Unwinding Modal Paradoxes on Digraphs

Abstract

The unwinding that Cook (J. Symbol. Log. 69(3), 767–774 2004) proposed is a simple but powerful method of generating new paradoxes from known ones. This paper extends Cook’s unwinding to a larger class of paradoxes and studies further the basic properties of the unwinding. The unwinding we study is a procedure, by which when inputting a Boolean modal net together with a definable digraph, we get a set of sentences in which we have a ‘counterpart’ for each sentence of the Boolean modal net and each point of the digraph. What is more, whenever a sentence of the Boolean modal net says another sentence is necessary, then the counterpart of the first sentence at a point correspondingly says the counterparts of the second one at all accessible points of that point are all true. The output of the procedure is called ‘the unwinding of a Boolean modal net on a definable digraph’. We prove that the unwinding procedure preserves paradoxicality: a Boolean modal net is paradoxical on a definable digraph, iff the unwinding of it on this digraph is also paradoxical. Besides, the dependence digraph for the unwinding of a Boolean modal net on a definable digraph is proved to be isomorphic to the unwinding of the dependence digraph for the Boolean modal net on the previous definable digraph. So the unwinding of a Boolean modal net on a digraph is self-referential, iff the Boolean modal net is self-referential and the digraph is cyclic. Thus, on the one hand, the unwinding of any Boolean modal net on an acyclic digraph is non-self-referential. In particular, the unwinding of any Boolean modal net on $\langle {\mathbb N}, <\rangle$ is non-self-referential. On the other hand, if a Boolean modal net is paradoxical on a locally finite digraph, the unwinding of it on that digraph must be self-referential. Hence, starting from a Boolean modal paradox, the unwinding can output a non-self-referential paradox only if the digraph is not locally finite.

Appendix: An Application to Modal Theories

It is well-known that some modal theories, even very weak, may contain a contradiction provided that the notion of necessity is treated as a predicate of sentences. We also know that the inconsistency of these modal theories is due to modal paradoxes. What is more, for different modal theories, we may have to employ different modal paradoxes to prove inconsistency, or else we may miss the contradiction. In this appendix, we will apply our approach to giving a semantical analysis of some modal theories in the language ${\mathscr L}_{\Box }$. In particular, we focus on making it clear in what sense a modal paradox is sufficient to prove the inconsistency of a modal theory.¹

In this appendix, unless otherwise claimed, we only study sentences of ${\mathscr L}_{\Box }$. Thus, we always use A for a sentence (of ${\mathscr L}_{\Box }$), and Σ for a set of sentences. We will use $\langle {\mathfrak N}, X\rangle \models A$ rather than its abbreviation X⊧A, since the latter may be confused with the classical consequence relation.

1.1 ω-Logic and ω-Normal Theories

To establish the modal theories in ${\mathscr L}_{\Box }$, we must first formulate the notion of syntactic consequence. Since we always fix ${\mathfrak N}$ as the standard structure for ${\mathscr L}_{\Box }$, we will give the syntactic consequence relation based on ω-logic rather than first-order logic. A deduction of A from Σ in the ω-logic is a (possibly infinite) sequence of formulas, in which each formula is either a theorem of Peano arithmetic, a member of Σ, obtained from earlier formulas of the sequence by modus ponens, or obtained from earlier formulas of the sequence by the ω-rule (that is, from $A\left (\bar {n}\right )$ for any $n\in {\mathbb N}$, to infer ∀xA(x)). We use Σ ⊩^ωA, if there exists a deduction of A from Σ. The corresponding semantic consequence relation ⊧^ω is defined as follows: Σ⊧^ωA, iff for any $X\subseteq {\mathbb N}$, $\langle {\mathfrak N}, X\rangle \models {\Sigma }$ implies $\langle {\mathfrak N}, X\rangle \models A$. Theorem 8 is the fundamental result about the two consequences, which is also known as Henkin and Orey’s ω-completeness theorem.

Theorem 1

Σ ⊩^ωA, iff Σ⊧^ωA.

Before proving the above theorem, we first give some notions. Define Σ is closed under the first-order logical consequence if each first-order logical consequence of Σ belongs to Σ. In that case, we also say Σ is a theory. Σ is closed under the ω-rule, if whenever $A\left (\bar {n}\right )\in {\Sigma }$ for any $n\in {\mathbb N}$, ∀xA(x) ∈Σ. Σ is closed under the necessitation, if whenever A ∈Σ, $\Box \ulcorner {A}\urcorner \in {\Sigma }$. When Σ is closed under the ω-rule, we also say that it is ω-complete. The notation $\Box ^{-}\ulcorner {\Sigma }\urcorner$ stands for the set $\{n\mid \Box \left (\bar {n}\right )\in {\Sigma }\}$. The following lemma is the key to Theorem 8.

Lemma 1

Let Σ be an ω-complete and consistent theory. Then there exists an ω-complete and maximally consistent Γ, such that ${\Gamma }\supseteq {\Sigma }$, and for any sentence A, $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models A$, iff A ∈Γ.

Proof

Enumerate all sentences of ${\mathscr L}_{\Box }$ in a list A₀, A₁, …. We construct an increasing sequence of sets Γ₀, Γ₁, … as follows: let Γ₀ = Σ,

• (a) if Γ_n ∪{A_n} is inconsistent, let Γ_n+ 1 = Γ_n ∪{¬A_n},

• (b) if Γ_n ∪{A_n} is consistent and A_n is form ∃xA(x), we shall show that there exists a number i_n, such that ${\Gamma }_{n} \cup \left \{A\left (\overline {i_{n}}\right )\right \}$ is consistent, and thus we can set ${\Gamma }_{n+1} = {\Gamma }_{n} \cup \left \{A\left (\overline {i_{n}}\right )\right \}$,

• (c) otherwise, let Γ_n+ 1 = Γ_n ∪{A_n}.

To prove the existence of the number i_n, we first notice that in Γ_n, only finitely many sentences are added to Σ. Let B be the conjunction of these sentences. Assume that ${\Gamma }_{n} \cup \left \{A_{n}, A\left (\bar {i}\right )\right \}$ is inconsistent for any $i\in {\mathbb N}$, then $B\to \neg A\left (\bar {i}\right )$ belongs to Σ. By the ω-completeness of Σ, we can get $\forall x\left (B\to \neg A\left (x\right )\right )$ belongs to Σ, and thus $B\to \forall x\neg A\left (x\right )$ belongs to Σ. It follows that Γ_n ∪{∃xA(x)} is inconsistent, a contradiction.

Let Γ be the union of all Γ_n ($n\in {\mathbb N}$). Γ is clearly maximally consistent. To see it is ω-complete. Suppose for any $n\in {\mathbb N}$, $A\left (\overline {n}\right ) \in {\Gamma }$, we want ∀xA(x) ∈Γ. Assume ∀xA(x)∉Γ, then ∃x¬A(x) ∈Γ. Let ∃x¬A(x) be the k-th sentence in our enumeration list, that is, A_k = ∃x¬A(x). Then A_k ∈Γ_k+ 1, and by our construction, $\neg A\left (\overline {i_{k}}\right ) \in {\Gamma }_{k+1}$. Hence, $\neg A\left (\overline {i_{k}}\right ) \in {\Gamma }$, or equivalently, $A\left (\overline {i_{k}}\right ) \notin {\Gamma }$, a contradiction.

It remains to prove that for any sentence A, $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle {\models } A$, iff A ∈Γ. The proof is a routine induction on the complexity of A. The atomic sentence of form t₁ = t₂ is obvious. For the atomic sentence of form $\Box \left (t\right )$, we notice that $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models \Box \left (t\right )$, iff $t^{{\mathfrak N}}\in \Box ^{-}\ulcorner {\Gamma }\urcorner$. Let $t^{{\mathfrak N}} = n$, then $t = \bar {n}$ is provable in Peano arithmetic, and by the indiscernibility of identicals, $\Box \left (t\right ) \leftrightarrow \Box \left (\bar {n}\right )$ is also provable in Peano arithmetic. Now, we have $t^{{\mathfrak N}}\in \Box ^{-}\ulcorner {\Gamma }\urcorner$, iff $\Box \left (\bar {n}\right )\in {\Gamma }$, which is precisely equivalent to $\Box (t)\in {\Gamma }$. The cases for connectives are immediate by the maximal consistency of Γ. For the sentence of form ∀xA(x), $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models \forall x A(x)$, iff $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models A\left (\bar {n}\right )$ for any $n\in {\mathbb N}$. The inductive hypothesis implies that $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models A\left (\bar {n}\right )$, iff $A\left (\bar {n}\right )\in {\Gamma }$. At last, by the ω-completeness of Γ, $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models \forall x A(x)$, iff ∀xA(x) ∈Γ. □

It should be noted that a consistent (and even ω-consistent) set cannot necessarily be extended to a maximally consistent and ω-complete set. See [34].

Proof Proof of Theorem 8

The necessity can be proved by a transfinite induction on the lengths of the ω-logic deductions. For the sufficiency, assume ${\Sigma }\nvdash ^{\omega }A$ and let ${\Sigma }^{\prime }$ be the set of all sentences that are the ω-logical consequence of Σ ∪{¬A}, then ${\Sigma }^{\prime }$ is an ω-complete and consistent theory. By Lemma 6, there exists a set ${\Gamma }\supseteq {\Sigma }^{\prime }$ such that for any sentence B, $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models B$, iff B ∈Γ. It follows $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \models {\Sigma }$ but $\langle {\mathfrak N}, \Box ^{-}\ulcorner {\Gamma }\urcorner \rangle \not \models A$. Hence, Σ⊮^ωA. □

A more general version of Theorem 8 is Henkin and Orey’s ${\mathfrak M}$-completeness theorem for any structure ${\mathfrak M}$ of ${\mathscr L}$. In turn, Henkin and Orey’s ${\mathfrak M}$-completeness theorem is a particular case of the omitting types theorem. We refer the reader to [25, p. 151] or [6, pp. 80 ff.] for more details.

There is nothing special about the predicate $\Box$ in ω-logic. In the following, we introduce the notion of digraph validity for the language ${\mathscr L}_{\Box }$, and so associate the predicate $\Box$ with the notion of necessity. First, by an admissible model for ${\mathscr L}_{\Box }$ we mean a pair $\langle {\mathcal G}, \sigma \rangle$, where ${\mathcal G}$ is a digraph, and σ is an admissible assignment for ${\mathscr L}_{\Box }$ (i.e., the set of all sentences) on ${\mathcal G}$. A sentence A is valid on an admissible model $\langle {\mathcal G}, \sigma \rangle$, if for any w in the domain of ${\mathcal G}$, $\langle {\mathfrak N}, \sigma (w)\rangle \models A$. A is valid on ${\mathcal G}$, if it is valid on any admissible model, which is based upon ${\mathcal G}$. A schema is valid on ${\mathcal G}$, if all of its instances are valid on ${\mathcal G}$. Note that since ${\mathscr L}_{\Box }$ is a first-order language, we also have the well-known notion of validity based on Tarski’s notion of satisfaction. One should not confuse the two notions of validity.

The following notion, proposed by Halbach et al. [14, Definition 39] (who called it ‘the $\Box$-closed set’ though), is an analogue to the normal system in modal logic.

Definition 1 ([14, p. 210])

A theory is ω-normal, if

• (a) it contains Peano arithmetic, and is closed under the necessitation.

• (b) it contains (all instances from) the schema $\Box \ulcorner {A\to B}\urcorner \to \Box \ulcorner {A}\urcorner \to \Box \ulcorner {B}\urcorner$ (K for short), and

• (c) it contains the schema $\forall x\Box \ulcorner {A(\dot {x})}\urcorner \to \Box \ulcorner {\forall x A(x)}\urcorner$ (BF for short), and

• (d) it is ω-complete.

Theorem 2 ([14, p. 211])

An ω-normal theory is consistent, iff there exists an admissible model, on which each sentence in this theory is valid.²

Proof

The sufficiency is apparent. To see the necessity, let Σ be an ω-normal theory, and we construct a digraph 〈W,R〉, and an assignment σ as follows:

• (a) W is the collection of all sets $X\subseteq {\mathbb N}$, such that $\langle {\mathfrak N}, X\rangle {\models }{\Sigma }$, that is, each sentence is true in the structure $\langle {\mathfrak N}, X\rangle \models {\Sigma }$.

• (b) R is a binary relation on W such that uRv, iff $\langle {\mathfrak N}, v\rangle {\models } u$, that is, each sentence whose Gödel number belongs to u is true on the structure $\langle {\mathfrak N}, v\rangle$.

• (c) σ(u) = u for any u ∈ W.

By Lemma 6, we can choose $X = \Box ^{-}\ulcorner {\Gamma }\urcorner$, where Γ is the set we have proved it exists. X is an element of W, and thus W is non-empty. By the definition of W and σ, $\langle {\mathfrak N}, u\rangle \models {\Sigma }$ holds for any u ∈ W. Hence, each sentence of Σ is valid on the model 〈W,R,σ〉.

To see 〈W,R,σ〉 is an admissible model, we prove for any sentence A and for any u ∈ W, $\langle {\mathfrak N}, u\rangle {\models } \Box \ulcorner {A}\urcorner$, iff for any v with uRv, $\langle {\mathfrak N}, v\rangle \models A$. By the definition of R, the right side is equivalent to u⊧^ωA, which, by Theorem 8, is equivalent to u ⊩^ωA. Thus, it suffices to prove that A ∈ u, iff u ⊩^ωA.The non-trivial case is the right-to-left direction. For this, we must verify that u contains Peano arithmetic, and is closed under both modus ponens and the ω-rule.

By (a) of Definition 16 and the definition of W, we can deduce that u contains Peano arithmetic. By the fact that schema K is true on $\langle {\mathfrak N}, u\rangle$, we can see immediately that u is closed under modus ponens. Similarly, the fact that schema BF is true on $\langle {\mathfrak N}, u\rangle$ says precisely that u is closed under the ω-rule. □

We will use K^ω for the smallest ω-normal theory. For any modal schema φ, such as $\Box \ulcorner {A}\urcorner \to A$, we use K^ω + φ for the smallest ω-normal theory that contains K^ω and all instances of φ. We will also use K + φ for the theory, which is the same as K^ω + φ except that the ω-rule is no longer available (and correspondingly, the deduction length can be only finite).

1.2 (In)consistency of Some Modal Theories

As Montague [26] first discovered in 1963, modal theories, even if containing only very weak modal principles (such as the schema T), can lead to inconsistency, provided that these theories are built on the elementary arithmetic, and the necessity is treated as a predicate of sentences. One famous theorem Montague [26] proved is that $\mathbf {K}+ \Box \ulcorner {A}\urcorner \to A$ is inconsistent.³ Montague proved this result by use of the modal liar. His proof is syntactical. A similar result we will prove is that $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is inconsistent. Our proof is semantical, and it will provide more information: as we will see, any paradox is a witness to the inconsistency of the theory $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$. We first give two lemmas.

Lemma 2

$\Box \ulcorner {A}\urcorner \rightarrow A$ is valid on a digraph, iff this digraph is reflexive.

There is an analogous result to Lemma 7 in modal logic. Their proofs are similar. We omit the details of the proof.

Lemma 3

Any paradoxical set is paradoxical on any reflexive digraph.

Before proving Lemma 8, we prove the paradoxicality is invariant under p-morphism (Lemma 9). Recall that a p-morphism from ${\mathcal G} = \langle W, R\rangle$ to ${\mathcal G}^{\prime }=\langle W', R'\rangle$ is a function π from W onto W^′, satisfying the following property: for any u,v ∈ W, if uRv, π(u)R^′π(v); conversely, if π(u)R^′π(v), then there exists w ∈ W such that uRw and π(w) = π(v).

Lemma 4

If there exists a p-morphism π from ${\mathcal G}$ to ${\mathcal G}^{\prime }$, then a set of sentences is paradoxical on ${\mathcal G}$, iff it is so on ${\mathcal G}^{\prime }$.

Proof

Let $\sigma ^{\prime }$ be an assignment on ${\mathcal G}^{\prime }$. Define an assignment σ on ${\mathcal G}$ as follows: for any u ∈ W, $\sigma (u) = \sigma ^{\prime }(\pi (u))$. Then for any sentence A, we can prove by a routine induction on the complexity of A that $\langle {\mathfrak N}, \sigma (u)\rangle \models A$, iff $\langle {\mathfrak N}, \sigma ^{\prime }(\pi (u))\rangle \models A$. We leave the details to the reader. We now prove that if $\sigma ^{\prime }$ is admissible for a sentence A, then σ is also admissible for A. For this, we need to verify that for any u ∈ W, $\langle {\mathfrak N}, \sigma (u)\rangle \models \Box \ulcorner {A}\urcorner$, iff for all v ∈ W with uRv, $\langle {\mathfrak N}, \sigma (v)\rangle \models A$.

First, suppose $\langle {\mathfrak N}, \sigma (u)\rangle {\models } \Box \ulcorner {A}\urcorner$ and uRv, then by the result we just mentioned, $\langle {\mathfrak N}, \sigma ^{\prime }(\pi (u))\rangle {\models } \Box \ulcorner {A}\urcorner$. Since π is a p-morphism, and $\sigma ^{\prime }$ is admissible for A, we have π(u)R^′π(v), and $\langle {\mathfrak N}, \sigma ^{\prime }(\pi (v))\rangle \models A$. Hence, $\langle {\mathfrak N}, v\rangle \models A$.

Conversely, suppose $\langle {\mathfrak N}, \sigma (u)\rangle {\not \models } \Box \ulcorner {A}\urcorner$, then $\langle {\mathfrak N}, \sigma ^{\prime }(\pi (u))\rangle {\not \models } \Box \ulcorner {A}\urcorner$. Since $\sigma ^{\prime }$ is admissible for A and π is a surjection, there exists v ∈ W, such that π(u)R^′π(v) and $\langle {\mathfrak N}, \sigma ^{\prime }\left (\pi (v)\right )\rangle \not \models A$. Also, from the fact that π is a p-morphism, it follows that there is w ∈ W such that uRw and π(w) = π(v). That means $\langle {\mathfrak N}, \sigma ^{\prime }\left (\pi (w)\right )\rangle \not \models A$. To sum up, the point w satisfies uRw and $\langle {\mathfrak N}, \sigma \left (w\right )\rangle \not \models A$.

We have proved that if ${\mathcal G}^{\prime }$ has an assignment admissible for A, so does ${\mathcal G}$. The converse is also right. We leave the proof to the reader. Now the desired result follows immediately. □

Proof Proof of Lemma 8

By Definition 11, a set is paradoxical, iff it is so on the minimal reflexive digraph ${\mathcal G}_{0}=\langle \{0\}, =\rangle$. For any digraph ${\mathcal G}$, the map** u↦0 is clearly a p-morphism from ${\mathcal G}$ to ${\mathcal G}_{0}$. Thus, the result follows from Lemma 9. □

Theorem 3

${\mathbf K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is inconsistent.

Proof

Assume ${\mathbf K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is consistent, then by Theorem 9, there exists an admissible model 〈W,R,σ〉, on which $\Box \ulcorner {A}\urcorner \to A$ is valid. By Lemma 7, R is reflexive. Now, choose any paradoxical set, say Σ. By Lemma 8, Σ is paradoxical on the model 〈W,R,σ〉. However, that model is admissible for ${\mathscr L}_{\Box }$ (and in particular, for Σ), a contradiction! □

Note that we can easily verify that the modal liar is paradoxical on any reflexive digraph. From this result, together with Lemma 7, we can also obtain Theorem 10. However, such proof only shows that the inconsistency of $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is caused by the modal liar. Our proof of Theorem 10 is mainly based on Lemma 8, which tells that the inconsistency of $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ can be caused by any paradox.

A generalization of Theorem 10 is as follows: ${\mathbf K}^{\omega }+ \Box ^{n}\ulcorner {A}\urcorner \to A$ is inconsistent. This result follows from the following results: $\Box ^{n}\ulcorner {A}\urcorner \to A$ is valid on a digraph, iff this digraph is n-reflexive (that is, for any u ∈ W, uRⁿu); the n-cycle modal liar is paradoxical on any n-reflexive digraph. Note that the latter is a modification of Halbach et al.’s Example 6 in [14, p. 189].

Next, we turn to the schema $\Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$.

Lemma 5

$\Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is valid on a digraph, iff this digraph is serial.

The following result is a generalization of Example 9 in [14, p. 190].

Lemma 6

McGee’s modal paradox is paradoxical on any conversely non-well-founded digraph. In particular, it is paradoxical on any serial digraph.

Proof

Recall that at the end of Section ??, we denote McGee’s modal paradox by the set $\{\delta _{i}{\ \mid \ } i\in {\mathbb N}\}$, in which $\delta _{0} \equiv \exists x\neg \Box \ulcorner {\delta _{\dot {x}}}\urcorner$, and for all i ≥ 0, $\delta _{i+1}\equiv \Box \ulcorner {\delta _{i}}\urcorner$. Assume McGee’s modal paradox is not paradoxical on a conversely non-well-founded digraph, say ${\mathcal G} = \langle W, R\rangle$. First, we can find infinite points in W, say u₀, u₁, …, such that u_kRu_k+ 1 for any $k\in {\mathbb N}$. Let σ be an assignment on ${\mathcal G}$, which is admissible for McGee’s modal paradox.

Case 1: for some number k, $\langle {\mathfrak N}, \sigma (u_{k})\rangle {\not \models } \delta _{0}$. It follows that for any $i\in {\mathbb N}$, $\langle {\mathfrak N}, \sigma (u_{k})\rangle {\models } \Box \ulcorner {\delta _{i}}\urcorner$. Hence, on the one hand, we have $\langle {\mathfrak N}, \sigma (u_{k+1})\rangle \models \delta _{0}$. On the other hand, for any $i \in {\mathbb N}$, $\langle {\mathfrak N}, \sigma (u_{k+1})\rangle \models \delta _{i+1}$, that is, $\langle {\mathfrak N}, \sigma (u_{k+1})\rangle \models \Box \ulcorner {\delta _{i}}\urcorner$. Thus, $\langle {\mathfrak N}, \sigma (u_{k+1})\rangle \not \models \delta _{0}$, a contradiction.

Case 2: for any number k, $\langle {\mathfrak N}, \sigma (u_{k})\rangle {\models } \delta _{0}$. From $\langle {\mathfrak N}, \sigma (u_{0})\rangle {\models } \delta _{0}$, we can see that for some $i\in {\mathbb N}$, $\langle {\mathfrak N}, \sigma (u_{0})\rangle {\not \models } \Box \ulcorner {\delta _{i}}\urcorner$. But u₀Ru₁R…Ru_i, $\delta _{1}\equiv \Box \ulcorner {\delta _{0}}\urcorner$, $\delta _{2}\equiv \Box \ulcorner {\delta _{1}}\urcorner$, and so on. We can see $\langle {\mathfrak N}, \sigma (u_{i+1})\rangle {\not \models }\delta _{0}$, a contradiction again! □

Theorem 4

${\mathbf K}^{\omega }+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is inconsistent.

Theorem 11 is immediate from Lemma 10 and 11. For a different proof, we refer the reader to [14, p. 212].

The origin of Theorem 11 is McGee’s theorem, which shows that ${\mathbf K}+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is ω-inconsistent.⁴ Like Montague, McGee also proved his theorem by a proof-theoretical approach. Note that $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is the ω-rule closure of $\mathbf {K}+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$, and thus $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is consistent, iff $\mathbf {K}+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$ is consistent in ω-logic. We also know that the consistency in ω-logic implies the ω-consistency. Hence, we can obtain Theorem 11 from McGee’s theorem. It also shows that McGee’s theorem is stronger than Theorem 11. However, our proof of Theorem 11 is a semantical one, providing us more information. The following notion is helpful for our understanding of this point.

Definition 2 ([17, pp. 241, 249, 254])

Let Σ and Γ be two sets of sentences. A digraph is said to be a characterization digraph of Σ, if Σ is paradoxical on this digraph. Σ has the same degree of paradoxicality as Γ, if they have precisely the same characterization digraphs. Σ has the degree of paradoxicality no higher than Γ, if the characterization digraphs of Σ is included in those of Γ.

Our proof of Theorem 11 employs McGee’s modal paradox. According to Definition 17, we can also use any paradox whose degree of paradoxicality is not less than McGee’s modal paradox. In contrast, the following result shows that there exists a serial model which is admissible for any Boolean modal paradox. It indicates that we cannot prove Theorem 11 by the use of any Boolean paradox. Recall the inconsistency of $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is caused by any modal paradox. In this sense, we can take the inconsistency of $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to A$ is stronger than that of $\mathbf {K}^{\omega }+ \Box \ulcorner {A}\urcorner \to \Diamond \ulcorner {A}\urcorner$.

Lemma 7

Any Boolean modal net is not paradoxical on the digraph $\langle {\mathbb N}, \textup {Suc}\rangle$, where S is the successor relation: 〈n,m〉∈Suc, iff m = n + 1.

Proof

Since the digraph $\langle {\mathbb N}, \textup {Suc}\rangle$ is acyclic, by Proposition 6, the unwinding of a Boolean modal net is not self-referential. We notice that $\langle {\mathbb N}, \textup {Suc}\rangle$ is locally finite. By Theorem 4, if the unwinding of a Boolean modal net on $\langle {\mathbb N}, \textup {Suc}\rangle$ is not self-referential, then that Boolean modal net cannot be paradoxical on $\langle {\mathbb N}, \textup {Suc}\rangle$. □

The above results on modal theories are all negative: these modal theories are inconsistent due to the occurrence of some modal paradox. From Theorem 9, we can also establish positive results about the modal theories. In this respect, we should mention that the first two conclusions about the characterization of consistent modal theories are about K^ω itself and $\mathbf {K}^{\omega }+\Box \ulcorner {A}\urcorner \to \Box ^{2}\ulcorner {A}\urcorner$, which was proved by Halbach et al. [14, pp. 212, 213]. What they proved is K^ω and $\mathbf {K}^{\omega }+\Box \ulcorner {A}\urcorner \to \Box ^{2}\ulcorner {A}\urcorner$ are sound and complete to the corresponding digraphs ($\mathbf {K}^{\omega }+\Box \ulcorner {A}\urcorner \to \Box ^{2}\ulcorner {A}\urcorner$, the digraphs are transitive, and for K^ω, just the digraphs without any requirement). Again, thanks to Theorem 9 that they gave, which is a cornerstone of exploring the consistency of modal theories and their adequacy. We will give a generalization of Halbach et al.’s characterization. As we will see, for those consistent modal theories, it is not surprising that they are sound and complete to the corresponding digraphs, just as the analogous modal systems in modal (operator) logic.

For numbers m, n, k, and l, we define schema $G^{m, n}_{k, l}$ as follows:

$$ \Diamond^{m}\Box^{n}\ulcorner{A}\urcorner\to\Box^{k}\Diamond^{l}\ulcorner{A}\urcorner. $$

Lemma 8

$G^{m, n}_{k, l}$ is valid on a digraph, iff this digraph is $\left ({m, n}_{k, l}\right )$-convergent: for all u,v,w ∈ W, if uR^mv and uR^kw, there exists x ∈ W, such that vRⁿx and wR^lx.

The proof of Lemma 13 is similar to that of the analogue in modal logic. We refer the reader to [4, p. 89] or [22, p. 52].

Theorem 5

${\mathbf K}^{\omega }+G^{m, n}_{k, l}$ is inconsistent, iff m = k = 0, but not n = l = 0. What is more, if ${\mathbf K}^{\omega }+G^{m, n}_{k, l}$ is consistent, then for any sentence A, ${\mathbf K}+G^{m, n}_{k, l}\vdash ^{\omega } A$, iff A is valid on all $\left ({m, n}_{k, l}\right )$-convergent digraphs.

Proof

If m = k = 0, $G^{m, n}_{k, l}$ is collapsed to $\Box ^{n}\ulcorner {A}\urcorner \to \Diamond ^{l}\ulcorner {A}\urcorner$. The corresponding convergence condition is as follows: for all u ∈ W, there exists v ∈ W, such that uRⁿv and uR^lv. Suppose further either n≠ 0 or l≠ 0, then clearly the above convergence condition implies the seriality condition. By Lemma 11, we can see $\mathbf {K}^{\omega }+G^{m, n}_{k, l}$ is inconsistent.

Conversely, we show ${\mathbf K}^{\omega }+G^{m, n}_{k, l}$ is consistent whenever either n = l = 0, or not m = k = 0. By Theorem 9, it suffices to find an admissible model on which $G^{m, n}_{k, l}$ is valid for those m,k,n,l satisfying the above condition. And, by Proposition 2, we have known that there is a (unique) admissible assignment on any conversely well-founded digraph. In particular, there is an admissible assignment on the minimal irreflexive digraph 〈{0},≠〉 (or any digraph with the empty binary relation). We shall show $G^{m, n}_{k, l}$ is valid on this digraph (provided that m,k,n,l satisfies the above condition).

Notice that the convergence condition for $G^{m, n}_{k, l}$ can be expressed as in first-order formula $\forall u\forall v\left (uR^{m}v\to \ldots \right )$ or $\forall u\forall w\left (uR^{k}w\to \ldots \right )$. Thus, if not both m and k are equal to 0, the formula is trivially true when the binary relation R is empty. Besides, if both n and l are equal to 0, $G^{m, n}_{k, l}$ is collapsed to $\Diamond ^{m}\ulcorner {A}\urcorner \to \Box ^{k}\ulcorner {A}\urcorner$, and the corresponding condition is: $\forall u\forall v\forall w\left (uR^{m}v\wedge uR^{k}w\to v=w\right )$. Again this condition is trivially true for the empty relation R. To sum up, whenever either n = l = 0, or not m = k = 0, the condition for $G^{m, n}_{k, l}$ always holds on the minimal irreflexive digraph. Consequently, by Lemma 13, $G^{m, n}_{k, l}$ is valid on the minimal irreflexive digraph.

It remains to show that when ${\mathbf K}^{\omega }+G^{m, n}_{k, l}$ is consistent, ${\mathbf K}+G^{m, n}_{k, l}\vdash ^{\omega } A$, iff A is valid on all $\left ({m, n}_{k, l}\right )$-convergent digraphs. The necessity can be proved by (transfinite) induction on the (possibly infinite) proof length of A. For the sufficiency, first note $\mathbf {K}^{\omega }+G^{m, n}_{k, l}$ is exactly the set $\left \{A\mid \mathbf {K}+G^{m, n}_{k, l}\vdash ^{\omega } A\right \}$. Thus, $\mathbf {K}+G^{m, n}_{k, l}\vdash ^{\omega } A$, iff $\mathbf {K}^{\omega }+G^{m, n}_{k, l}\vdash A$. That means, when $\mathbf {K}^{\omega }+G^{m, n}_{k, l}$ is consistent, $\mathbf {K}+G^{m, n}_{k, l}$ is consistent in ω-logic.

Suppose ${\mathbf K}+G^{m, n}_{k, l}\nvdash ^{\omega } A$, then ${\mathbf K}^{\omega }+G^{m, n}_{k, l}\wedge \neg A$ is consistent. By Theorem 9, there is an admissible model, say 〈W,R,σ〉, such that $G^{m, n}_{k, l}\wedge \neg A$ is valid on this model. By Lemma 13, the validity of $G^{m, n}_{k, l}$ on this model implies 〈W,R〉 is $\left ({m, n}_{k, l}\right )$-convergent. A is invalid on 〈W,R〉 for its negation is valid on the model 〈W,R,σ〉. In conclusion, we find an $\left ({m, n}_{k, l}\right )$-convergent digraph, on which A is invalid. □

From Theorem 12, we can see that among schema $G^{m, n}_{k, l}$, only schema of form $\Box ^{n}\ulcorner {A}\urcorner \to \Diamond ^{l}\ulcorner {A}\urcorner$ implies the paradoxical contradiction on the background theory K^ω. In this sense, Montague’s theorem, and especially, McGee’s theorem, are two prototypes about the inconsistent modal theories. Also, Theorem 12 suggests that when we take the notion of necessity as a predicate of sentences, there do be some modal theories that ‘must be sacrificed’ due to their inconsistency, but what must be sacrificed, not at all as Montague [26, p. 294] had declared, are not ‘all of modal logic’, but only a small part of modal logic.