Proof-Theory and Ontological Commitment

The Cut rule read from bottom to top says that for any sentence ϕ if Γ ñ Σ is a coherent position then one of Γ, ϕ ñ Σ and Γ ñ ϕ, Σ is coherent. Cut entails the claim that any coherent position can be filled in to a coherent position that either 6_{Bonevac [5] suggests that this account of quantification is a formal treatment of a view endorsed}

by Sellars [62]. This is suggested by the following passage from that (1948) article.

It has not always, however, been realized that this train of thought leads directly to the conclusion that our language claims somehow to contain a designation for every element in every state of affairs, past, present, and future; that, in other words, it claims to mirror the world by a complete and systematic one-to-one correspondence of designations with individuals. If it is obvious that our language does not explicitly contain such designations (and it would hardly be illuminating to say that it contains them implicitly), it is equally clear that our language behaves as though it contained them. [62, pg. 603]

More evidence that Sellars would have endorsed such a view can be found in Sellars [67]. There he argues that an adequate account of quantification should reduce the “indefinite reference” of quantified sentences to the “definite reference” of unquantified ones. In a footnote, he suggests that this is best accomplished by considering expansions of a language. Sellars [63, 67] spends considerable efforts arguing against Quine’s dictum. Similar remarks to those made by Sellars [63] are made by Prior [45]. The results of this chapter can be seen as a formal explanation of the insights of those philosophers.

asserts or denies any sentence ofL . The LD rule read from bottom to top says that if Γ, Dxϕ ñ Σ is coherent then it is coherent to introduce intoL a term t to serve as a witness for that existential, i.e. the position Γ, Dxϕ, ϕrt{xs ñ Σ is coherent where t does not occur in ΓŤtDxϕuŤ Σ.

A position Γ ñ Σ is maximal iff for any sentence ϕ it either asserts ϕ or it denies ϕ, i.e. ϕ P Γ or ϕ P Σ. One position Γ ñ Σ is a superposition of another ∆ ñ Λ iff Γ ñ Σ asserts all the sentences that ∆ ñ Λ asserts and denies all the sentences that ∆ ñ Λ denies, i.e. ∆ Ď Γ and Λ Ď Σ. The set of maximal coherent superpositions of a position Γ ñ Σ, written OpΓ ñ Σq, are used to determine the ontological commitments of Γ ñ Σ.

Above it was noted that the rule of Cut entails that if a position Γ ñ Σ is coherent then there is a coherent position ∆ ñ Λ such that ∆ ñ Λ is a maximal superposition of Γ ñ Σ. The Cut rule entails the stronger claim that if Γ ñ Σ is a coherent position ofL then if L1 _{is an expansion of} L there is a maximal coherent superposition of Γ ñ Σ in the language of L1_{. All the rules of fig. 5.1 are valid in} any expansion of the language L by additional expressive resources.

In order to generate OpΓ ñ Σq a tree is constructed with Γ ñ Σ at its root. Let W be a denumerably infinite set of witness terms. The witness terms are those expressions which are not in L but are in expansions of L . The elements of W will extend the language L in the course of the construction. They serve, as in standard completeness proofs, as the witnesses to the truth of existentially quantified sentences. Let c1, . . . , cn, . . . be an ordering on the set N and ϕ1, . . . , ϕn, . . . be an ordering on the set of sentences ofL . A leaf ∆ ñ Λ of the tree under construction

is closed iff $ ∆ ñ Λ. A leaf is open iff it is not closed.

The construction of the tree proceeds in stages. Each stage consists of sub- stages which themselves consist of sub-sub-stages. Let ϕi be the ith sentence in the ordering. At stage i consider the left-most open leaf of the construction that has not been considered at stage i. Let ∆ ñ Λ be the nth _{such open leaf, call this sub-stage} i.n. At this stage extend the tree according to the following rules.

(a) If ϕj is of the form Dxψ, ϕj P ∆, and there is no sentence of the form ψrw{xs in ∆, then take the least element of W that does not appear in ∆Ť Λ, wk, and extend the branch by

∆, ϕrwk{xs ñ Λ ∆ ñ Λ Call this stage i.n.j.

(b) If & ∆, ϕi ñ Σ and & ∆ ñ ϕi, Σ then extend the branch by ∆, ϕi ñ Λ ∆ ñ ϕi, Λ

∆ ñ Λ Call this stage i.n.0.

(c) If $ ∆, ϕi ñ Λ then extend the branch by ∆ ñ ϕi, Λ

∆ ñ Λ Call this stage i.n.0

∆, ϕi ñ Λ ∆ ñ Λ Call this stage i.n.0

Let τ pΓ ñ Σq be a tree constructed in the above way with Γ ñ Σ as its root. Let Γ ñ Σ, ∆1 ñ Λ1, . . . , ∆n ñ Λn, . . . be an open branch in this tree. Call Ť

i∆i ñ Ť

iΛi the maximal leaf of this branch.

Definition 24 (Maximal Coherent Superposition). If Γ ñ Σ is a position then the set of maximal coherent superpositions of that position OpΓ ñ Σq is the set of maximal leaves of open branches in τ pΓ ñ Σq.

A position is a reckoning of how things are. A maximal position, if coherent, is an account of how everything is. The maximal coherent positions leave nothing unsaid. They are complete stories of how the world could be. The set OpΓ ñ Σq is the set of complete stories of the world that the position Γ ñ Σ could tell. The complete stories that a position could tell reveal the ontological commitments of that position. If on every complete way of filling in a position a sentence of the form F a is asserted, then that position is committed to asserting that there are F ’s.

Theorem 5.3.2. For any sentence ϕ that does not contain elements of W , it holds that $ Γ ñ ϕ, Σ iff for every ∆ ñ Λ P OpΓ ñ Σq, ϕ P ∆.

Theorem 5.3.2 says that for a position Γ ñ Σ it is incoherent to deny a sentence ϕ ofL while asserting all of Γ and denying all of Σ iff ϕ is asserted in every maximal coherent superposition of Γ ñ Σ. The sentences that a position rules out denying are the sentences that each of its maximal coherent superpositions asserts. Restall [52]

suggests that a position Γ ñ Σ is committed to a sentence ϕ when it is incoherent to maintain that position and deny ϕ, i.e. $ Γ ñ ϕ, Σ. For instance, any position asserting ϕ^ψ is committed to the sentence ϕ because it is incoherent to assert ϕ^ψ and deny ϕ, $ ϕ ^ ψ ñ ϕ. This account of commitment avoids it being the case that a person must know, assert, or explicitly believe what they are committed to. A person is committed to what, given their assertions and denials, it is incoherent for them to deny.

One of the benefits of this account of commitment is that it makes clear the com- mitments of a position that asserts a disjunction. A position that asserts ϕ _ ψ is not committed to either ϕ or to ψ, though it is incoherent to deny both at the same time. It is natural to move from commitment to a sentence to commitment to a set of sentences. Any position that asserts ϕ _ ψ is set-committed to tϕ, ψu. The disad- vantage of this approach is made clear by the consideration of quantified sentences. In both of the above examples the commitments of a sentence could be expressed in terms of their sub-sentences. A position that asserts Dxϕ is not committed to any sentence of the form ϕrt{xs and not even set-committed to any set of the form tϕrt0{xs, ϕrt1{xs, . . .u. The simple substitutional approach to quantification enforces the latter set-commitment of an existential sentence. Theorem 5.3.2shows that Re- stall’s account of commitment can be captured by considering the set of maximal coherent superpositions of a position. For instance, it follows from theorem 5.3.2

that if a position Γ ñ Σ asserts ϕ ^ ψ then ϕ appears in every maximal coherent superposition of Γ ñ Σ. The commitments of a position which asserts an existen- tial sentence can be described in terms of the instances of that existential sentence

appearing in maximal coherent superpositions of the original position. The ability to adequately capture the commitments incurred by asserting a sentence in terms of its sub-sentences is what makes it possible to generate an account of the ontological commitments of a position.

Definition 25 (Ontological Commitment). A position Γ ñ Σ is ontologically com- mitted to there being entities of which K is true iff for each ∆ ñ Λ P OpΓ ñ Σq there is a name or witness t such that Kvt{ξw P ∆.

On this account of ontological commitment a position is committed to there being an entity of which K is true iff on every maximally coherent way of filling in that position a sentence of the form Kvt{ξw is true, i.e. K is true of something. Ontological commitment to a kind is commitment to naming something of which K is true if one were to tell the complete story of the world. According to this view to be assumed as an entity is to be reckoned nameable in all maximal coherent superpositions. This account of ontological commitment does not require that variables be given values or even that variables have “ranges”. In order to learn the ontological commitments of a position, one does not need to explore the ways that variables are matched to objects in a domain, but to explore what objects are named in the maximal coherent superpositions of that position.7

Expanding substitutional quantifiers when paired with this account of ontological 7_{This account of ontological commitment shows formally a way of accomplishing what Sellars}

[67] was hoping to accomplish. He argues that the ontological commitments of a quantified sentence (one that he says “refers indeterminately”) ultimately must rest on the non-quantified instances of that sentence (sentences that he says “refer determinately”). In Sellars’s terminology this account of ontological commitment is an explanation of how ‘indirect reference’ can be explained in terms of ‘direct reference’.

commitment treat variables as little more than bookkeeping devices. There is no spe- cial relation that variables have to domains in a model or to ontological commitment generally. Variables, and the quantifiers that bind them, mark which sentences may be asserted or denied in complete stories of the world. The above construction makes plain that the ontological commitments of a theory whose quantifiers are interpreted in this way need not bear a special relation to a domain of quantification.

Call the account of ontological commitment described in this section proof-theoretic ontological commitment and call the account of the previous section (section 5.2) model-theoretic ontological commitment. Before considering extensions of first-order logic the proof theoretic and the Quinean account of ontological commitment agree with one another.

Theorem 5.3.3. A set of sentences Γ has a model-theoretic ontological commitment to K iff the position Γ ñ has a proof-theoretic commitment to K.