Truth Without World

Truth Without World

“In the natural sciences, when a theory is devised in some idealized domain, we ask whether the theory is true and try to answer the questions in various ways. Of course, we expect that the theory is probably false.”

¾ Noam Chomsky, Rules and Representations (p.104)

The problem

(i) Truth as external ground: Truth appears to be concept on which the very notion of rational thought pivots. The notions of right/wrong, justified/unjustified, valid/invalid, reliable/unreliable, meaningful/meaningless, etc. all seem to involve truth as an antecedent notion. In particular, truth is the external ground to which thought is answerable. The structure of a thought makes it apt to enter into the ‘space of reasons’, to stand in truth relevant relations with other thoughts.

(ii) Linguistic capacity - the language faculty - is a species-specific structure of the human brain. For theoretical purposes (science): a) to be linguistically competent is for one’s brain to realise an abstract function that recursively maps from a lexicon to an infinite set of structures that pair ‘instructions’ to phonological systems and intentional-conceptual systems; and b) a language is a steady state of an individual’s brain. Two consequences: b1) we each have our own individual language, qua brain state; b2) there are no constitutive external conditions on the language state.

The Thesis

Possession of the concept of truth is possession of the means to opaquely metarepresent bare phrase structures. A BPS becomes a representation as an element of a system which can be represented as a representation.

Truth: Deflationism

Varieties of deflationism

1. Redundancy theory (Ramsey, Ayer)

2. Performative theory (Strawson, Wittgenstein?)

3. Disquotationism (Tarski?, Quine, Leeds, Field)

4. Anaphoric theories (Grover, Brandom)

5. Propositional deflation theories (Horwich, Soames, M. Williams)

6. Rich Disquotation? (Frege, Davidson, McDowell, Wright, McGinn, Hornsby)

The Exhaustion Thesis

The content (sense/cognitive significance) of the concept of truth is exhausted by the content of that to which it applies.

Or:

Truths have no more in common than the adventitious occurrence of the same concepts in the complex contents to which the concept rightly applies.

The Good of Deflationism

(1) The Fundamental Intuition about Truth (FIT)

The truth of a sentence/proposition/etc. S depends upon two factors: (i) what it says and (ii) how the world is. If the world is as S says, then S is true, otherwise false.

Tarski (1933/36) first showed how FIT can be respected without appeal to (i) an ontology of facts/wordly state of affairs or (ii) an ideology of correspondence/saying.

(i) ‘Snow is white’ is true iff ‘Snow is white’ says that snow is white & it is a fact that snow is white.

(ii) (Pp)(It is a fact that p iff p).

(iii) (Pp)(‘p’ means that p).

(iv) ($x)(true(x) iff x = ‘Snow is white’ and snow is white).

(v) ‘Snow is white’ is true iff ‘Snow is white’ = ‘Snow is white’ and snow is white.

(vi) ‘Snow is white’ is true iff Snow is white.

Consequence I: The concept of truth is independent of metaphysical/epistemological dispute.

Consequence II: The articulation of the truth of a sentence S involves no metalinguistic predication over the lexical items of S; the truth conditions of S are articulated via the predication contained in S itself. But…

Consequence III: The concept of truth enshrines a metalinguistic role such that we can sally from language-talk to world-talk.

(2) Generality and Inference

A truth concept enables us generalise over clausal position; in effect, truth provides the service of a predicate for an otherwise stranded bound variable ranging over some set, all of whose members we wish to assert.

(i) If ‘2+2 = 4’ is a theorem of PA, then 2+2 = 4.

(ii) For all x, if x is a theorem of PA, then x.

(iii) *("x)(PA(x) → x)

(iv) ("x)(PA(x) → True(x))

Pronouns allow for a similar service in generalising over nominal position; where, regimented, the pronouns function as variables.

(i) If Bill is over 6ft, then Bill is tall.

(ii) If one is over 6ft, then one is tall.

(iii) ("x)(FT(x) → Tall(x))

Generalising…

In ND:

1 (1) ("x)(F(x) → True(x)) A

2 (2) F(p) A

1 (3) F(p) → True(p) 1UE

1,2(4) True(p) 2,3MPP

1,2(5) True(p) ↔ p 4Tdf

1,2(6) True(p) → p ↔df

1,2(7) p 4,5MPP

A similar derivation for existential generalisation is obviously invalid (at line (7)):

1(1) ($x)(F(x) Ù True(x)) A

2(2) F(p) Ù True(p) A

2(3) True(p) 2ÙE

2(4) True(p) ↔ p 3Tdf

2(5) True(p) → p ↔df

2(6) p 3,5MPP

1(7) p 1,2,6EE

The thought is that the concept of truth plays the functional role of encapsulating infinite conjunctions/disjunctions. Content-wise, the two generalisations give way to (1) and (2) respectively, in line with the exhaustion thesis:

(1) If F(p₁), then p₁; and if F(p₂), then p₂;…

(2) Either F(p₁) and p₁; or F(p₂) and p₂;…

The Problem with Deflationism: Transparency

Transparency thesis: Truth is only (at best) syntactically metarepresentational; conceptually, it is flat with that to which it applies.

Problem I: Incorporation

(i) ‘Der Schnee ist weiss’ is true.

(ii) ‘Every semi-stable elliptical curve is a modular form’ is true.

(iii) The Continuum Hypothesis is true.

Problem II: Opaque Generalisation

A Triviality: To generalise is to relinquish any independent access to the objects of the domain over which the generalisation holds.

Consequence: Content-wise, for the cognitive agent, truth generalisations are not flat with that to which truth is applied.

To see this explicitly/formally…

Deflationary assumption I: Generalisations are functional bindings of positions in a Boolean structure, e.g., an open first-order formula.

Deflationary assumption II: QNPs are incomplete at logical form.

(A) All QNP structures are conservative.

Where D is a natural language quantifier, and D is a function from a nominal set to a predicate set to a cardinal value:

(CONSERVE) If D_k is a k-place determiner expressing a function f: P(U) µ D, then D_k is conservative iff, for all nominal arguments N₁,…N_k and all VP arguments A, B, if │N_i ∩ A│ = │N_i ∩ B│, then D[N_i](A) ↔ D[Ni](B).

In simple terms: if two predicate sets A, B have the same intersection with the same nominal sets, then the D-function defined on the nominals over the sets takes the same value with predicate A as it does with predicate B. So, where k = 1, for all sets X, Y, (i) holds:

(i) D[X](Y) ↔ D[X](X ∩ Y).

(ii)a Every boy runs ↔ Every boy is a boy that runs.

b Most boys run ↔ Most boys are boys that run.

c Bob’s son runs ↔ Bob’s son is a son that runs.

Consequence: The truth of a generalisation “lives on” the character of the members of the nominal set.

Might non-members be relevant? No…

(B) All QNP structures are extended.

(EXT) If D is a Det expressing a function f : P(U) µ D(U), for each universe U, then D is extended iff "U, U*, where X, Y f U and X, Y f U*, D_U[X](Y) ↔ D_U_*[X](Y).

In simple terms: the value of every QNP structure is invariant over universes so long as its argument sets are invariant over the universes.

Consider:

(1) Everything but boys run ↔ Everything but boys are boys that run

(2)a NALL[X](Y) is T iff {U - X} f Y

b f_NALL[U - X](Y) ↔ f_NALL[U - X]( U - X ∩ Y)

c Every non-boy runs ↔ Every non-boy is a non-boy that runs.

(D) QNPs are not sortally reducible

(SR) If D is a Det defined over P(U), then D[X](Y) is sortally reducible just if there is

some Boolean function f such that D[X](Y) ↔ D(U)f(X, Y).

In simple terms, this means that the value of the formula just depends on either {X ∩ Y} or {X -Y}. Consequently, the nominal restriction can be canceled by a Boolean compound of the two sets expressed by some first-order relation between open sentences. Sortal reducibility, however, is not a defining semantic characteristic of Det+NP constructions; indeed, it does not even equate with logicality, for logicality does not equate with being first-order.

Consequence I: QNPs are not LF incomplete

Consequence II: QNP formulae do not give way to their ‘instances’ as a matter of LF syntax.

Moral: Set theoretical analysis of the structure of QNPs shows that first-order representation is inadequate - the representation upon which deflationism is based.

(Radically) Bare Phrase Structures

{{some, boy} {{some, boy} {{likes, v} {likes, himself}}}}

Second Phase

{some, boy} {{some, boy} {{likes, v} {likes, himself}}}

Some boy {some, boy} {{likes, v} {likes, himself}}

<some boy> likes+v {likes, himself}

Copying from the edge of the first phase to [SPEC CP]

[SPEC CP]

Copying from V head to

incorporation with v

<likes> himself

Derivation by Phase

A phase of a derivation is a segment of structure that satisfies interface conditions and so can spell-out to external systems. Chomsky’s hypothesis is that phases are either

(i) v*P (an abstract element that supports all argument roles of a verb)

(ii) CP (a clause that supports a v*P and tense and discourse features).

Think of v*P as the first phase which, once constructed, is frozen, but its edge is open to copying higher up to form the second phase. The elements of the first phase are interpreted conceptually, e.g., compositional event structure. The elements of the second phase determine morphological form (semantically irrelevant) and discourse parameters of both phases. Phases iterate.

The Architecture of the Language faculty

LEXICON

[NUMERATION]

First Phase Ü Verb Structure

Discourse Structure Þ Second Phase

The Role of Truth

A hypothesis that might be true: Truth is that mechanism that turns structures (RBPSs) into representations.

The mechanism makes available a structure to carry assertoric, interrogative, imperative, opative, content.

A structure accessible to truth becomes a representation - it is representational only as apt to be metarepresented.

TM: The representation of a representation as a representation.

OM: The representation of a representation as a structure.

TM: Carruthers, Dennett, Recanati, Grice, Sperber & Wilson, Carston, (more or less) every deflationist.

OM: John Collins, Tarski.

Thesis: TM f OM

OM = TM iff the represented structure is transferable.

Reasons for Optimism

1. Truth is opaque: bare clausal form supports the attribution of truth.

2. Truth incorporates alien structures, again in bare form.

3. The disquotational effect drops out for those structures non-incorporated.

4. The failure of the deflationary construal of generalisations is respected: the general

terms are copied high and fix the domain over which truth may opaquely operate.

5. Tarski’s ‘truth from meaning platitude’ is respected.