How Does Frege's Indication Principle Relate Propositions to Domain Indicators?

[MathematicalPhysicist]

im searching for the philosophical proof of the next statement (according to my text the source is from frege):
if:
1)propsitons aren't complex singular terms (definite descriptions) and they are domain indicators for definite descriptions.
2)for equivalent singular terms there's the same domain indicator.
3) a singular term doesn't change its indicator. if a singular term is contained in/substituted by another with the same indicator;
then:
4) for all the propsitions with the same truth value there's the same indicator.

 

[NickJ]

First, the use of "singular term" and "definite description" strikes me as odd: these are properties that terms can have (like nouns or pronouns).

Better to say (1) as: "Propositions are not sentences; but they are domain indicators for sentences".

This captures the fact that the same proposition can be expressed by different sentences (e.g., by an English sentence and a French one). The domain of what is talked about by a sentence is captured by teh proposition the sentence expresses.

Then (2) becomes: sentences that "are equivalent" express the same proposition.

Of course, this claim about equivalence needs to be made more precise. But let's ignore complications that might arise for sentences that contain indexicals and non-rigid designators and so on.

(3) is more clearly written as: if one sentence is replaced by a sentence that expresses the same proposition, then:

(4) the proposition retains its truth-value.Now on to the main question: how to show that (4) follows from (1)-(3)?

a. Consider a sentence token "s". Suppose that "s" expresses the proposition "S", and suppose that "p" is a sentence-token that is "equivalent" to "s". (Example: let "s" = "Snow is white", "p" = (however one says "snow is white" in some other language", and "S" is the proposition that snow is white.

b. Suppose, for the sake of argument, that "s" is true. (We could give a parallel argument for the case in which "s" is false.)

c. Then "S" is true, since "s" expresses the proposition "S". (I take this to follow from the idea that (1) is trying to capture.)

d. Since "p" is equivalent to "s", "p" also expresses "S" (via (2)).

e. Since "p" is equivalent to "s", "p" is also true. (from whatever "equivalence" is supposed to mean?)

f. Hence, "S" is true, since "p" expresses the proposition "S". (I take this to follow from the idea that (1) is trying to capture.)

g. Thus, whenever "s" is replaced by "p" (as supposed by (3)), the proposition expressed by "s" retains its truth-value. (Follows from the above argument.) QED.

----------

Perhaps you are trying to show something different, namely, that whenever one term in a sentence is replaced by a term that refers to the same thing, the sentence retains its truth value.

This thesis is false. Example: It is true that Lois Lane knows that Clark Kent is Clark Kent, because (we can suppose that) she knows that everything is identical to itself. But Lois Lane does not know that Clark Kent is identical to Superman. Yet "Superman" and "Clark Kent" refer to the same person. QED.

Example 2: It is necessarily true that nine is greater than seven. But it is not necessarily true that the number of planets is greater than seven -- the number of planets might have only been three or six or one. But "nine" and "the number of planets" refer to the same thing, namely, how many planets there are in our actual solar system. QED.

Hope this helps!

 

[Architeuthis Dux]

The Frege proof you are searching for is known as the "Indirect Proof" or "Proof by Contradiction." It is a method of proof commonly used in philosophy and mathematics, where one assumes the opposite of what they are trying to prove and then shows that this leads to a contradiction. In this case, the statement you are referring to is known as the "Indication Principle" and it is a fundamental principle in Frege's theory of meaning.

The Indication Principle states that propositions are not complex singular terms (such as definite descriptions) and they serve as domain indicators for definite descriptions. This means that propositions do not refer to specific objects, but rather they indicate the domain in which the definite description is true. For example, the proposition "the tallest building in the world" does not refer to a specific building, but rather it indicates the domain of all buildings and their heights.

The second premise of the Indication Principle states that for equivalent singular terms, there is the same domain indicator. This means that if two singular terms refer to the same object, they will also have the same domain indicator. For example, the terms "the tallest building in the world" and "the building that is the tallest in the world" refer to the same building and thus have the same domain indicator.

The third premise states that a singular term does not change its indicator when it is contained in or substituted by another term with the same indicator. This means that if a singular term is replaced by another term that refers to the same object, the domain indicator remains the same. For example, if we replace "the tallest building in the world" with "the building that is the tallest in the world," the domain indicator remains the same.

From these premises, we can conclude that for all propositions with the same truth value, there is the same domain indicator. This is because if two propositions have the same truth value, it means that they refer to the same domain and thus have the same domain indicator.

In conclusion, the Indication Principle is a crucial part of Frege's theory of meaning and it provides a philosophical proof for the relationship between propositions and definite descriptions. It shows that propositions do not refer to specific objects, but rather they indicate the domain in which the definite description is true.