§22. Frege's idea that every assertion contains an assumption, which is the thing that is asserted, really rests on the possibility found in our language of writing every statement in the form: "It is asserted that such-and-such is the case."—But "that such-and-such is the case" is not a sentence in our language—so far it is not a move in the language-game. And if I write, not "It is asserted that....", but "It is asserted: such-and-such is the case", the words "It is asserted" simply become superfluous.
We might very well also write every statement in the form of question followed by a "Yes"; for instance: "Is it raining? Yes!" Would this show that every statement contained a question?
Of course we have the right to use an assertion sign in contrast with a question-mark, for example, or if we want to distinguish an assertion from a fiction or a supposition. It is only a mistake if one thinks that the assertion consists of two actions, entertaining and asserting (assigning the truth-value, or something of the kind), and that in performing these actions we follow the propositional sign roughly as we sing from the musical score. Reading the written sentence loud or soft is indeed comparable with singing from a musical score, but 'meaning' (thinking) the sentence that is read is not.
Frege's assertion sign marks the beginning of the sentence. Thus its function is like that of the full-stop. It distinguishes the whole period from a clause within the period. If I hear someone say "it's raining" but do not know whether I have heard the beginning and end of the period, so far this sentence does not serve to tell me anything.
Imagine a picture representing a boxer in a
particular stance. Now, this picture can be used to tell someone how he should
stand, should hold himself; or how he should not hold himself; or how a
particular man did stand in such-and-such a place; and so on. One might (using
the language of chemistry) call this picture a proposition-radical. This will be
how Frege thought of the "assumption".
Compare §65, where he starts his anti-essentialist discussion of 'game' with the question whether all language-games have something in common (so that we could explain the essence of all language-games). 'Language-game' does not have an essential meaning; it refers to a loosely-boundaried area of stuff, shading into stuff such as playing chess silently, which we would call 'a game' rather than a 'language-game'. Is the picture of the boxer part of a language? Would, constructing an IKEA table from the pack of bits, using the instructions, be part of a language-game? (Constructing an object from a description (a drawing) in §23) Must there be some involvement of communication for something to be a 'language'? Must there be some symbolic representation? W. would object to all these attempts to pin down what 'language' and 'language-game' refer to.
The Frege reference is to the general idea that the words that we use, such as the nouns, gain their meaning (sense) by referring to things. This leads to puzzles over '"The present king of France is bald", which seems to be meaningful, but has no reference.