face of this conception, we should remind ourselves why “truth” is quite unequivocal between “analytic truth” and “synthetic truth.”
We can start by considering a standard disquotational principle for truth (where both occurrences of “P” are to be replaced by a declarative sentence):
(T) “P” is true if and only if P.
If “true” is ambiguous between analytic truth and synthetic truth,
(T) must itself be disambiguated. Nevertheless, the left-to-right direction holds for both notions:
(Talr) “P” is analytically true only if P.
(Tslr) “P” is synthetically true only if P.
Obviously, “Bachelors are unmarried” is analytically true only if bachelors are unmarried, just as “Bachelors are untidy” is synthetically true only if bachelors are untidy. The exact parallelism of (Talr) and (Tslr) already casts doubt on the supposed ambiguity. Indeed, they are jointly equivalent to a single principle about the disjunction of analytic truth and synthetic truth (“simple truth”):
(Taslr) “P” is analytically true or synthetically true only if P.
Worse, the right-to-left direction fails for both notions:
(Tarl) “P” is analytically true if P.
(Tsrl) “P” is synthetically true if P.
For (Tarl) has a false instance when a synthetic truth is substituted for “P”; (Tsrl) has a false instance when an analytic truth is substituted for “P.” There are no natural substitutes for the right-to-left direction of (T) in the form of separate principles for analytic truth and synthetic truth. Rather, the natural substitute for the right-to-left direction disjoins the two notions:
(Tasrl) “P” is analytically true or synthetically true if P.
But (Tasrl) reinstates simple truth as the theoretically important characteristic.
One cannot avoid the problem by qualifying “true” in (T) with “analytic” for “the relevant kind of sentence” and with “synthetic” for the rest. For the sentences of the relevant kind are presumably just the analytic truths and analytic falsehoods. Thus the schemas for analytic and synthetic truth amount to these:
(Ta) If “P” is analytically true or analytically false, then “P” is analytically true if and only if P.
(Ts) If “P” is neither analytically true nor analytically false, then “P” is synthetically true if and only if P.
But (Ta) and (Ts) follow from (Taslr), (Tasrl) and the analogue for falsity of (Taslr):7
(Faslr) “P” is analytically false or synthetically false only if not P.
Thus the information in (Ta) and (Ts) is in effect just information about the disjunction of analytic truth and synthetic truth. The attempt to treat analytic truth and synthetic truth separately just confuses the theory of “true.” The same happens for other theoretically important applications of “true.”
Consider the standard two-valued truth-table for the material conditional:
| A | B | A →B |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
If “true” is ambiguous between analytic truth and synthetic truth, what does “T” mean in that table? We might try subscripting it as Tanalytic and Tsynthetic, multiplying the possibilities in the first two columns accordingly and adding the appropriate subscript in the third column. “F” will require corresponding subscripts too. Since the possibilities Tanalytic, Tsynthetic, Fanalytic and Fsynthetic arise for both A and B, the new truth-table will have sixteen lines. Worse, consider this case:
| A | B | A →B |
| Tsynthetic | Tsynthetic | T? |
What subscript is appropriate for the third column? Suppose that Barbara is a barrister, and therefore a lawyer. Of the following four sentences, (1), (2) and (4) are synthetic while (3) is analytic (with “if” read as →):
1 (1) Barbara is a barrister.
2 (2) Barbara is a lawyer.
3 (3) If Barbara is a barrister, Barbara is a lawyer.
4 (4) If Barbara is a lawyer, Barbara is a barrister.
Since Barbara could easily not have been a lawyer at all, (1) and (2) are synthetic. If there are analytic truths, (3) is one of them; “barrister” simply means a lawyer with certain qualifications. Thus we cannot put “synthetic” for the missing subscript in that line of the truth-table, for that gives the wrong result when we read A as (1) and B as (2). Since Barbara could easily have been a lawyer without being a barrister, by being a solicitor, (4) is synthetic too. Thus we also cannot put “analytic” for the missing subscript, since that gives the wrong result when we read A as (2) and B as (1). Therefore the truth-table cannot be completed. Whether a material conditional is analytically true and whether it is synthetically true are not a function of whether its antecedent is analytically true, whether its antecedent is synthetically true, whether its consequent is analytically true and whether its consequent is synthetically true.
The best we can do is to put the disjunction of Tanalytic and Tsynthetic in the third column. But then in order to apply the truth-table iteratively, when one occurrence of → is embedded inside another, we shall need further lines in which such disjunctions appear in the first two columns as well as the third. In effect, we have merely recovered a single sense of “true,” applicable to both analytic truths and synthetic truths, albeit awkwardly defined by a disjunction. The same conclusion can be reached by looking at combinations of other logical constants, such as conjunction and negation. What does the central work in the compositional semantics is that indiscriminate notion of truth, not the more specific notions of analytic truth and synthetic truth.
A corresponding result holds for the theory of logical consequence. Valid arguments preserve truth from premises to conclusion. What can we say if “truth” must be disambiguated between analytic truth and synthetic truth? A valid argument whose premise is a synthetic truth may have either a synthetic truth or an analytic truth as its conclusion. For example, the conjunction of a synthetic truth with an analytic truth is itself a synthetic truth, and has each conjunct as a logical consequence. For logic, the significant generalizations concern the indiscriminate disjunction of analytic truth with synthetic truth, not either disjunct separately.8
Analytic truths and synthetic truths are true in exactly the same central sense of “true.” That is compatible with their being true in very different ways, just as being a mother and being a father are two very different ways of being a parent; “parent” is not ambiguous between mothers and fathers. But truth-conditional semantics undermines even that idea. For how are (3) and (4) true in very different ways? Each is a material conditional; the antecedent
