The Supermodel Argument

Bays(2008) notes that the key idea behind the ‘just-more-theory’ manoeuvre consists in the fact that any additional requirement imposed on a language’s referential re- lations can be viewed as just a new sentential theoretical or operational constraint addable to the language itself and up for reinterpretation. Realist attempts to fix the intended interpretation of the language have been met with a first-order regimentation

of their proposal later reinterpreted with Putnam’s favourite model theory. Against this too liberal use of the just-more-theory manoeuvre, Bays(2008) gives a trivial se- mantics accountable by just-more-theory-like reasoning – his Supermodel Argument. The argument is used as a dismissal of Putnam’s manoeuvre: either the supermodel is accounted by the manoeuvre working as a reductio, or the supermodel and the just-more-theory are refuted on the same grounds.

We first build Bays’supermodel G:

• Define a satisfaction relation for |=g that agrees with the classical first-order

recursive clauses for all the logical operators except for negation, i.e. for any model M and variable assignment v, the clause for ¬ is defined as M, v |=g

¬ϕ⇔M, v|=gϕ.

• Define a modelG=hD, Iisuch thatD={d}and all relation symbols are inter- preted maximally; i.e. for everyn-ary relationRandn-tupleshI(t1), ...I(tn)i ∈

Dn : hI(t1), ...I(tn)i ∈I(R).

Claim: Under the satisfaction relation|=g,Gis a trivial model: for allϕ,G|=gϕ.

Proof. The proof is by induction on the complexity of ϕ. Every atom is satisfiable since relation symbols have a maximal interpretation. For the negation case, we have by induction hypothesis G |=g ϕ which by the redundant negation clause implies

G |=g ¬ϕ. For conjunction, the truth of the conjuncts implies the truth of the

conjunction. Finally, for quantifiers, since the domain of G has only one element, G|=g∀xϕ⇔G|=g ∃xϕ⇔G|=gϕ(x/d).

If ‘satisfaction’ is understood as |=g, then G will trivially satisfy every sentence.

Consequently, it will satisfy all the theoretical-cum-operational constraints imposed on the language’s referential relations. It then follows that theoretical-cum-operational constraints do not seem to commit to the existence of more than one object; the supermodel G is such an example of an interpretation satisfying our constraints in a one-element model. Of course we may add additional requirements relative to, say, cardinality or imposing a more ‘natural’ semantic clause for negation. But by the ‘just-more-theory’ manoeuvre the extra requirements may, after being first-order regimented, be added to the total collection of sentences in the language and, after, subject to multiple reinterpretations. SinceGsatisfies everything, it will also satisfy those new requirements. For example, we may add a Multiplicity Constraint of the form:

Multiplicity Constraint: An intended interpretation must have more than one element.

Obviously, we may formalize the above claim by ∃x∃y(x 6=y), and, indeed, G will satisfy the requirement – after all, G |=g ∃x∃y(x6= y). Similar arguments may be

rehearsed for other cases. In the end, Bays(2008) concludes that the new constraints do not dismissGas unintended and, quite on the contrary, show that every theoretical- cum-operation constraint is trivially satisfiable inG.

The Supermodel Argument is meant as a reductio of Putnam’s manoeuvre. But to appreciate exactly what the argument is supposed to be a reductio of we need to look more closely at Putnam’s way to reinterpret additional theoretical constraints. Put- nam’s argument is dependent on the privileged semantics within which non-intended

interpretations are to be formed – for exploring the basic limitative meta-properties of first-order logic, classical standard model theory is required. Still, to model a lan- guage’s referential relations we may prefer to work with a stronger semantics, say, first-order model theory plus a Causality Constraint:

Causality Constraint: An intended interpretation must respect all the intended causal relations between words and referents.

Since, presumably, we sometimes wish to use different names for different objects, the interpretation G that assigns to all names the same denotation dG _{will fail to}

respect the causal links embedded in a speaker’s referential practice. When specifying the semantic clauses for a satisfaction relation we must account for these relevant causal links. Such a requirement can then be seen as specifying which kind of model theory should be used when interpreting our theoretical constraints, which can be immediately recognized as a substantially different project than that of merely adding to the language new theoretical constraints reinterpreted in the standard classical model theory that Putnam chooses to use. Putnam’s manoeuvre tends to confuse these distinct tasks, taking the Causality Constraint as an example of the latter and not of the former:

[...] we can view Putnam’s just-more-theory defence as an attempt to close the gap between the kinds of strong background semantics preferred by realists [e.g. classical first-order semantics plus a causality constraint] and the substantially weaker background semantics for the model-theoretic argument [...] by reducing the realist’s strong semantics to the first-order semantics needed for Putnam’s model theory. (Bays, 2008 : 202-203)

After Putnam has formalised the relevant constraints in first-order logic, there will be many possible interpretations of ‘Causality fixes reference’ (as there will be many interpretations of what ‘Causality’, ‘fixes’ and ‘reference’ denote), such that multiple deviant models will satisfy the requirement. The Supermodel Argument then shows that there is nothing unique about Putnam’s reduction: just as we may reduce real- ist’s strong semantics to Putnam’s preferred first-order semantics, so we may reduce Putnam’s semantics to|=g, the satisfiability relation with redundant negation.

For the model-theoretic sceptic the challenge is then to provide a way to refute G that does not refute Putnam’s manoeuvre at the same time. More precisely, to justify the reduction from the realist’s strong semantics to standard first-order semantics without at the same time accounting for the supermodel semantics, or any other kind of reduction.

Tim Button(private correspondence) locates the problem with the Supermodel G in its redundant negation operator. For let truthG

g be the property relation that

is formed by considering whatGsatisfiesg (where satisfiabilityg is satisfiability |=g).

Now, Button claims7that ‘Whatever truth is, it’s nottruthG_g’ After all: every sentence istrueGg, but weaccept some sentences andreject others’. If truth wastruthGg then

the assertion of ϕ would imply the assertion of ¬ϕ. However, asserting ϕ implies (though not always) the rejection of ¬ϕ. Essentially, truthG

g just does not conform

with our normal assertion and rejection practice. Hence, truth is not truthG g. 7_{Thanks to Tim here for permission to quote his comments.}