(The introduction of even a single dyadic predicate variable, however, would make the system undecidable, and, in fact, even the system that contains only a single dyadic predicate variable and no other predicate variables at all has been shown to be undecidable).
If one negates just a predicate, that leaves open the possibility that some other predicate can suitably be applied to the subject in question.
Indeed, Quine could also have given the (radical) Meinongian the word 'is' and any other predicate that purports to impose an ontological restriction on the quantifier; for what is central to Quine's criterion is that one cannot quantify over entities without incurring ontological commitment to those entities.
Leśniewski knew that other predicates than 'ε' could be taken as primitive, a fact emphasized later by Lejewski.
However, a feature contains both a predicate and an object, and some predicates appear frequently connecting many different objects whereas other predicates appear seldomly.
In order to treat truth like other predicates, one will add the axiom ∀A(T[TA] ↔ TA) (where ∀A ranges over all sentences).
Some predicates (which are the ones like "green") can be used for this; other predicates (the ones like "grue") must be excluded, if induction is supposed to make any sense.
As shown in the last predicate in the rule in Listing 2, a predicate can embed other predicates to show more complicated relations: the predicate "greater_than_or_equal(Ceiling_height,quantity 7,feet))" represents the quantitative relation between ceiling height and the quantity 7 feet, where the predicate "quantity 7,feet))" is embedded as an argument.
In terms of consequence relations, circumscription allows us to define, for each predicate P, a non-monotonic relation A(P) φ that holds precisely when A*(P) ⊨ φ. (This basic form of circumscription has been generalized, for, in practice, one needs to minimize the extension of a predicate, while allowing the extension of certain other predicates to vary).
Goodman recognizes the temptation to dismiss predicates like "schmelicious" as artificial or contrived, but insists that such artificiality is simply a consequence of the "entrenchment" in our language of other predicates we accept, rather than a language-independent fact about the world.
In (GH) a mixed sentence φ is derived from a set of purely descriptive sentences (i.e., sentences free of O) only if φ is completely O-irrelevant (that is, predicates in φ within the scope of O can be replaced by other predicates salva valididate).
