open the question whether there could be intrinsically intentional abstract items.  I tend to think that there are no intrinsically intentional abstract items.  John's position, assuming I understand it, is that some abstract items are intrinsically intentional, and that some  intrinsically intentional items are not abstract, mental states being examples of the latter.

The bare bones of the debate between John and I may be set forth as an aporetic triad:

1) Fregean senses are intrinsically intentional items. 

2) Only conscious items are intrinsically intentional.

3) Fregean senses are not conscious. 

It is easy to see that this threesome is not logically consistent: the propositions cannot all be true. John and I assume that the Law of Non-Contradiction holds across the board: we are not dialetheists.  So something has to give. Which limb of the triad should we reject?  (3) is not in dispute and presumably will be accepted by all: no abstract item is conscious, and senses are abstract.  'Abstract' was defined in earlier entries, and John and I agree on its meaning. The dispute concerns (1) and (2). I reject (1) while accepting the other two propositions; John rejects (2) while accepting the other two propositions.

I argue from the conjunction of (2) and (3) to the negation of (1), while John argues from the conjunction of (1) and (3) to the negation of (2).

My rejection of (1) entails that there are no Fregean senses (Sinne).  This is because Fregean senses, by definition, are intrinsically intentional. It follows that they are essentially intrinsically intentional. So if they can't be intrinsically intentional, then they can't exist. Why are senses essentially intrinsically intentional?  Well, as platonica, senses are necessarily existent: they exist in all metaphysically possible worlds. It follows that they exist in worlds in which there are no finite minds.* Now a sense, by definition, is a mode of presentation (Darstellungsweise) of its object.  It mediates between minds and things.  Reference, whether thinking reference or linguistic reference, is routed though sense. The (re)presentational power of a sense is essential to it, and it has this power even in worlds in which there are no words to express the sense, no things to be presented by  the sense, and no minds to refer to things via senses.  For example, consider a possible world W in which there are no languages, no minds, and no planet Venus. In W the sense that 'Phosphorus' — 'Morning Star,' Morgenstern — expresses in our world exists (because it exists in every world) and has its (re)presentational power there in W. Thus its intentionality is intrinsic to it and does not depend on any relations to words or to things or to minds.  It (re)presents non-linguistically and non-mentally and without the need for physical embodiment.

I think it follows that there is no distinction in reality — although there is one notionally — between the power of a Fregean sense to represent and its exercise of this power.  There is, in other words, no distinction in reality between the power of a sense to represent and its actually representing.  I say this because the existence of what an intrinsically intentional item is of or about has no effect whatsoever on the aboutness of the item.  Suppose I am thinking about the Washington monument, but that while I am thinking about it, it ceases to exist. That change in objective reality in no way affects the aboutness of the intentional state.  Thus the power of an intrinsically intentional item to represent does not need an external, objectively real, 'trigger' to actualize the power.  The extramental existence of the Washington monument is not  a necessary condition of the aboutness of my thinking about it.  The content and aboutness of my thinking is exactly the same whether or not the monument exists 'outside the mind.' The same goes for senses. The sense of 'Phosphorus' presents Venus whether or not Venus exists. And the content of the sense is exactly the same whether or not Venus exists.

There is an important difference, however, between an intrinsically intentional mental state and a Fregean sense.   The occurrent mental state or 'act' — in the terminology of Twardowksi, Husserl, et al. — is the state of a mind. It is the act of a subject, the cogitatio of an ego, where the last three occurrences of 'of' all express the genitivus subjectivus.  This is essentially, not accidentally, the case.  There has to be an ego behind the cogitatio for the cogitatio to be a cogitatio of a cogitatum.  But there needn't be an ego 'behind' the sense for the sense to be a sense of or about a thing. If a Fregean sense mediates a reference between a mind and a thing, it is not essential to the mediation that there be a mind 'behind' the sense.

Here then is an argument against Fregean senses:

4) Every instance of intrinsic intentionality has both a subject and an object.
5) Some instances of reference-mediation by a sense do not have both a subject and an object.
Therefore
6) Some instances of reference-mediation by a sense are not instances of intrinsic intentionality.

When I reject the proposition that Fregean senses are intrinsically intentional items, I thereby reject the very existence of Fregean senses. I am not maintaining that Fregean senses exist but are derivatively intentional items.  I do hold, however, that there are derivatively intentional items, maps for example.  Maps get their meaning and aboutness from us original Sinn-ers. A map is not about a chunk of terrain just in virtue of the map's physical and geometrical properties. Consider the contour lines on a topographical map. The closer together, the steeper the terrain. But that closer together should mean steeper is a meaning assigned by the community of map-makers and map-users. This meaning is not intrinsic to the map qua physical object. Closer together might have meant anything, e.g., that the likelihood of falling into an abandoned mine shaft is greater.

So some things derive their referential and semantic properties from other things.  That is also true of language. Words and phrases don't mean anything in and of themselves. Mind is king: no minds, no meaning. I subscribe to the primacy of the intentional over the linguistic. 

John and I agree that Fregean senses, whether propositional or sub-propositional, are explanatory posits.  They are not 'datanic' as I like to say. Thus it is a pre-analytic or pre-theoretical datum that the sentences 'The sky is blue' and Der Himmel ist blau 'say the same thing' or can by used to say the same thing. But that this same thing is a Fregean proposition goes beyond the given and enters the explanatory realm.  One forsakes phenomenology for dialectics. Now what am I claiming exactly? That there is no need for these posits, that to posit senses is to 'multiply entities beyond necessity in violation of Occam's Razor: Entia non sunt multiplicanda praeter necessitatem?  Or am I saying something stronger, namely, that there cannot be any such items as Fregean senses? I believe my view is the latter, and not merely the former.  If senses cannot exist, then they cannot be reasonably posited either.  

John's view, I take it, is that both Fregean senses and some conscious items are intrinsically intentional or object-directed. He is not maintaining that only third-world entities (abstracta) are intrinsically intentional. By contrast, I maintain that only second-world entities (mental items, both minds and some of their occurrent states) are intrinsically intentional.

I assume that John intends 'intrinsically intentional' to be taken univocally and not analogically.  Thus he is not saying that Fregean senses are of or about first-world items in a manner that is analogous to the way second-world items are of or about first-world items.    

Fregean senses are intrinsically intentional, necessarily existent, abstract entities.  By its very nature a sense presents or represents something apart from itself, something that may or may not exist. It is a natural, not conventional, sign.  

Do I have a compelling argument against Fregean senses?  Above I mentioned the following argument:

2) Only conscious items are intrinsically intentional.

3) Fregean senses are not conscious. Therefore:

1) It is not the case that Fregean senses are intrinsically intentional.

But this argument appears to beg the question at (2).  Why can't there be intrinsically intentional items that are not conscious?  If there can be intentionality below the level of conscious mind in the form of dispositionality — see Intentionality, Potentiality, and Dispositionality — why con't there be intentionality above the level of conscious mind in platonica?

Nevertheless, there seems to me something incoherent about Fregean senses. They actually represent even in worlds in which there is nothing to represent and no one to whom to represent.  Consider again the sense of 'Phosphorus.'  It exists in every world including worlds in which Venus does not exist and no mind exists. In those worlds, the sense in question actually represents but does not represent anything to anyone.  It is therefore a non-representing representation, and thus an impossibility.

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*A finite mind is the kind of mind that needs such intermediary items as Fregean senses or Husserlian noemata to mediate its reference (both thinking reference and linguistic reference) to things that it cannot get completely before its mind in all their parts, properties, and relations. An archetypal intellect such as the divine mind can get at the whole of the thing 'in one blow.'  As an infinite mind it has an infinite grasp of the infinitely-propertied thing.  An infinite mind has no need of senses. The existence of senses therefore reflects the finitude of our minds.  That the reflections of this finitude should be installed in Plato's heavenly place (topos ouranos) seems strange.  It looks to be an illict hypostatization. But this thought needs a further post for its adequate deployment.