Begriffsschrift, First Chapter
Explanation of the symbols. All the symbols are divided into
- those that can be taken to mean various things
- those that have a fully determinate sense The problem here is that what does it mean that a symbol can be “taken to mean various things”. When specifying that a symbol can be taken to mean something, a universe where this “something” resides should be presented simultaneously. and also the meaning of this particular symbol should be able to ranged over this whole universe. Is it possible for one to even think about this universe and the ranging-over by a symbol throughout this universe?
Judgement. Here Frege seems to be distinguishing between sentences and propositions. Sentences are sequence of words of which the meaning is always “recognized as” true, while propositions are those sequence of words for which a truth-value can be assigned. It doesn’t seem like that Frege is distinguishing semantics and syntax here. It is tempting to think that the “turnside” symbol is the valuation in the common sense and the content stroke is just an indicator of a propositon.
No distinction of subject and predicate. Propositions are defined up to conceptual content, so propositions presenting the same conceptual content are not distinguished.
The Function. In fact this section looks like Frege is introducing the notion of currying. The proposition ‘Cato killed Cato’, when the first occuring ‘Cato’ is a variable, then ‘killing Cato’ is the function: $f(\cdot,\cdot): X \times Y \rightarrow Z$ becomes $ f(\cdot,y): X \rightarrow Z^Y$; also when the second occurence is regarded as a variable, we have the function “being killed by Cato” i.e. $f(x,\cdot): Y\rightarrow Z^X$. This is how cartesian closedness arises.
Function and Concept
Note that Frege did something very much like Lambda calculus.
On Concept and Object
The word “concept” for Frege is purely logical in its sense. He says that “what is logically simple cannot have a proper definition” but it is unclear whether is a claimed logically simple “object” really simple or even what “simple” means here.
Double usage of the word is.
- It is green. It is a mammal. Here “is” serves as a copula, a verbal sign of predication. Here the word appearing after “is” is a reference of a grammatical predicative and hence is a concept in the sense of Frege.
- A thing is Alexander the Great. Here “is” is used like the “equals” sign in arithmetic, to express an equality (the same as, no other than, identical with). Here the word appearing after “is” is a proper name, a name of an object in the sense of Frege.
Hence for Frege there can be a “concept” of morning star (=Venus): You are my morning star. This concept is something that is embodied by the appearance or even the sense of the name of the morning star. Note that this is related to the origin of words referencing colors: (it is said that) there is no color blue in the Old testament. I have a functionalist interpretation of this phenomenom (namely a word for a particular color originates only when it is needed). The word lapis lazuli and azul etc.
He says ’the concept “horse”’ do designate an object but in that account not a concept. Now this looks like heavily functionalist. I say ‘The concept “horse”’ designates an object since it is used in such a manner that it designates the concpet “horse” as an object while in ‘it is a horse’ the concept is used in such a manner that it designates the concpet ‘horse’ as a concept. In Grundlagen he proposed a simple criterion: the singular definite article always indicates an object whereas the indefinite article accompanies a concept-word (with some easily recognizable special cases as exceptions).
“There is such good accord between the linguistic distinction and the real one”. However this can be that “the real one” is constructed and limited by the linguistic distinction, provided that there is really a “real one”, since Logos is through whom God creates.
Now something interesting: subject-concept as in the sentence ‘all mammals have red blood’. However we could say instead “whatever is a mammal has red blood” or “if anything is a mammal then it has red blood”. Hence these are still concepts. Quantifiers such like ‘all,’ ‘any,’ ’no,’ ‘some,’ are prefixed to concept-words. In universal and particular affirmative and negative sentences we are expressing relations between concepts.
When negating ‘all mammals are land-dwellers’, one must negate as ‘all mammals are not land-dwellers’ if here ‘all mammals’ are logical subject of the predicate ‘are land-dwellers’. Instead we use ’not all mammals…’ so ‘all’ logically belongs with the predicate, i.e. ‘mammals are not all land-dwellers’.
Existence. Frege have called existence a property of a concept (note the similarity between Kierkegaard’s rejection to the ontological arguments for God’s existence). In ’there is at least one square root of 4’ it is about a concept ‘square root of 4’, now ’the concept square root of 4 is realized’ then ’the concept square root of 4’ form the proper name of an object - which is a concept - and it is about this object that something is asserted. Now think about does not exist, is it legal to say ‘the concept A cannot be realized’ (instead of ‘A does not exist’). If ‘A’ doen’t exist, what is being said of when we say ’the concept A’? Plato, sophist. See the chapter ’negation’ where he discuss a similar problem
The thought itself does not yet determine what is to be regarded as the subject.
On Sense and Reference
Here Frege does some distinction between semantics (extension) and syntax (a priori/analytic). Sense looks like more related to synatx and reference looks like more related to semantics.
Frege holds that equality is a relation between names or signs of objects rather than a relation between objects. In contrast to analytic or a priori statment $a=a$, $a=b$ inteds to say that the signs or names $a$ and $b$ designate the same thing so that those signs themselves would be under discussion. If the sign $a$ is distinguished from the sign $b$ only as object (e.g. by means of its shape) not as sign (i.e. not by the manner in which it designates something), the cognitive value of $a=a$ becomes essentially equal to that of $a=b$ if $a=b$ is true.
The sense of the sign: wherein the mode of presentation is contained: let $a,b,c$ be the lines conecting the vertices of a triangle with the midpoints of the opposite sides,
- the reference of the expressions ’the point of intersection of $a$ and $b$’ and ’the point of intersection of $b$ and $c$ would be the same
- but not their senses.
I say that the sameness of the reference of the expressions arises from some kind of ambient logic chosen. When the notion of sameness is chosen such that two objects are the same when they are equal as objects in a space, or when the universe of discourse is that of ‘space-points’, their so-called references should be the same. When the notion of sameness is chosen such that two objects are the same when their signifiers - expressions signiying these two objects - are the same, then these two ‘points’ (we should not be allowed to speak of them as points since the universe of discourse is no longer that of space-points) are not the same. Maybe one should ask whether there is a notion of internal/proper semantics associated to a given syntax. Also in between different universes of discourse there should be a transformation or morphism between objects rather than regarding these ‘objects’ as some definite and given underlying objects referenced by expressions inside a standard, metaphysical universe.
To a definite totality of signs there corresponds a definite sense. In grasping a sense one is not certainly assured of a reference, like the expression ’the least rapidly convergen series’ has a sense but demonstrably has no reference (i.e. the corresponding object). To a given reference (an object) there does not belong only a single sign.
I still cannot comprehend what they mean when they are talking about ‘objects’ (those which are referenced) since for an ‘object’ to be percepted or thought of as existing an essense should already be given to it. I hold that there is no distinction between essence and existence since for one to think of an existing object he must construct an essence for it so that the existence of the object can, at least, be comprehended, and when changing the perspective ’the thing’ that exists just become different beings, in fact beings in literally different universes. If there is always a definite sense, as it was claimed by Frege, corresponidng to a definite totality of signs (i.e. an expression), then an essence should be already given, and hence the correponding existence. Here it would be better to say that in the universe of discourse when the objects are just expressions per se the ‘object’ referenced by the expression exists, but in a universe of discourse where mathematical analysis makes sense the expression is just meaningless and a corresponding sense is nonexistent.
A proper name (word, sign, sign combination, expression) expresses its sense, stands for or disignates its reference. By means of a sign we express its sense and designate its reference.
Objection to idealists: ’the Moon is smaller than the Earth’ is different from ’the idea of the Moon is smaller than the idea of the Moon’.
I say: same as above. This distinction is problematic (Frege seems like a complete Platonist in this repect). What is important for the sentence ’the Moon is smaller than the Earth’ is its syntax, its structure. In ’the Moon […]’ it is just referencing a relation (as an object) ‘A (is) R (than) B’ and one confirms that this sentence structure makes sense (grammatically). If the universe is switched to that with a physical meaning then it is no longer a matter of grammar. It is not a matter of standing for or signifying but a matter of relations between different universes. One should take the reality of some universe of signs seriously.
Then Frege went on to discuss the sense and reference for an entire declarative sentence, etc. He says that the truth value of a sentence constitutes its reference, so I will say that the truth value itself depends on a valuation given, and different valuations can only be done on ‘a’ sentence presented in different universes. Nothing interesting.
It is not that there is no reference for a sentence like ’the triangle $\Delta$ has two angles’ (not a good example but strangely for the time being I cannot think of an example which is contingent or not analytical, while I doubt if there is any difference), the reference of the sentence per se is just the grammatical form of the expression, not some sense. It is just that there is no way to take this sentence to a universe where mathematical statements make sense, or more precisely a universe where ordinary mathematical statements make sense cannot be constructed from this sentence. Proving the ’non-existence’ (this is a bad word) of a seemingly existing object is finding a way to prove that it cannot be constructed at all.
Maybe it can be put briefly as follows: subjective logic is a special case of objective logic, hence $a=a$ is a special case of $a=b$, of which the bijection is trivial in the universe related to some syntax.
Grasping the sense of a sentence would at the same time be an act of judging, namely truth value comes with the meaning, or meaning is the way in which the truth-value of a sentence can be decided.