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Internal and Intuitive Truth

The next question to consider is whether the validity of the (T) Schema picks out a unique notion of truth. Tarski seems to hold that it does. In [21] he expresses the conviction that the material adequacy condition imposed onto the definition of truth, is capable to select the classical Aristotelian notion of truth as correspondence. The conviction is not explicitly stated, but it can be inferred from the following facts:

(i) In section I.3 Tarski expresses an intention:

We should like our definition to do justice to the intuitions which adhere to the classical Aristotelian conception of truth […] we could perhaps express this conception by means of the familiar formula:

The truth of a sentence is its agreement with (or correspondence to) reality.17

(ii) In section I.4 the same intention is made precise by requiring that the definition satisfies the material adequacy condition. Hence, the material adequacy condition 'does justice' to the intuitive notion of truth as correspondence. The question whether the intuitive notion of truth as correspondence is bivalent is not explicitly addressed by Tarski; an affirmative answer from him is suggested by the fact that in [20] he derives the principle of bivalence from the (materially adequate) definition of truth.18 However, the derivation crucially uses Excluded Middle, as is made clear by the following steps:

(10)

(i)
α
+ −α [the law of Excluded Middle]

(ii) T αα [the (T) Schema]

(iii) T α + T −α [from (i) and (ii) by Replacement].

So, Tarski's conviction is correct only under the premiss that the metalanguage is associated to a metatheory whose semantics validates Excluded Middle. If this principle is not valid in the metatheory, it is possible to exhibit counterexamples to Tarski's conviction, namely it is possible to define a non-bivalent notion of truth satisfying the (T) Schema. I shall now show how.19

Let us adopt a metatheory in which the logical constants are read according to the (revised) BHK-explanation (which, of course, does not validate the Excluded Middle). We are looking for a materially adequate definition, i.e. such that all the equivalences

(11) T αα

are logical consequences of it (where T is the intended truth operator). The definition I suggest is the following20:

Definition 4 T α =def. K α,

where the meaning of K is defined by Definition 2. Definition 4 is materially adequate: (2t) is valid for the reasons explained above; and the converse

(12) K αα

is valid as well: it expresses the expectation of a function h associating, to every observation o that what one is presented with is a proof of α, a proof h(o) of α, and h is warranted to exist by the factivity of proof observation.21 In conclusion, the knowledge operator K is a truth operator, and of course this operator does not satisfy the principle of bivalence.

Concluding, it is true both (i) that the validity of the (T) schema expresses our essential intuition about the notion of truth, and (ii) that our most common intuitive notion of truth is realistic; but the reason why (ii) holds is bivalence, not the (T) schema: the validity of the (T) schema is neutral among different intuitive notions of truth. It is therefore possible, and necessary, to introduce a clear distinction between

the condition at which an operator is a truth operator and the condition at which an operator reflects our realistic intuitions about the notion of truth; the former consists in the validity of the (T) schema,22 the latter may be epitomized into the slogan of truth as correspondence and consequently into the validity of the law of bivalence. A notion satisfying the former condition is capable to play (at least some of) the roles of the notion of truth; truth as correspondence constitutes our predominant common-sense notion of truth. Between these two extremes there is a variety of truth notions, of which knowability and existence of a verification are two instances. I shall call “internal” these theoretical notions of truth, to stress the fact that each of them is capable to play the, or at least some of the, conceptual roles of the notion of truth within the framework of the related theory of meaning and of the formal semantics that adopts it. In this terminology we can say that bivalent truth is both the predominant intuitive notion of truth and the internal notion of classical logic; and that, besides it, there are several other internal notions of truth.

At this point it should be clear that an intuitive sense of the principle (RAR), according to which it becomes acceptable, does exist: for, if the logical constants are understood according to the BHK-explanation, and truth is understood according to Definition 4, then (RAR) is a tautology, saying that every known statement is known. The intuitionistic solution of the paradox consists therefore in accepting (RAR) as obvious when the logical constants are understood intuitionistically and truth as internal.

Is the idea of equating truth to knowledge consistent? There is an argument— called by [16] “The Standard Argument”—that purports to show that it is not.23 It consists in the following derivation of ∃α(α ∧¬ K α) from the assumptions p ∨¬ p and ¬ K p ∧¬ K ¬ p:

(13)

D [ p ∧¬ K p]3 [¬ p ∧¬ K ¬ p]4

( p ¬K p ) (¬ p ¬K ¬ p ) α¬K α) α¬K α) 3, 4

α(α ∧¬ K α)

where D is: [ p]1 ¬ K p ¬K ¬ p

¬ K pp]2 ¬ K p ¬K ¬ p

¬ K ¬ p

p ¬K p ¬ p ¬K ¬ p

p ¬p ( p ¬K p ) (¬ p ¬K ¬ p ) ( p ¬K p ) (¬ p ¬K ¬ p ) 1, 2

( p ∧¬ K p)(¬ p ∧¬ K ¬ p)

22The claim that an operator O is a truth operator iff it satisfies the schema (7) should not be confounded with the minimalist claim that (7) is a good definition of the meaning of O. The former claim is perfectly compatible with the idea, embraced above, that the validity of (7) is not the definition, but the material adequacy condition of the definition, of O.

23See [16], p. 275.

Now, if we observe that there are statements p that the intuitionist acknowledges as being decidable (i.e. such that p ∨¬ p is assertible), and that, as a matter of fact, are unknown (i.e., such that ¬ K p ∧¬ K ¬ p is true),24 we obtain that ∃α(α ∧¬ K α) is assertible.

My answer consists in observing that the argument is valid but unsound, since

¬ K p ∧¬ K ¬ p is intuitionistically inconsistent. Assume that ¬ K p ∧¬ K ¬ p is assertible, and reason in the following way:

[¬ K p ∧¬ K ¬ p]1 [ p]2 [¬ K p ∧¬ K ¬ p]1 [¬ p]3

(14) ¬ K p K p

⊥ 2 ¬ K ¬ p K ¬ p

⊥ 3

¬ p ¬¬ p

1

¬(¬ K p ∧¬ K ¬ p)

The formula ∃α(α ∧¬ K α) may therefore be false. In order to show that it is actually false, let us wonder whether there could be a proof of it, i.e. a procedure p whose execution yields, after a finite time, a pair (c,π ), where c is a proposition and π is a proof of c ∧¬ K c. A proof of c ∧¬ K c is a pair (π1, π2), where π1 is a proof of c and π2 is a proof of ¬ K c; such a pair cannot exist, on pain of contradiction: being presented with π1, one can effect the observation that what one is presented with is a proof of c, thereby obtaining a proof π3 of K c; coupling π3 with π2 we obtain a proof of K c ∧¬ K c: a contradiction; p cannot therefore exist. In conclusion, the intuitionist cannot assert ∃α(α ∧¬ K α), and the idea of statements that, being unknown, are not yet true nor false is not inconsistent.

The intuitionistic inconsistency of ¬ K p ∧¬ K ¬ p may sound unacceptable from the intuitive standpoint, since it seems to conflict with the idea, which also an intu-

itionist should accept, that there are undecided, hence unknown, statements. Here it is important, again, to pay attention to the intuitionistic meaning of the logical constants, in particular of negation. The assertibility of ¬(¬ K α ∧¬ K ¬α) means that a method is known to transform every proof of ¬ K α ∧¬ K ¬α into a contradiction, hence that a logical obstacle is known to the possibility that there is a proof of ¬ K α ∧¬ K ¬α; it does not exclude the fact that neither α nor ¬α are known. We will see in a moment whether the existence of such a fact can be acknowledged within the intuitionistic conceptual framework. Before, I want to comment upon the existence of a logical obstacle to the possibility that there is a proof of ¬ K α ∧¬ K ¬α. This is neither unacceptable nor unexpected if we keep present that the operator K is, in intuitionistic logic, a truth-operator; for it is a principle valid in general, i.e. for every internal notion of truth, that the formula expressing the proposition “ p is neither true nor false” is inconsistent. Take for instance the formula − T α & − T −α, expressing the same proposition within classical logic, and reason exactly in the same way as in (14), simply replacing ¬ with −, and ∧ with &. The crucial step is the

24An example is “Prime(n)”, where n is some very large number.

inference of T α from α; in other terms, the inconsistency of the formula expressing the proposition “α is neither true nor false” depends on the validity of the principle α → T α (together with propositional laws that are common to classical and intuitionistic logic). We have seen that the reason why that principle is intuitionistically valid is the assumption that proofs are epistemically transparent; of course this very assumption may be questioned,25 but the issue of its truth or falsity is utterly different from the question whether there are intuitive truths that, as a matter of fact, are unknown.

 
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