(p.252) Appendix
(p.252) Appendix
§A.1 Minimax Rules
If we take (single conclusion) sequent rules to have the schematic form:
X ╞ C iff in all models M if all of X are true in M then C is true and if C is false and all of X but P are true, then P is false.)
For if all of $\underset{0\u3008i\le n}{\cup}}{X}_{i$ are true in valuation v then so are all the P_{i}, since all the premiss sequents are neoclassically correct; hence so is C by the minimax entailment. Whilst if C is false in v but all of the wffs in $\underset{0\u3008i\le n}{\cup}}{X}_{i$ but A are true in v then since the P_{i} minimax entail C, at least one of the premiss succeedents is false, say P_{k}; in which case by the neoclassical correctness of X_{k} ⇒ P_{k}, A ∊ X_{k} and is false in v.
Minimax soundness is a mechanically decidable principle, for finite sequent rules. If we wish to set out rules for it, one set has the general form:
X
(1) φ
Given
X
(2) φ[P/Q]
1 MM
(‘MM’ for Minimax) where φ[P/Q] results from φ by uniform substitution of a subformula P by Q and where (p.253)
P is
A
and Q is
∼∼A
or P is
∼∼A
and Q is
A
or P is
A & B
and Q is
B & A
or P is
A ∨ B
and Q is
B ∨ A
or P is
A & B
and Q is
∼(∼A ∨ ∼B)
or P is
∼(∼A ∨ ∼B)
and Q is
A & B
or P is
A ∨ B
and Q is
∼(∼A & ∼B)
or P is
∼(∼A & ∼B)
and Q is
A ∨ B
or P is
(A & (B & C))
and Q is
((A & B) & C)
or P is
((A & B) & C)
and Q is
(A & (B & C))
or P is
(A ∨ (B ∨ C))
and Q is
((A ∨ B) ∨ C)
or P is
((A ∨ B) ∨ C)
and Q is
(A ∨ (B ∨ C))
or P is
(A & (B ∨ C))
and Q is
((A & B) ∨ (A & C))
or P is
((A & B) ∨ (A & C))
and Q is
(A & (B ∨ C))
or P is
(A ∨ (B & C))
and Q is
((A ∨ B) & (A ∨ C))
or P is
((A ∨ B) & (A ∨ C))
and Q is
(A ∨ (B & C))
Minimax soundness is easily established by showing that the formula on the left has the same value, in every model, as the formula on the right. In addition standard sequent form natural deduction ∨I, &E and &I rules (without any determinacy restriction on overlapping assumptions in both premiss antecedents) are minimax sound as is the Mingle rule:
which is a derived rule of the logic RM which extends relevant logic R by addition of the Mingle axiom scheme.^{1}
X
(1) A & ∼A
Given
X
(2) B ∨ ∼B
1 Mingle
Most of the minimax rules generalize straightforwardly to the infinitary case. (The distributivity rules are an exception: to extend completeness results for standard languages any significant way into the transfinite one needs extra rules such as the ‘ordinary’ or ‘Chang’ distributivity laws (Karp 1964, p. 41; Dickmann 1975, p. 421). The distributivity schema, for all γ 〈 κ, κ the index of the language, is :
§A.2 Soundness of Strict Neoclassical Logic
The proof is of the usual inductive type. The semantics for the operators is the Lukasiewicz/Kleene semantics of Chapter 7. As illustration, the inductive step for the ∨E rule:
where (X ∩ (Y ∪ Z)) = Ø is as follows:
X
(1) P ∨ Q
Given
Y,P
(2) C
Given
Z,Q
(3) C
Given
X,Y,Z,
(5) C
1,2,3 ∨E
Proof: (i) Truth preservation: this is exactly as in the classical case. (ii) Falsity preservation (upwards): suppose that C is false in v and all of X,Y,Z, are true in v but A.
Case (a): A ∉ X. Then, by the (strict) neoclassical correctness of line (1) (inductive hypothesis) P ∨ Q is true in v hence one of the disjuncts is; suppose without loss of generality that it is P. By the correctness of line (2) in the rule above, A ∊ Y and is false.
Case (b): A ∉ Y ∪ Z. By the correctness of lines (2) and (3) both P and Q are false in v hence so is P ∨ Q. By the correctness of line (1), A ∊ X and is false in v. By the overlap clause (X ∩ (Y ∪ Z)) = Ø, these two options are exhaustive.^{2} ⃞
A similar, somewhat less convoluted, proof establishes the soundness of ∼E. The proof for ∨E generalizes to the parallel ∃E rule:
with the usual clauses on t and where X ∩ Y = Ø. Again truthpreservation is classical.
X
(1) ∃xφx
Given
Y, φx/t
(2) C
Given
X,Y,
(4) C
1,2 ∃E
Falsity preservation. Suppose all of X,Y, but A are satisfied by σ (in model M) but C is false relative to σ.
Case (a) A ∉ X. Then ∃xφx is satisfied by σ hence for some α in the domain of M, φx is satisfied by σ[x/α] hence, since t is not in φ, φx/t is satisfied by σ[t/α]. Since t is not in Y or C, all of Y but A are satisfied by σ[t/α] along with φx/t whilst C is falsified by that variant assignment (this is provable by induction on sentence complexity). Hence by the correctness of line 2, A ∊ Y and is falsified by σ[t/α] hence, using the same lemma, by σ.
Case (b) A ∉ Y. Then by the correctness of 2, φx/t is falsified by σ hence so is φx (since t is new to φ) by σ[x/σ(t)]. Since t does not occur in Y or C the same holds (p.255) for any tvariant σ′ which agrees with σ on all variables and parameters except perhaps t. Thus φx is falsified by every σ[x/α] for all α in the domain so that ∃xφx is falsified by σ. By the correctness of line 1, A ∊ X and is falsified by σ.
By the overlap clause (X ∩ Y) = Ø, these two options are exhaustive. □
Soundness for the conditional rules is complicated by the fact that we require them to be sound under two interpretations of the conditional. Firstly, the nonbivalent conditional where a conditional with true antecedent and untrue consequent, or false consequent and unfalse antecedent, takes the gap value. Secondly, the bivalent conditional in which, in those two cases, the conditional is false. However the soundness proofs for the strict neoclassical rules:
→I:
→E:
P
(1) Q
Given
—
(2) P → Q
1 → I
where as usual X ∩ Y = Ø, goes through in essentially the same way in either case. The nonclassical cases are the falsitypreservation clauses:
X
(1) P → Q
Given
Y
(2) P
Given
X, Y,
(3) Q
1,2 →E

(a) Falsitypreservation and truth preservation, →I: suppose P → Q is untrue in valuation v (in the bivalent case, it will be false). Then either P is true at v and Q untrue, or Q is false at v and P unfalse; both these possibilities contradict the neoclassical correctness of line 1, hence P → Q cannot be untrue at v.

(b) Falsity preservation →E: suppose Q is false and all of X, Y, but A are true.

Case (i): A ∉ X. Then, by the correctness of line (1) P → Q is true in v so that (on either interpretation) P is false. By the correctness of line (2) in the rule above, A ∊ Y and is false.

Case (ii): A ∉ Y. By the correctness of line (2), P is true in v. Since Q is false, so too (on either interpretation) is P → Q. By the correctness of line (1), A ∊ X and is false in v. This exhausts the possibilities. ≤

The two other principles taken as primitive for the conditional^{3} are:
Contraposition:
X
(1) P → Q
Given
X
(2) ∼Q → ∼P
1, Contrap.
(p.256) Transitivity:
X
(1) P → Q
Given
Y
(2) Q → R
Given
X, Y,
(3) P → R
1,2 Trans. (where X ∩ Y = Ø.)
Soundness for contraposition is straightforward since the respective succeedents in each rule take the same value in every neoclassical model, on either interpretation. For transitivity, in the truthpreservation direction, suppose all of X, Y, are true in v. Then P → Q and Q → R are true. On either interpretation of →, if P is true in v, so is R, if P is gappy, Q is not false hence neither is R. Thus in either case, and also in the third case in which P is false in v, P → R is true (again on either interpretation of →).
For the falsity preservation direction, take the nonbivalent interpretation first of all. If P → R is false in v then P is true in v, R false. Suppose all of X, Y are true there but A. There are two cases:
Case (i): A ∉ X. Then, by the correctness of line (1) P → Q is true in v hence so is Q, so Q → R is false. By the correctness of line (2), A ∊ Y and is false.
Case (ii): A ∉ Y. By the correctness of line (2), Q → R is true in v, hence Q is false so by the correctness of line (1), A ∊ X and is false in v. This exhausts the possibilities.
Next the bivalent interpretation of →. If P → R is false in v then P is not false and R not true, nor do they have the same truth value in v. There are two cases:
Case (i): A ∉ X. Then, by the correctness of line (1) P → Q is true in v. If P is true, Q is true but R is not hence Q → R is false in v. If P is gappy, Q is not false but R is hence once again, Q → R is false in v. Either way, by the correctness of line (2), A ∊ Y and is false.
Case (ii): A ∉ Y. By the correctness of line (2), Q → R is true in v. Hence Q is not true. If Q is false then since P is not false, P → Q is false in v. If Q is gappy, R is gappy, hence P is true so once again P → Q is false. Either way, by the correctness of line (1), A ∊ X and is false in v. This exhausts the possibilities. □
§A.3 Infinitary Rules
The infinitary generalizations of &E and ∨I are obvious. For the other two rules they are ∧I:
whilst ∨E is:
X_{α}
(1.α) A_{α}
Given for all α 〈 β
$\underset{\text{\alpha}\text{\u3008}\beta}{\cup}}{\text{X}}_{\text{\alpha}$
(2) ∧A_{α α 〈 β}
1.α α〈β ∧I
(p.257) where $\text{X}\cap {\displaystyle \underset{\text{\alpha}\u3008\beta}{\cup}}{\text{Y}}_{\text{\alpha}}=\xd8$.
X
(1) ∨A_{α α 〈 β}
Given
Y_{α}, A_{α}
(2.α) C
Given, α 〈 β
$\text{X},{\displaystyle \underset{\text{\alpha}\u3008\beta}{\cup}}{\text{Y}}_{\text{\alpha}}$
(3) C
1, 2.α, α 〈 β ∨E
The soundness steps in the soundness proof are straightforward generalizations of the finitary case. Thus for ∨E:

(i) Truth preservation is exactly as in the classical case.

(ii) Falsity preservation (upwards): suppose that C is false in v and all of X and all in $\underset{\text{\alpha}\u3008\beta}{\cup}}{\text{Y}}_{\text{\alpha}$ are true in v but P.

Case (a): P ∉ X. Then, by the (strict) neoclassical correctness of line (1) (inductive hypothesis) ∨A_{α α} _{〈 β} is true in v hence one of the disjuncts is; suppose without loss of generality that it is A_{γ}. By the correctness of line (2.γ) in the rule above, P ∊ Y_{γ} and is false.

Case (b): $\text{P}\notin {\displaystyle \underset{\text{\alpha}\u3008\beta}{\cup}}{\text{Y}}_{\text{\alpha}}$. By the correctness of all of lines (2.α), α 〈 β, all of the A_{α}, α 〈 β are false in v hence so is ∨A_{α α} _{〈 β}. By the correctness of line (1), P ∊ X and is false in v.
By the overlap clause $\text{X}\cap {\displaystyle \underset{\text{\alpha}\u3008\beta}{\cup}}{\text{Y}}_{\text{\alpha}}=\xd8$, these two options are exhaustive. □

§A.4 IS
As set out in Chapter 8, §VI, with h a bijection from the θ_{β}sized set of singular terms of the language L_{κ} of IS onto V_{θβ}, we write t_{i} for h(t_{i}). To prove negationcompleteness for the atoms of L_{κ} we prove something stronger: that every truth of the diagram of V_{θβ}, every atomic truth or falsehood concerning identity or membership, is provable.
Firstly, the positive truths. If u_{j} ∊ t_{i}^{4} then u_{j} occurs in a disjunct in the right hand side of the ith list comprehension axiom:
We then have ├ u_{j} ∊ t_{i} by strict neoclassical reasoning using ∀E (∧E), =I, ∨I and ↔E (i.e. &E and →E).^{5} Moreover since each set in V_{θβ} has only one name, the positive identity truths are all provable by =I, reflexivity of identity.
For the negative truths, we reason by induction on the rank of terms t_{i} in u_{j} ≠ t_{i} and u_{j} ∉ t_{i} (where the rank of t_{i} is just the rank of h(t_{i}) in V_{θβ}). For rank zero, we have ├ t_{i} ∉ t_{0}, for all t_{i} by a strict neoclassically legitimate reductio, using the empty set list axiom and =I.
Turning to identity, u_{j} = t_{0} is false, for all u_{j} distinct from the empty set, but in that case u_{j} has at least one member t_{k} so that t_{k} features in the righthand side of the ith (p.258) comprehension axiom and we thereby have ├ t_{k} ∊ u_{j}. Since ├ t_{k} ∉ t_{0}, identity laws and ∼I give us a neoclassical proof of u_{j} ≠ t_{0}.
For the inductive step of this part of the proof, we assume completeness for all identity and membership sentences for terms of rank 〈 α, α 〉 0, (each is provable if true in V_{θβ}); we have then to show (i) ├ u_{k} ∉ t_{α} and (ii) ├ u_{j} ≠ t_{α}, where u_{k} ∉ t_{α} and u_{j} ≠ t_{α}, for terms u_{j} and u_{k} of rank 〈 α.
(i) Since u_{k} ∉ t_{α}, u_{k} is distinct from every u_{l} ∊ t_{α}. By IH, ├ u_{k} ≠ u_{l} for every such t_{l}, l ∊ L, L indexing the members of t_{α}. Hence ├ u_{k} ∉ t_{α} by the following neoclassically correct proof (here the α^{th} comprehension axiom is:
1 
(1) u_{k} ∊ t_{α} 
H 
1 
(2) ∨(u_{k} = u_{l}) l ∊ L 
1 CE 
3.l 
(3.l) u_{k} = u_{l} 
l ∊ L 
— 
(4.l) u_{k} ≠ u_{l} 
Given l ∊ L 
3.l 
(5.l) ⊥ 
3.l, 4.l ∼E 
1 
(6) ⊥ 
2, 5.l l ∊ L, ∨E 
— 
(7) u_{k} ∉ t_{α} 
6 ∼I. 
(ii) If u_{j} ≠ t_{α} then they have different members. Suppose that t_{k} ∊ u_{j} but t_{k} ∉ t_{α}. By what we have just established, ├ t_{k} ∊ u_{j} and ├ t_{k} ∉ t_{α}. A neoclassically reductio establishes ├ u_{j} ≠ t_{α}. If instead t_{k} ∉ u_{j} but t_{k} ∊ t_{α} the argument is the same since u_{j} is of rank 〈 α. Hence it follows by induction (using symmetry of = and the proof in (i) above for u_{j}, t_{k} j 〉 k) that every atomic sentence and every negation of an atomic sentence, whatever the ranks of the terms in the sentences, is provable when true.
The argument that all V_{θβ} truths are provable generalizes from the proof for atomic sentences by the same inductive argument as in Chapter 8, §VI.
§A.5 Prime Extensions
Start with a theory T which is satisfied by a neoclassical model M. Then it can be expanded to a prime theory T* also satisfied by M.
Proof: At initial stage ⟨ 0,0⟩ we set Δ_{⟨ 0,0⟩} = T. Wellorder all the disjunctions in the language (so we require choice). At stage ⟨ 0,α + 1⟩ we consider the αth disjunction ∨P_{i}. If Δ_{⟨ 0,α⟩} ├ ∨P_{i} then (using choice again) select an arbitrary disjunct P_{k} such that P_{k} is true in M (an arbitrary sentence otherwise—but we will prove there is no otherwise) and let Δ_{⟨ 0,α + 1⟩} = Δ_{⟨ 0,α⟩}, P_{k}. If it is not the case that Δ_{⟨ 0,α⟩} ├ ∨P_{i} then Δ _{⟨ 0,α + 1⟩} = Δ_{⟨ 0,α⟩}. At limit stages E9⟨ 0, λ⟩ Δ_{〈 0,λ〉} = $\underset{\beta \u3008\lambda}{\cup}}{\Delta}_{\u30080,\beta \u3009$ whilst Δ_{〈1,0〉} = $\underset{\beta \u3008\kappa}{\cup}}{\Delta}_{\u30080,\beta \u3009$.
An inductive proof shows that at each stage α up to and including 〈 1,0〉 the associated set Δ_{〈 0,α〉} is satisfied by M and hence if Δ_{〈 0,α〉} ├ ∨P_{i} there always is at least one disjunct P_{k} such that Δ_{〈 0,α〉}, P_{k} is true in M, since Δ_{〈 0,α〉} is satisfied by M.
(p.259) IH: We have by IH that Δ_{〈 0,α〉} is satisfied by M. If it is not the case that Δ_{〈 0,α〉} ├ ∨P_{i} (as before, this the α^{th} disjunction) then Δ_{〈 0,α+1〉} = Δ_{〈 0,α〉} and so all its members are true in M. If Δ_{〈0,α〉} ├ ∨P_{i} then ∨P_{i} is true in M, hence at least one disjunct is true in M. Whichever true disjunct P_{k} is selected, Δ_{〈 0,α+1〉} = Δ_{〈 0,α〉}, P_{k} is satisfied by M. At limit stages we have by IH that every member of Δ_{〈 0,λ〉} = $\underset{\beta \u3008\lambda}{\cup}}{\Delta}_{\u30080,\beta \u3009$ is true in M hence Δ_{〈 0,λ〉} is satisfied by M, likewise for Δ_{〈 1,0〉}.
At stage 〈 1,0 〉 we proceed as before up to stage 〈 2,0 〉 and so on, with at stage 〈 μ, 0〉, Δ_{〈 μ,0〉} = $\underset{\eta \u3008\mu ,\beta \u3008\kappa}{\cup}}{\Delta}_{\u3008\eta ,\beta \u3009$. Since our language is of a standard settheoretic size κ, cardinality considerations show that we must reach a fixed point π such that Δ_{〈 π+1,0〉} = Δ_{〈 π,0〉}. Call this theory Δ_{〈 π+1,0〉}, which we have seen is also satisfied by M, T*. By dint of the construction, if T* ├ ∨Q_{i}, ∨Q_{i} any disjunction, then for some disjunct Q_{k}, T* ├ Q_{k}.
Note that if φ is gappy in M then, since T* is satisfied by M, we have neither T* ├ φ nor T* ├ ∼φ nor T* ├ (φ ∨ ∼φ) since T* is prime. As an example, take ZFCU and consider a classical model M (no gaps, so the positive and negative extensions for each predicate exhaust the domain) with a countable infinity of urelements so there is a bijection from the subdomain of urelements onto the natural numbers. Expand the language by adding individual constant 0, one oneplace function term S, two twoplace functions + and × and two oneplace predicates N and C. Amend M to a neoclassical model M* by interpreting N and C classically (i.e. with its negative extension the complement of its positive extension) with the urelements as the positive extension of N and the sets as the positive extension of C. Assign S^{n}0 to the urelement assigned n and interpret + and × by plus and times (thus as fully defined functions) over the extension of N. Now we reinterpret identity. Its positive extension is to stay the same and for α, β both sets or both urelements, 〈α, β〉 is in the negative extension of = iff α ≠ β; otherwise, where one is a set and the other is an urelement, the pair is not in the negative extension.
We thus have to change the theory of identity. We can still have ∀x x = x as an axiom but we amend Leibniz' law to:
∀xy((Cx & Cy) → (φx & x = y) → φy)) ditto for the restriction to N.
We retain the axioms of ZFC but with all quantifiers restricted to C and add the firstorder Peano–Dedekind axioms, all quantifiers restricted to N. The resultant theory, call it NS for numbers/sets, is clearly true in M*. By the previous result we can expand it to a prime theory NS* also satisfied by M*. Since crosscategorical identities such as SS0 = {{Ø}} (set brackets as abbreviations) are indeterminate in M* all of the following fail:
(p.260) NS* ├ SS0 = {{Ø}}; NS* ├ SS0 ≠ {{Ø}};
NS* ├ (SS0 = {{Ø}} ∨ SS0 ≠ {{Ø}})
If our theory T is expressed in a →free sublanguage and is satisfied by neoclassical M, there must exist a classical model M_{C} which satisfies T. This is generated by assigning arbitrarily either T or F to every atom in the language of T which is gappy in M. By the monotonicity of the →free language, every sentence with a determinate value in M has the same value in M_{C}. The above Henkinstyle construction then generates from M_{C} a theory T** which is prime and which proves every instance of LEM (since M_{C} is classical) hence is negationcomplete.
§A.6 Limitative Results in Infinitary Logics
There are pleasing completeness results for some infinitary languages, for example completeness results for standard infinitary propositional languages L_{κ}, κ inaccessible (Karp 1964, §5.5, pp. 51–3) and for (objectual) quantifier languages of the form L_{κ,κ} (again κ inaccessible—Karp 1964, p. 131)^{6} and also, a rather special case, L_{ωI},_{ω0}. Nevertheless, as remarked Chapter 8, §VII, limitative theorems such as Scott's Undefinability Theorems (Scott 1965; Karp 1964, ch. 14) hold for many other cardinals (e.g. successor cardinals κ^{+}). Scott's theorem says that, for these languages, the set of all (codes) of logically true formulae of the language is indefinable in the language and this in turn (given that the length of proofs is bounded above by the index of the language) renders the systems incomplete.
One important point to make is that one necessary condition of application for Scott's results, the definability of the basic syntactic notions inside the object language itself, does not apply in our quasisubstitutional case, essentially generalized propositional logic. Although there is no inconsistency in the neoformalist engaging in some model theory, for example assigning a domain of referents for each singular term in L_{κ}, note that every such domain of individuals must be of cardinality 〈 κ. For generalizations are infinitary regular conjunctions and disjunctions and each such is of length 〈 κ.
Since the cardinality of any such language, the number of its wffs, is ≥ κ (= κ, for κ inaccessible) this means that we cannot characterize the syntax of the language up to isomorphism by a theory expressible in the language: we cannot even quantify over all symbols or terms of the language. What this shows is that the neat generalization of firstorder language to the vastly more expressive infinitary languages L_{κ}, κ (p.261) inaccessible, has come, in our case, at a lethal cost. We patently can describe the syntax of English in English (albeit our theories thereof are far from perfect) and construct selfreferential ‘diagonalization’ attributions. Hence the languages L_{κ} fail to regiment crucial aspects of real language. The paradoxes are not even stateable. Again the moral I draw is the need to return to naïve set theory.
Notes:
(1) See Anderson and Belnap (1975, §29.5), especially the theorem ∼(A → A) → (B → B) together with theorem RM67 (p. 397) (A → A) ⇔ (∼A ∨ A).
(2) In the more liberal system in which we treat axioms as if they were part of the logic (see Chapter 8, §III), the overlap condition is amended to the requirement that only axioms feature in X ∩ (Y ∪ Z). Since axioms are true in the admissible models (by definition) we can still rule out the third case, in the falsitypreservation direction, that A ∊ X ∩ (Y ∪ Z). For then A is an axiom and so true, along with all the other premisses, contradicting downward truthpreservation, given the falsity of C. The same argument applies in the other cases.
(3) By complicating the → I rule to allow conditional sentences only in the antecedent of the premiss we can render the transitivity and contraposition rules redundant whilst preserving soundness.
(4) Here ‘∊’ is a term both of the object language and one of the informal metalanguage, with its usual meaning.
(5) I will annotate the derivability of one side of an instance of a comprehension axiom from the other by CE. It is assumed, as before, that the mathematical axioms are immune from the overlap restrictions.
(6) These results depend on extensions of distributivity principles (laws of dependent and independent choices) to govern the interaction between quantifiers and sentential operators.