By Rudolf Carnap

A transparent, accomplished, and rigorous therapy develops the topic from basic recommendations to the development and research of really advanced logical languages. It then considers the appliance of symbolic common sense to the rationalization and axiomatization of theories in arithmetic, physics, and biology. countless numbers of difficulties, examples, and routines. 1958 variation.

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1), let w ∈ Ea. Then if w ∈ |∀xϕ|f [a/y], there exists X ∈ Prop with Eb ⇒ |ϕ|f [a/y][b/x]. w∈X⊆ b∈U In particular, when b = a, since w ∈ Ea we get w ∈ |ϕ|f [a/y][a/x]. 3, |ϕ|f [a/y][a/x] = |ϕ(y/x)|f [a/y] because y M f [a/y] = a. Thus w ∈ |∀xϕ|f [a/y] ⇒ |ϕ(y/x)|f [a/y] = |∀xϕ → ϕ(y/x)|f [a/y]. 2. Suppose that x has no free occurrence in ϕ. If ϕ is admissible in M, then M |= ϕ → ∀xϕ. Proof. 2, |ϕ|f = |ϕ|f [a/x] ⊆ Ea ⇒ |ϕ|f [a/x]. But |ϕ|f ∈ Prop by M-admissibility of ϕ, so ❧ Ea ⇒ |ϕ|f [a/x] = |∀xϕ|f.

2. A model validates CQ if any of the following hold: Commutativity of Quantiﬁers in Varying-Domain Kripke Models 21 (1) Prop is an atomic Boolean algebra. (2) Prop is ﬁnite. (3) The universe U is ﬁnite. Proof. (1) Put B = Prop. For any f , all sets |ϕ|f [a/x, b/y], |∀xϕ|f [b/y], |∀yϕ|f [a/x] are in B by admissibility. 1, hence as B is atomic this makes |∀x∀yϕ|f = |∀y∀xϕ|f . (2) By (1), as any ﬁnite Boolean algebra is atomic. (3) If U is ﬁnite, then for any f , {|∀x∀yϕ|f, |∀y∀xϕ|f } ∪ {|ϕ|f [a/x, b/y], |∀xϕ|f [b/y], |∀yϕ|f [a/x] : a, b ∈ U } is a ﬁnite subset of Prop, so it generates a Boolean subalgebra B of Prop that is ﬁnite, hence atomic.

Goldblatt and I. Hodkinson • if Y = ∅, then |∃xϕ|f is Q if all the f y are ∼-equivalent and no f z is ∼-equivalent to them: for then, the set inside the square brackets is a single ∼-equivalence class, so its ↑ is Q. Otherwise, |∃xϕ|f is ∅. Thus, for any f ∈ ω U , |∃xϕ|f = y∼y ∧ y,y ∈Y ¬(y ∼ z) f. y∈Y,z∈Z So ∃xϕ is equivalent to this quantiﬁer-free formula if Y = ∅ (and, as one can see, if Y = ∅ as well). 3, and hence the proof that M is a model. 6. Completeness and the Barcan Formulas Let L be any (consistent) normal propositional modal logic.

### An Introduction to Symbolic Logic and Its Applications by Rudolf Carnap

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