We present LogAG, an algebraic language for reasoning about graded propositions. LogAG is algebraic in that it is a language of only terms, some of which denote propositions. Both propositions and their grades are taken as individuals in the LogAG ontology. Thus, the language includes terms denoting graded propositions, grades of propositions, grading propositions, and graded grading propositions in an arbitrary compositional structure. In this paper, we present the syntax and semantics of LogAG, defining an infinite sequence of graded logical consequence relations, each corresponding to accepting graded propositions at some nesting depth. We show the utility of LogAG in default reasoning, reasoning about information provided by a chain of sources with varying degrees of trust, and representing the dilemma one is in when facing paradoxical liar-like sentences.
Graded, or weighted, logics have witnessed increased attention and interest over
the years, which may be attested by the sheer length of the bibliography of a
recent comprehensive survey [
of the Liar” in this paper is not true. Having read the previous sentence,
should you believe it or not? (cf. [
Weighted logics have something to say about each of the above cases (or, at least, so we claim). In this report, we present the syntax and semantics of a family of logical languages, LogAG, for reasoning about graded propositions and, in Section 4, we demonstrate how the above cases are treated within LogAG theories. While most of the weighted logics we are aware of employ some form of non-classical, possible-worlds, semantics, basically assigning some notion of grade to possible worlds or truth values, LogAG is a non-modal logic, with classical notions of worlds and truth values. This is not to say that LogAG is a common classical logic—it surely is not—but it is closer in spirit to classical nonmonotonic logics in artificial intelligence [9, 7, for example].3 In such formalisms, as in LogAG, there is a classical logical consequence relation on top of which we define a non-classical relation which is more restrictive, selecting only a subset of the classical models. We achieve this by taking the algebraic, rather than the modal, route.
LogAG is algebraic in the sense that it only contains terms, algebraically
constructed from function symbols. No sentences are included in a LogAG
language; instead, there are terms of a distinguished syntactic type that are taken
to denote propositions. LogAG is a variant of LogAB [
LogAG is a class of many-sorted languages that share a common core of logical
symbols and differ in a signature of non-logical symbols. In what follows, we
identify a sort σ with the set of symbols of sort σ. A LogAG language is a set of
terms partitioned into three base syntactic sorts, σP , σD and σI . Intuitively, σP
3 But it is neither second-order like circumscriptive theories [
As is customary in many-sorted languages, an alphabet of LogAG is made up of a set of syncategorematic punctuation symbols and a set of denoting symbols each from a set σ = {σP , σD, σI } ∪ {τ1 −→ τ2|τ1 ∈ {σP , σD, σI } and τ2 ∈ σ} of syntactic sorts. Intuitively, τ1 −→ τ2 is the syntactic sort of function symbols that take a single argument of sort σP , σD, or σI and produce a functional term of sort τ2. Given the restriction of the first argument of function symbols to base sorts, LogAG is, in a sense, a first-order language.
A LogAG alphabet is a union of four disjoint sets: Ω ∪ Ξ ∪ Σ ∪ Λ. The set Ω, the signature of the language, is a non-empty, countable set of constant and function symbols. Each symbol in the signature has a designated syntactic type from σ. The set Ξ = {xi, di, pi}i∈N is a countably infinite set of variables, where xi ∈ σI , di ∈ σD, and pi ∈ σP , for i ∈ N. Σ is a set of syncategorematic symbols, including the comma, various matching pairs of brackets and parentheses, and the symbol ∀. The set Λ is the set of logical symbols of LogAG, defined as the union of the following sets. 1. {¬} ⊆ σP −→ σP 2. {∧, ∨.} ⊆ σP −→ σP −→ σP 3. {≺, =} ⊆ σD −→ σD −→ σP 4. {G} ⊆ σP −→ σD −→ σP A LogAG language with signature Ω is denoted by LΩ. It is the smallest set of terms formed according to the following rules; as usual, terms involving ⇒, ⇔, and ∃ may be introduced as abbreviations in the standard way.
– Ξ ⊂ LΩ – c ∈ LΩ, where c ∈ Ω is a constant symbol. – f (t1, . . . , tn) ∈ LΩ, where f ∈ Ω is of sort τ1 −→ . . . −→ τn −→ τ (n > 0) and ti is of sort τi. . – {¬t1, (t1 ∧ t2), (t1 ∨ t2), ∀x(t1), G(t1, t2), t3 ≺ t4, t3 = t4} ⊂ LΩ; where t1, t2 ∈ σP ; t3, t4 ∈ σD; and x ∈ Ξ.
Definition 1. A LogAG structure is a quintuple S = D, A, g, <, e , where – D, the domain of discourse, is a set with two disjoint, non-empty, countable subsets P and G. – A = P, +, ·, −, ⊥,
sums), non-degenerate ( – g : P × G −→ P. – <: G ×G −→ P satisfies the following properties for every distinct g1, g2, g3 ∈ G: O1. g1 < g2 = −(g2 < g1).
O2. [(g1 < g2) · (g2 < g3)] + g1 < g3 = g1 < g3.
is a complete (closed under arbitrary products and
= ⊥) Boolean algebra [
O4. g1 < g =
g < g1 = .
g∈G g∈G – e : G × G −→ {⊥, }, where, for every g 1, g2 ∈ G, e(g1, g2) = , if g 1 = g2, and e(g1, g2) = ⊥, otherwise.
tions P, structured as a Boolean algebra; (ii) a set of grades G, and (iii) a set of individuals P ∪ G. These stand in correspondence to the syntactic sorts of LogAG. In what follows, we let DσP = P, DσD = G, and DσI = P ∪ G. g is a function which maps a proposition p and a grade g to the proposition that p is a proposition of grade g. By refraining from imposing any constraints on g (other than functionality), we are admitting virtually any intuitive interpretation of grading. Properties O1–O4 require propositions in the range of < to give rise to an irreflexive linear order on G which is serial in both directions. Similarly, the rigid definition of e gives rise to the identity relation on G.
Definition 2. A valuation V of a LogAG language LΩ is a triple where S, VΩ, VΞ , – S = D, A, g, <, e is a LoAg G structure; – VΩ is a function that assigns to each constant of sort τ in Ω an element of Dτ , and to each function symbol f ∈ Ω of sort τ1 −→ . . . −→ τn −→ τ an n n-adic function VΩ(f ) : × Dτi −→ Dτ ; and
i=1 – VΞ : Ξ −→ D is a variable assignment, where, for every i ∈ N, vΞ(pi) ∈ DσP , vΞ(di) ∈ DσD , and vΞ(xi) ∈ DσI .
In what follows, for a valuation V = S, VΩ, VΞ with x ∈ Ξ of sort τ and a ∈ Dτ , V[a/x] = S, VΩ, VΞ[a/x] , where VΞ[a/x](x) = a, and VΞ[a/x](y) = VΞ(y) for every y = x.
Definition 3. Let LΩ be a LogAG language and let V be a valuation of LΩ. An interpretation of the terms of LΩ is given by a function [[·]]V : – [[x]]V = VΞ(x), for x ∈ Ξ – [[c]]V = VΩ(c), for a constant c ∈ Ω – [[f (t1, . . . , tn)]]V = VΩ(f )([[t1]]V , . . . , [[tn]]V ), for an n-adic (n ≥ 1) function symbol f ∈ Ω – [[(t1 ∧ t2)]]V = [[t1]]V · [[t2]]V – [[(t1 ∨ t2)]]V = [[t1]]V + [[t2]]V – [[¬t]]V = −[[t]]V – [[∀x(t)]]V = a∈Dτ [[t]]V[a/x] – [[G(t1, t2)]]V = g([[t1]]V , [[t2]]V ) – [[t1 ≺. t2]]V = [[t1]]V < [[t2]]V – [[t1 = t2]]V = e([[t1]]V , [[t2]]V )
ing to the notion of truth. This is achieved using the natural partial order ≤
associated with A [
[[γ]]V ≤ [[φ]]V . 3
In [
g(p, g) is a grading proposition. Moreover, if g(p, g) ∈ Q ⊆ P, we say that p is graded in Q. We define the set of p graders in Q to be the set G(p, Q) = {q|q ∈ Q and q grades p}. Throughout, we assume a LogAG structure S = D, A, g, <, e .
terms corresponds to the filter F ([[Γ]]) generated by the set [[Γ ]] of denotations
of members of Γ [
Embedding and Chains
Definition 5. A proposition p ∈ P is embedded in Q ⊆ P if (i) p ∈ Q or (ii) for some g ∈ G, g(p, g) is embedded in Q. Henceforth, let E(Q) = {p|p is embedded in Q}.
Definition 6. For Q ⊆ P, let δQ : E(Q) −→ N, where 1. if p ∈ Q, then δQ(p) = 0; and 2. if p ∈/ Q, then δQ(p) = e + 1, where e = minq∈G(p,E(Q)){δQ(q)}. δQ(p) is referred to as the degree of embedding of p in Q.
In the sequel, we let En(Q) = {p ∈ E(Q) | δQ(p) ≤ n}, for every n ∈ N. Definition 7. A grading chain of p ∈ P is a finite sequence q0, q1, . . . , qn of grading propositions such that qn grades p and qi grades qi+1, for 0 ≤ i < n. q0, q1, . . . , qn is a grading chain if it is a grading chain of some p ∈ P.
A grading chain C2 = q0, q1, . . . , qn extends a grading chain C1 if C1 is a grading chain of q0. The grading chain C1 C 2 is said to be an extension of C1 (where denotes sequence concatenation).
that we impose no special restrictions on the function g and the proposition algebra, grading chains in a set Q may, in general, be quite counter-intuitive.
Definition 8. Let Q ⊆ P. 1. A grading chain is well-founded if all its extensions are well-founded. Q is well-founded if every grading chain in Q is well-founded. 2. A grading chain q0, q1, . . . , qn is acyclic if, for every 0 ≤ i, j ≤ n, qi = qj only if i = j. Q is acyclic if every grading chain in Q is acyclic. 3. Q is depth-bounded if there is some d ∈ N such that every grading chain in
Q has at most d distinct grading propositions. 4. Q is fan-out-bounded if there is some fout ∈ N such that every grading proposition in Q grades at most fout propositions. 5. Q is fan-in bounded if there is some fin ∈ N where |G(p, Q)| ≤ fin, for every p ∈ Q. 6. Q is non-explosive if for every R ⊆ Q, if R has finitely-many grading propositions, then so does F (R).
longest grading chain of p in Q if
Proposition 1. 5 Let Q ⊆ P. 1. If P is depth-bounded, then it is not acyclic. 2. If Q is depth-bounded and acyclic, then Q is well-founded. 3. If E(Q) is depth-bounded, then there is some n ∈ N such that δQ(p) ≤ n, for every p ∈ E(Q). 4. If P is fan-out-bounded and Q is non-explosive with finitely-many grading propositions, then En(F (Q)) has finitely-many grading propositions, for every n ∈ N. Further, if E(F (Q)) is depth-bounded, then it has finitely-many grading propositions. 5. If Q is depth-bounded, then every graded p ∈ Q has a longest grading chain in Q. Further, if Q is fan-in-bounded, then every graded p ∈ Q has finitelymany longest grading chains in Q.
5 Proofs of observations and propositions are omitted for space limitations. A longer version of the paper includes all the proofs. 3.2
Telescoping
proposition set Q may be further enriched by telescoping Q and accepting some
of the propositions embedded therein. For this, we need to define (i) the process of
telescoping, which is a step-wise process that considers propositions at increasing
degrees of embedding, and (ii) a criterion for accepting embedded propositions
which, as should be expected, depends on the grades of said propositions.
Definition 9. Let S be a LogAG structure with a depth- and fan-out-bounded
P. A telescoping structure for S is a quadruple T = T , O, ⊗, ⊕ , where
– T ⊆ P;
– O is an ultrafilter of the subalgebra induced by Range(<) (see [
Definition 10. Let ⊗ : i∞=1 Gi −→ G, and let C = q0, q1, . . . , qn be a grading chain of p ∈ P. The fused ⊗-grade of p with respect to C is the grade f⊗(p, C) = ⊗( g0, . . . , gn ), where qi = g(qi+1, gi), for 0 ≤ i < n, and qn = g(p, gn). Definition 11. Let T be a telescoping structure. If p ∈ Q, for a fan-in-bounded Q ⊂ P, then the T-fused grade of p in Q is defined as fT(p, Q) = f⊗(p, Ck) kn=1 where
Ck kn=1 is a permutation of the set of longest grading chains of p in Q.6
of LogAG structures, we say that a kernel of Q ⊆ P is a subset-minimal X ⊆ Q such that F (X ) is improper ( = P). Let Q⊥ be the set of Q kernels. Definition 12. For a telescoping structure T = T , O, ⊗, ⊕ bounded Q ⊆ P, if X ⊆ Q, then p ∈ X survives X in T if 1. p ∈ T ; or 2. G(p, Q) = ∅ and there is some q ∈ X , with G(q, Q)
(fT(q, Q) < fT(p, Q)) ∈ O.
The set of kernel survivors of Q in T is the set and a fan-in= ∅, such that κ(Q, T) = {p ∈ Q| if p ∈ X ∈ Q ⊥ then p survives X given T}.
Observation 1 If F (T ) is proper, then F (κ(Q, T)) is proper.
Definition 13. Let Q, T ⊆ P. We say that p is supported in Q given T if 6 Note that T-fusion is well-defined given the fan-in-boundedness of Q and the final clause of Proposition 1. 1. p ∈ F (T ); or 2. there is a grading chain q0, q1, . . . , qn member of R is supported in Q.
of p in Q with q0 ∈ F (R) where every The set of propositions supported in Q given T is denoted by ς(Q, T ).
Observation 2 ς(Q, T ) = F (T ) ∪ G, for some set G of propositions graded in Q.
Definition 14. Let T be a telescoping structure for S. If Q ⊂ P such that E1(F (Q)) is fan-in-bounded, then the T-induced telescoping of Q is given by τT(Q) = ς(κ(E1(F (Q)), T), T ) Proposition 2. For a telescoping structure T, τT is a function from fan-inbounded sets in 2P to sets in 2P .
sitions, then τT(Q) is defined, for every telescoping structure T. On the other hand, if F (Q) is improper, then τT(Q) is undefined. In what follows, provided that the right-hand side is defined, let
n τT (Q) =
Q if n = 0 τT(τTn−1(Q)) otherwise Definition 15. Let T be a telescoping structure. We refer to F (τTn(T )) as a degree n(∈ N) graded filter of T, denoted Fn(T).
fixed-point for graded filters is not secured. (Check Example 3 in Section 4.) We can only prove a weaker property for a special class of telescoping structures. Theorem 1. Let T be a telescoping structure where T is finite. There is some n ∈ N such that if Fi(T) is defined and Fi(T) ∩ Range(g) = τ i(T) ∩ Range(g) for every i ≤ n, then for every j ∈ N, there is some k ≤ n such that Fn+j (T) = Fk(T).
Corollary 1. If, in Theorem 1, Fn(T) = F (E(T )) for some n < 2|E(T )| + 1, then Fn+k(T) = Fn(T), for every k ∈ N.
Theorem 2. If T is a telescoping structure where F (T ) is proper, then, if defined, Fn(T) is proper, for every n ∈ N. 4
Given the definition of a LogAG structure, we impose some reasonable constraints on which sets of LogAG terms qualify as LogAG theories. A LogAG theory is a finite set T ⊆ σP such that E ∪ O ⊆ T, where – E is the smallest set containing the following terms:
. 1. ∀d[d = d] . . 2. ∀d1, d2[d1 = d2.⇒ d2 = d1.] . 3. ∀d1, d2, d3[(d1 .= d2 ∧ d2 = d3) ⇒ d1 = d3] 4. ∀p, d1, d2[(d1 = d2 ∧ G(p, d1)) ⇒ G(p, d2)] and – O is the smallest set containing the following terms: . 1. ∀d1, d2[¬(d1 ≺ d2) ⇔ (d2 ≺ d1 ∨ d2 = d1)] 2. ∀d1, d2, d3[(d1 ≺ d2 ∧ d2 ≺ d3) ⇒ d1 ≺ d3] Given a LogAG theory T and a valuation V = S, VΩ, VΞ , let V(T) = {[[φ]V] | φ ∈ T}). Further, for a LogAG structure S, an S grading canon is a triple C = ⊗, ⊕, n where n ∈ N and ⊗ and ⊕ are as indicated in Definition 9. Definition 16. Let T be a LogAG theory and V = S, VΩ, VΞ a valuation, where S has a set P which is depth- and fan-out-bounded, for some LogAG language LΩ. For every φ ∈ σP and S grading canon C = ⊗, ⊕, n , φ is a graded consequence of T with respect to C, denoted T | C φ, if Fn(T) is defined and [[φ]]V ∈ Fn(T), for every telescoping structure T = V(T), O, ⊗, ⊕ for S, where O extends F (V(T) ∩ Range(<)).7 It should be clear that | C, where C = ⊗, ⊕, n , reduces to |= if n = 0 or if
φ is a graded consequence of T with respect to C if F (V(T)) is not proper. In what follows, let TC = {φ | T | C φ}. When we are considering a set of canons which only differ in the value of n, we write Tn instead of TC.
Unlike |=, | C is, in general, non-monotonic. (In the sequel, we interpret grades by the rational numbers, with their natural order remaining implicit.) Example 1 (Opus and Tweety). We can represent the case of Opus and Tweety from Section 1 using a LogAG theory TOT1 = E ∪ O ∪ ΓOT1, where ΓOT1 is made up of the following terms. 1. ∀x[Bird(x) ⇒ G(F lies(x), 5)] 2. ∀x[P enguin(x) ⇒ G(¬F lies(x), 10)] 3. ∀x[P enguin(x) ⇒ Bird(x)] 4. P enguin(Opus) 5. Bird(T weety) Figure 1 displays the relevant graded consequences of TOT1 with respect to a series of canons, with 0 ≤ n ≤ 2. Upon telescoping to n = 1, we believe 7 An ultrafilter U extends a filter F , if F ⊆ U . that Tweety flies and Opus does not fly. The embedded proposition that Opus flies does not survive telescoping since we trust that Opus does not fly, being a 1 penguin, more than we trust that it flies, being a bird. TOT1 is a fixed point. Now, consider the theory TOT2 = E ∪ O ∪ ΓOT2, where ΓOT2 is similar to ΓOT1, but with propositions (1) and (2) replaced by “G(∀x[Bird(x) ⇒ F lies(x), 5)” and “G(∀x[P enguin(x) ⇒ ¬F lies(x), 10)”, respectively. Thus, we trade the “de re” representation of TOT1 for the “de dicto” representation in TOT2. This change 1 1 results in a change in the fixed point that we reach. In TOT2, as in TOT1, we 1 end up believing that Opus does not fly. Unlike TOT1 however, we give up our belief in the proposition that birds fly and, hence, cannot conclude that Tweety flies.
describe the situation using LogAG in at least two theories. Consider the theory TSM1 = E ∪ O ∪ ΓSM1, where ΓSM1 is made up of the following terms. 1. ∀p[Source(p, LL) ⇒ G(p, 11)] 2. ∀p[Source(p, DP ) ⇒ G(p, 4)] 3. ∀p[P erceive(p) ⇒ G(p, 15)] 4. ∀l, t[G(At(KC, l, t) ⇔ At(SM, l, t), 10.5)] 5. ∀l1, l2, t, x[(Disjoint(l1, l2) ∧ At(x, l1, t)) ⇒ ¬At(x, l2, t)] 6. P erceive(Source(Source(At(SM, DT, 12 : 00), LL), DP )) 7. P erceive(At(KC, Of f ice, 12 : 00)) 8. Disjoint(Of f ice, DT ) n Figure 2 displays relevant members of TSM1 with respect to a series of canons, with ⊗ = mean and ⊕ = max, and with 0 ≤ n ≤ 3. Note that, for 1 ≤ n ≤ 2, we trust the proposition that Superman was at the office at noon (1.6). However, upon telescoping to n = 3, we lose our trust in said proposition, since we trust what Lois Lane says more than we trust our belief in the identity of Superman and Clark Kent (1.3).
Alternatively, consider the theory TSM2 = E ∪ O ∪ ΓSM2, where ΓSM2 is made up of the following terms. 1. G(G(G(At(SM, DT, 12 : 00), 11), 4), 15) 0.1. TSM1 0.2. G(Source(Source(At(SM, MDT, 12 : 00), LL), DP ), 15) 0.3. G(At(KC, Office, 12 : 00) ⇔ At(SM, Office, 12 : 00), 10.5) 0.4. G(At(KC, Office, 12 : 00), 15) 3. ∀l, t[G(At(KC, l, t) ⇔ At(SM, l, t), 10.5)] 4. ∀l1, l2, t, x[(Disjoint(l1, l2) ∧ At(x, l1, t)) ⇒ ¬At(x, l2, t)] employed with TSM1. In this case, we get a different fixed point; we end up (at n = 2) believing that Superman was at the office at noon, contrary to what has been reported by Lois Lane in the Daily Planet. The reason is that, due to the nesting of grading propositions, the fused grade of “AT (SM, DT, 12 : 00)” is now only 10 (which is less than 10.5, the grade of (1.4)), being pulled down by the low grade attributed to the Daily Planet.
n = 0 n = 1 n = 2 n = 3 0.1. TSM2 0.2. G(At(KC, Office, 12 : 00) ⇔ At(SM, Office, 12 : 00), 10.5) 1.1. T0SM2 1.2. G(G(At(SM, DT, 12 : 00), 11), 4) 1.3. At(KC, Office, 12 : 00) 1.4. At(KC, Office, 12 : 00) ⇔ At(SM, Office, 12 : 00) 1.5. At(SM, Office, 12 : 00) 1.6. ¬At(SM, DT, 12 : 00) 2.1. T1SM2 2.2. G(At(SM, DT, 12 : 00), 11) 3.1. T2SM2 0.1. TL 11..12.. Tφ0L 1.3. G(¬φ, 5)
Example 3 (The Liar). Consider the theory TL = E∪O∪{G(φ, 5), φ ⇔ G(¬φ, 5)}.
This is the closest we can get, within LogAG, to the case of the liar from
Section 1; given the non-degeneracy of the Boolean algebra, we can never have the
situation where [[φ]]V = −[[φ]]V (cf. [
Notwithstanding the abundance of weighted logics in the literature, it is our
conviction that LogAG provides an interesting alternative. While it has a
nonclassical semantics, LogAG is arguably intuitive, expressive, and quite similar in
syntax to first-order logic. We hope to have demonstrated the utility of LogAG
in default reasoning, reasoning with information reported through a chain of
sources, and even reasoning with paradoxical propositions. A careful examination
of how LogAG relates to other graded logics and non-monotonic formalisms is
called for. On a first pass, we believe that LogAG subsumes possibilistic logic
[
A.1
Proof of Theorem 1
Lemma 1. If T is a telescoping structure then, for every n ∈ N, if Fi(T) is defined and Fi(T) ∩ Range(g) = τ i(T) ∩ Range(g) for every i ≤ n, then Fn(T) = F (Q), for some Q ⊆ E(T ) and Fn(T) ∩ Range(g) ⊆ En(T ).
Proof. We prove the lemma by induction on n. For n = 0, F0(T ) = F (T ), where T ⊆ E(T ). Further, F0(T ) ∩ Range(g) = τ 0(T) ∩ Range((g)) = T ∩ Range(g) ⊆ E0(T ). Now assume that the statement holds for some k ∈ N. By the induction hEy1p(Fotkh(eTs)i)s, aFrek(iTn) E∩kR+a1(nTge).(gM)o⊆reoEvke(rT, )E. 1T(Fhke(rTef)o)re∩, Ralalnpgreo(pgo)si⊆tioEnks+g1r(aTd)e.dBiny Observation 2, τ k+1(T ) = τT(Fk(T)) = ς(κ(E1(Fk(T))), T), T ) = F (T ) ∪ G,
T where G is a set of propositions graded in κ(E1(Fk(T))). By Definition 12, κ(E1(Fk(T))) ⊆ E1(Fk(T)). Hence, G ⊆ Ek+1(T ). Thus, Fk+1(T) = F ([F (T ) ∪ G]) = F (T ∪ G), where T ∪ G ⊆ E(T ). Moreover, Fk+1(T) ∩ Range(g) = τ k+1(T) ∩ Range(g) ⊆ Ek+1(T ).
finite, then E(T ) is finite. Let b = |E(T )|. Now, taking n = 2b + 1, suppose that, for every i ≤ 2b + 1, Fi(T) is defined and Fi(T) ∩ Range(g) = τ i(T) ∩ Range(g). By Lemma 1, Fi(T) = F (Qi), for some Qi ⊆ E(T ). Since there are only 2b subsets of E(T ), then Fn(T) = Fj(T), for some j ≤ 2b. Further, by Proposition 2, for every j ∈ N, there is some k ≤ n such that Fn+j(T) = Fk(T). A.2
Proof of Corollary 1 We prove the case of k = 1 and the result follows by the same argument for all k ∈ N. = F (F (E(T ))) = F (E(T )) (Definition of E and the assumption of Theorem 1) (Since F (E(T )) is proper given that Fn+1(T) is defined) (Since every p ∈ F (E(T )) is supported given that
F (E(T )) = Fn(T) and Definitions 14 and 15) A.3
Proof of Theorem 2 For n = 0, the statement is trivial, since F0(T) = F (T ). Otherwise, the statement follows directly from Observation 1 since, by Definition 15, Fk+1(T) = F (K), for some K ⊆ κ(E1(Fk(T)), T).