1. ? Anti-realism, reductionism, reference
Reductionist thesis: statements of the given class have a translation to some reductive class of statements, and this translation confers meaning to the statements of the given class.
Examples: behaviorism, phenomenalism.
Reduction without anti-realism: Frege about directions and parallelism.
Directions are real objects: classes of parallel straight lines.
It eliminates reference (to directions) from statements about directions.
Phenomenalism as anti-realism: just for this reason.
Is it the same case? Can we have explicit definition for physical objects in terms of sense-data?
Three, main features of realism:
Bivalence, crucial role of reference, truth-conditional meaning-theory.
Reductionism gives up (?) the second one, but may keep the first and the third. ?

2. Reductive thesis: (Not necessarily a translation, but) a statement of the given class is made true (if it is true) by some statement or class of statements of the reductive class. This relation determines the meaning of any statement of the given class.
Types of reductive theses without reductionism:
(i) A weaker form of phenomenalism: A statement about material objects is made true by an infinite manifold of sense-datum statements.
(ii) The language of the reductive class cannot be given independently of the language of the given class.
Example: mathematical constructivism. A theorem T is made true by the statement „We have a proof of T”. We can’t eliminate the original terminology of the theorems by translating them into statements about possessing proofs.
Another example: statements in the past tense are made true by statements about present evidence.
(iii) We cannot effectively identify the member of the reductive class responsible for the truth of some statement of the given class.
Example: no translation from psychology to neurophysiology (Davidson).

3. Reductive thesis may lead to rejection of bivalence:
There are (or may be) statements A of the given class s. t. neither A, nor “Not A” is supported by some statement or set of statements of the reductive class.
Sophisticated realism: Every statement of the given class is being made true or false by some statement or set of statements of the reductive class. Example: central-state materialism about psychological statements.
Irreducibility thesis: no non-trivial general answer can be given to the question „What makes a statement of the given class true when it is true?”.
Example: mathematical realism.
Naive realism: realistic interpretation of the given class + irreducibility thesis.

4. Are subjunctive conditionals (SCs) allowed in the reductive class?
First group: yes (e.g. phenomenalism, behaviorism).
If we were to go to the center of the Sun with a thermometer, we would read 10^7 Kelvin grades on it.
Second group: no. E.g. mathematical constructivism.
Anti-realism about the past.
Bivalence: problem(s) with the first group.
If it were the case that A, it would be the case that B. (Example above.)
Let us suppose that it supports the statement „The temperature in the center of the Sun is 10^7 °K”
What should be the support of the statement „ „The temperature in the center of the Sun is not 10^7 °K”?
Obviously not the negation of the above SC: „… we would not necessarily read 10^7 °K”
Perhaps its opposite: „… we would not read 10^7 °K”

5. Strong bivalence (in the reductive class): one of the opposite SCs is true.
The realist may reduce the SCs to indicative (categorical) statements: „The temperature in the center of the Sun is/is not 10^7 °K”.
If we accept this sort reduction, then strong bivalence holds. But it is hardly acceptable on the other way.
There are other possible reductions of SCs: to observations and patterns in observations. But the don’t justify strong bivalence.
Dummett’s criticism on phenomenalists: they accepted bivalence for material-object statements (and therefore implicitly accepted strong bivalence for the underlying SCs).
Hence, phenomenalism is a sort of sophisticated realism, not of full-fledged anti-realism.

6. A reductive thesis is not a sufficient condition for anti-realism,
but not a necessary one, either.
Example: a (fictional) sort of neutralism about the future.
Neutralism: future tense statements are now true or false only in virtue of something that lies in the present (a reductive thesis).
But he does not regard the meanings of future-tense statements as given by the conditions under which they are (now) true. If they are now true, they are necessarily true.
Outline of a semantic theory for him:
Possible worlds consisting of the present state and past history of the world plus some possible future history.
True in a world W at t: true in every world that coincides with W up to t.
In this theory the reductive thesis falls out because there is no such thing as absolute truth of a statement.
Not realist, but objectivist: future-tense statements are determinately true or not true (at a given world and time).

7. Three (four) ways to interpret the thesis „a mathematical statement is true only if there exists a proof of it”:
(0) Maybe there is a proof although we never shall know it („Exists” understood platonistically. Nonsense.)
„Exists” understood as concrete, actual existence, i.e. we actually have a proof. Truth-value changes in time.
„Exists” (and „is true”) understood as tenseless, but constructively (We have or sometimes we shall have a proof of it.) It makes sense to suppose that a statement is true although we don’t know it.
We eliminate the concept of truth and substitute it with the relation „P is a proof for A” as the central concept of meaning-theory. In this case, no reductive thesis (because no truth).
We don’t end up with an objectivist theory of mathematical statements. The meaning of the statement is formulated in terms of our possessing a proof to it. That is why the reductive thesis is not a substantial component of the theory of meaning.