On the Semantics of Identity

 Revisiting Frege's Puzzling Dichotomy

Identifying a critical historical juncture in the philosophy of language:
the necessity of moving beyond rudimentary semantic theories to account for the
cognitive complexity of identity statements. The foundational challenge, as articulated
by Gottlob Frege in his seminal 1892 paper, "On Sense and Reference," is not merely
a linguistic quirk but a deep, structural problem regarding the relationship between
thought and reality.

I. The Formal Structure of the Puzzle

The initial proposition, "A = A ∧ A = B," succinctly captures the formal paradox.
Consider the following two sentences:

"The Morning Star is the Morning Star." (A=A)

"The Morning Star is the Evening Star." (A=B)

The problem is two-fold:

Epistemic Difference: Both statements are, in fact, true
(they share the same astronomical referent: the planet Venus). However,
their cognitive value is profoundly different. Statement (1) is a trivial,
a priori truth, a mere tautology that conveys no new information.
Statement (2), conversely, is a significant,
a posteriori discovery of astronomy. If a semantic theory holds that the only
meaning of a term is its referent, then (1) and (2) should be cognitively
identical—a conclusion empirically false.

The Problem of Informative Identity: If the referent (Bedeutung) alone
constitutes the meaning of a proper name, then the truth of an identity statement
A=B should be known simply by knowing the referents of A and B. Yet, as shown,
it requires empirical investigation.

Frege's solution was to posit a third semantic layer: Sense (Sinn). The sense is the mode of presentation of the referent. Both "Morning Star" and "Evening Star" refer to the same object (Venus), but they present that object via different descriptive criteria or conceptual content. The meaning of an identity statement A=B is thus the recognition that two different senses converge upon a single referent.

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II. Proper Names, Propositional Attitudes, and Cognitive Content

The examples—the favorite cow, Muhammad Ali/Cassius Clay, and the best friend—are excellent instances of how this Fregean framework extends to the problem of Propositional Attitude Ascriptions (beliefs, desires, knowledge).

When the ordinary use of names is considered, we enter into opaque contexts where the principle of substitutivity of identicals fails.

The Cow and the Social Masses:

• Agent 1 (The Farmer): Believes "Cow A has positive attributes."
  (Sense 1: favorite cow)

• Agent 2 (The Consumer): Desires "To eat Cow B." (Sense 2: edible substance)
  In reality, Cow A = Cow B (they are the same physical entity).
  However, substituting "Cow B" for "Cow A" in the farmer's belief leads
  to the false ascription: The Farmer believes "Cow B is inedible,"
  which may not hold in the context of the farm's eventual purpose.
  The mental meaning (the sense) determines the truth value of the
  propositional attitude, even when the referent remains constant.

Muhammad Ali / Cassius Clay: The identity Muhammad Ali = Cassius Clay is informative. The observer who knows the referent via the sense "the boxer Cassius Clay" may rationally and non-culpably deny that "the boxer Muhammad Ali" is the same individual, until they are presented with the empirical (or historical) fact of the name change. This highlights the crucial role of sense in determining cognitive content.

Your formulation concerning a Justified, True, Belief (JTB) is a precise application:
A and A1 differ in mental meaning if one can rationally hold A to be JTB while remaining skeptical of A1. The difference lies in the sense by which the two propositions are apprehended.

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III. The Limits of Reference: Abstracta and Empty Names

Our source rightly pivots to two other critical areas where
the Sense/Reference distinction is tested:

Empty Names (The Non-Referent): The classic example, often debated
by Russell, is "the present King of France."
This term clearly has a sense (a set of descriptive properties),
which allows us to meaningfully think about it, despite the fact that it currently
lacks a referent (Bedeutung). This reinforces the idea that sense is the primary
bearer of meaning, independent of the external existence of the referent.

Abstract Entities and Mental States: The question concerning whether names
and numbers are "abstract entities" is central to Fregean ontology.

• For Frege, the sense of a proper name (which includes numbers) is an objective,
  abstract entity—neither a physical object nor a subjective mental state—that
  exists in a "third realm" (the realm of thoughts or Gedanken). This objectivity
  is what ensures that two different people can grasp the same sense, even if
  their subjective mental images or ideas (your "mental states") differ.

• Introspection and "mental states" are relegated to the non-objective,
  psychological realm of Ideas (Vorstellungen), which are irrelevant
  to the truth and meaning of language. Sense is the necessary mediating
  layer that connects the subjective (ideas) to the objective (reference).

In sum, what is termed "Frege's Proper Puzzle" is not a flaw in his identity concept,
but his elegant and profound solution to the limitations of simple referential semantics.
The puzzle reveals that meaning is not merely reference, but the manner in which reference is achieved and cognitively assimilated.