2 The language of paradox $\mathcal {L}_{\mathsf {P}}$

Our first task is a brief review of a powerful framework for studying the patterns of sentential reference that generate semantic paradoxes: the language of paradox $\mathcal {L}_{\mathsf {P}}$ . $\mathcal {L}_{\mathsf {P}}$ contains conjunction $\wedge $ , a proper class of sentence names $\mathsf { S}_\alpha , \alpha \in \mathsf {On}$ , and a falsity predicate $\mathsf {F}$ .Footnote ² We allow arbitrarily long infinitary conjunction—hence, given any non-empty class $A \subseteq \mathsf {On}$ ,

$$ \begin{align*} \wedge\{\mathsf{F}(S_\alpha) : \alpha \in A\} \end{align*} $$

is a well-formed formula. This follows the treatment in [Reference Cook2, Reference Cook3]. For present purposes, however, we will have no need for anything beyond countably infinite conjunctions. The well-formed formulas $\mathsf {WFF}_{\mathcal {L}_{\mathsf {P}}}$ are all and only the conjunctions of expressions of the form $\mathsf {F}(\mathsf {S}_\alpha )$ , where a formula of the form $\mathsf {F}(\mathsf {S}_\alpha )$ alone is a treated as a one-conjunct conjunction and hence is in $\mathsf {WFF}_{\mathcal {L}_{\mathsf {P}}}$ .

A quick note on notation: Officially, only the $\mathsf {S}_\alpha $ s ( $\alpha \in \mathsf {On}$ ) are sentence names in $\mathcal {L}_{\mathsf {P}}$ . To improve clarity, however, we’ll sometimes use other letters in the same sans serif font for sentences letters (e.g., $\mathsf {L}$ for the sentence name in the Liar paradox, $\mathsf {Y}_\alpha $ s for sentence names in the Yablo paradox, and $\mathsf {M}_\alpha $ s and $\mathsf {M}^{\mathsf { D}}_\alpha $ s for sentence names in the McGee construction and its dual), and we will also use ordered n-tuples and other convenient constructions for indexing sentence letters. These more convenient notations should be understood as merely abbreviations for official $\mathcal {L}_{\mathsf {P}}$ constructions.

We obtain interesting (e.g., paradoxical) constructions within $\mathcal {L}_{\mathsf {P}}$ by applying a denotation function to $\mathcal {L}_{\mathsf {P}}$ . Denotation functions assign well-formed formulas to sentence names:

Definition 2.1. A denotation function is any function

$$ \begin{align*} \delta : \{\mathsf{S}_{\alpha}\}_{\alpha \in \mathsf{On}} \rightarrow \mathsf{WFF}_{\mathcal{L}_{\mathsf{P}}}, \end{align*} $$

mapping each sentence name in $\mathcal {L}_{\mathsf {P}}$ to a well-formed formula of $\mathcal {L}_{\mathsf {P}}$ .

Denotation functions allow us to reconstruct familiar semantic puzzles within $\mathcal {L}_{\mathsf {P}}$ . For example, the Liar paradox is obtained via selecting any denotation function $\delta _{\mathsf {L}}$ such that

$$ \begin{align*} \delta_{\mathsf{L}}(\mathsf{L}) = \mathsf{F}(\mathsf{L}). \end{align*} $$

Loosely put: Sentence name $\mathsf {L}$ names the sentence stating that (the sentence named by) $\mathsf {L}$ is false. We can express this more colloquially as

$$ \begin{align*} \mathsf{L}: \mathsf{F}(\mathsf{L}). \end{align*} $$

Similarly, we can reconstruct the Yablo paradox by selecting a denotation function $\delta _{\mathsf {Y}}$ where, for all $n \in \mathbb {N}$ ,

$$ \begin{align*} \delta_{\mathsf{Y}}(\mathsf{Y}_n) = \wedge\{\mathsf{F}(\mathsf{Y}_m) : m \in \mathbb{N} ~ \& ~ m> n\}, \end{align*} $$

and we can express this a bit more informally as

$$ \begin{align*} \mathsf{Y}_0: ~& \mathsf{F}(\mathsf{Y}_1) \wedge \mathsf{F}(\mathsf{Y}_2) \wedge \mathsf{F}(\mathsf{Y}_3) \wedge \cdots,\\ \mathsf{Y}_1: ~& \mathsf{F}(\mathsf{Y}_2) \wedge \mathsf{F}(\mathsf{Y}_3) \wedge \mathsf{F}(\mathsf{Y}_4) \wedge \cdots,\\ \mathsf{Y}_2: ~& \mathsf{F}(\mathsf{Y}_3) \wedge \mathsf{F}(\mathsf{Y}_4) \wedge \mathsf{F}(\mathsf{Y}_5) \wedge \cdots,\\ \text{etc.} \end{align*} $$

It will be useful to have some notation that tracks how a particular denotation function “behaves” with respect to a particular sentence or collection of sentences. Thus:

Definition 2.2. Given a denotation function $\delta $ , the dependency relation on $\delta $ is:Footnote ³

$$ \begin{align*} \mathsf{Dep}_\delta = \{\langle \mathsf{S}_\alpha, \mathsf{S}_\beta \rangle : \delta(\mathsf{S}_\alpha) = \wedge \{\mathsf{F}(\mathsf{S}_\gamma) : \gamma \in \Gamma\} \text{ and } \beta \in \Gamma\}. \end{align*} $$

The dependency relation $\mathsf {Dep}_\delta $ tracks dependencies amongst the sentences in $\mathcal {L}_{\mathsf {P}}$ induced by the denotation function $\delta $ : if $\langle \mathsf {S}_\alpha , \mathsf {S}_\beta \rangle \in \mathsf {Dep}_\delta $ , then the truth value of $\mathsf { S}_\alpha $ depends (in part) on the truth value of $\mathsf {S}_\beta $ , since $\mathsf {S}_\alpha $ refers to a conjunction containing $\mathsf {S}_\beta $ (according to $\delta $ ). We will write $\mathsf {Dep}_\delta (\mathsf {S}_\alpha , \mathsf {S}_\beta )$ as shorthand for $\langle \mathsf { S}_\alpha , \mathsf {S}_\beta \rangle \in \mathsf {Dep}_\delta $ . Figure 1 depicts (an initial segment of) the dependency relation for the $\mathcal {L}_{\mathsf {P}}$ version of the Yablo paradox (i.e., $\{ \mathsf {Y}_n\}_{ n \in \mathbb {N} }$ with $\delta _{\mathsf {Y}}$ ).Footnote ⁴

Fig. 1

An initial segment of $\mathsf {Dep}_{\delta _{\mathsf {Y}}}$ .

It will be convenient to restrict our attention to collections of sentence names (with a denotation function) that are closed in the following sense:

Informally: A class of sentence names (with a denotation function) is closed if, whenever it contains a sentence name $\mathsf {S}_\alpha $ , it contains all sentence names that $\mathsf {S}_\alpha $ depends on. All examples discussed in the remainder of this essay are closed.

We evaluate the status of various (usually closed) collections of sentence names relative to a particular denotation function via considering assignments:

Definition 2.4. An assignment is any (total) function $\sigma $ mapping the sentence names to truth values:

$$ \begin{align*} \sigma : \{\mathsf{S}_\alpha\}_{\alpha \in \mathsf{On}} \rightarrow \{\top, \bot\}. \end{align*} $$

$\mathcal {L}_{\mathsf {P}}$ -acceptability of an assignment is straightforward:

An assignment is $\mathcal {L}_{\mathsf {P}}$ -acceptable when it assigns truth to a sentence name $\mathsf {S}_\alpha $ if and only if it assigns falsity to every sentence name occurring in the conjunction denoted by $\mathsf {S}_\alpha $ (and hence, intuitively, assigns truth to the conjunction itself).

We can now define what it means to say that an $\mathcal {L}_{\mathsf {P}}$ construction is paradoxical, determinate, or indeterminate:

It is now straightforward to prove that the $\mathcal {L}_{\mathsf {P}}$ version of the Yablo paradox is paradoxical:

Actually, the paradoxicality of the Yablo paradox is merely a special case of the following phenomenon:Footnote ⁵

Note that this result is unifying, in the sense that it identifies a property shared by many (but not all—see below!) paradoxes. For example, the $\mathcal {L}_{\mathsf {P}}$ version of the Liar paradox is also transitive. Thus, $\mathcal {L}_{\mathsf {P}}$ is a powerful tool for identifying general properties of paradoxes, allowing us to shift our attention away from isolated examples, and towards general types.

The connections between the Liar paradox and the Yablo paradox are actually much deeper than the mere fact that both are transitive, however, since the Yablo paradox is (isomorphic to) the “unwinding” of the Liar paradox. First, a definition:

Definition 2.9. $\ll $ is the lexicographic ordering on pairs of ordinals:

$$ \begin{align*} \langle \alpha_1, \alpha_2 \rangle \ll \langle \beta_1, \beta_2 \rangle \text{ iff } &\text{either: } & \alpha_1 < \beta_1,\\ & \ \ \ \ \ \ \text{or: } & \alpha_1 = \beta_1 \text{ and } \alpha_2 < \beta_2. \end{align*} $$

Given an $\mathcal {L}_{\mathsf {P}}$ construction, we can “unwind” it, obtaining an infinite analogue that contains no loops in its dependency relation:

Definition 2.10. The unwinding of $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta $ is $\{\mathsf {S}_\alpha \}_{\alpha \in A}^{\mathsf {U}}$ with $\delta ^{\mathsf { U}}$ where:

$$ \begin{align*} \{\mathsf{S}_\alpha\}_{\alpha \in A}^{\mathsf{U}} = \{\mathsf{S}_{\langle n, \alpha \rangle} : n \in \mathbb{N} \text{ and } \alpha \in A\} \end{align*} $$

and

$$ \begin{align*} \delta^{\mathsf{U}}(\mathsf{S}_{\langle n, \alpha \rangle}) = \wedge\{\mathsf{S}_{\langle m, \beta \rangle} : \langle n, \alpha \rangle \ll \langle m, \beta \rangle \text{ and } \mathsf{Dep}_\delta(\mathsf{S}_\alpha, \mathsf{S}_\beta)\}. \end{align*} $$

Given an assignment $\sigma $ and a set of sentence names $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with denotation function $\delta $ , the following construction provides us with a new assignment that treats each sentence name in $\{\mathsf {S}_\alpha \}_{\alpha \in A}^{\mathsf {U}}$ with $\delta ^{\mathsf {U}}$ analogously to the way $\sigma $ treats its correlate in the original $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta $ :

Definition 2.11. The $\mathbb {N}$ -image of an assignment $\sigma $ is $\sigma ^{\mathbb {N}}$ where (for any $\alpha \in \mathsf {On}, n \in \mathbb {N})$ :

$$ \begin{align*} \sigma^{\mathbb{N}}(\mathsf{S}_{\langle n, \alpha\rangle}) = \sigma(\mathsf{S}_\alpha). \end{align*} $$

We now obtain the following striking result:Footnote ⁶

Further, it turns out that the assignments provided by Definition 2.11 are the only assignments (if any) that can be $\mathcal {L}_{\mathsf {P}}$ -acceptable on any unwinding:

Definition 2.13. An assignment $\sigma $ is recurrent on $\{\mathsf {S}_\alpha \}_{\alpha \in A}^{\mathsf {U}}$ with $\delta ^{\mathsf {U}}$ if and only if, for any $\alpha \in A$ and $n, m \in \mathbb {N}$ :

$$ \begin{align*} \sigma(\mathsf{S}_{\langle n, \alpha \rangle}) = \sigma(\mathsf{S}_{\langle m, \alpha \rangle}). \end{align*} $$

Thus, the following corollary is immediate:Footnote ⁷

This result allows us to further unify our explanation of at least some semantic paradoxes: Not only are the Liar and the Yablo paradox both instances of the transitivity-entails-paradoxicality phenomenon, but in addition, the latter is the unwinding of the former, and hence will have the same status vis-a-vis paradoxicality, determinacy, or indeterminacy. As we shall see, there are other well-known semantic paradoxes that can be easily re-formulated within $\mathcal {L}_{\mathsf {P}}$ , but which are neither transitive nor the unwindings of any simpler $\mathcal {L}_{\mathsf {P}}$ constructions.

Our next result concerns $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ —the dual of the language of paradox $\mathcal {L}_{\mathsf {P}}$ . We obtain $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ by replacing the conjunctions in $\mathcal {L}_{\mathsf {P}}$ with disjunctions. The definitions of denotation function, dependency relation, closure, and assignments remain the same, and the only modification required is adopting the following definition of $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ -acceptability, which reverses the role of truth and falsity:

Thus, an assignment is $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ -acceptable when it assigns falsity to a sentence name $\mathsf {S}_\alpha $ if and only if it assigns truth to every sentence name occurring in the disjunction denoted by $\mathsf {S}_\alpha $ (and hence, intuitively, assigns falsity to the disjunction itself).

The following definition and theorem provide the connection between constructions in $\mathcal {L}_{\mathsf {P}}$ and constructions in $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ .

Definition 2.17. Given an assignment $\sigma $ , $\sigma ^{\mathsf {D}}$ is the assignment:

$$ \begin{align*} \sigma^{\mathsf{D}}(\mathsf{S}_\alpha) & = \top, & \text{if } \sigma(\mathsf{S}_\alpha) = \bot,\\ & = \bot, & \text{if } \sigma(\mathsf{S}_\alpha) = \top. \end{align*} $$

Simply put, replacing the conjunctions in an $\mathcal {L}_{\mathsf {P}}$ construction with disjunctions (resulting in an $\mathcal {L}_{\mathsf { P}}^{\mathsf {D}}$ construction) preserves the status of the construction with respect to the paradoxicality/determinacy/indeterminacy trichotomy (and vice versa).

So far, we have only discussed results that are already well-known from the literature on $\mathcal {L}_{\mathsf {P}}$ . We now state a novel result, which we shall prove in the next section. First, some additional definitions:

Definition 2.19. A unary function:

$$ \begin{align*} f: \{\mathsf{S}_\alpha\}_{\alpha \in A} \rightarrow\{\mathsf{S}_\beta\}_{\beta \in B} \end{align*} $$

is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ to $\{\mathsf { S}_\beta \}_{\beta \in B}$ with $\delta _2$ if and only if:

• • Homo: For every $\mathsf {S}_{\alpha _1}, \mathsf {S}_{\alpha _2} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ , if $\mathsf { Dep}_{\delta _1}(\mathsf {S}_{\alpha _1}, \mathsf {S}_{\alpha _2})$ , then $\mathsf {Dep}_{\delta _2}(f(\mathsf {S}_{\alpha _1}), f(\mathsf { S}_{\alpha _2}))$ .

• • Strong: For every $\mathsf {S}_{\alpha _1} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}, \mathsf {S}_{\beta _1} \in \{\mathsf { S}_\beta \}_{\beta \in B}$ , if $\mathsf {Dep}_{\delta _2}(f(\mathsf {S}_{\alpha _1}), \mathsf {S}_{\beta _1})$ , then there is an $\mathsf { S}_{\alpha _2} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ such that $\mathsf {Dep}_{\delta _1}(\mathsf {S}_{\alpha _1}, \mathsf {S}_{\alpha _2})$ and $f(\mathsf {S}_{\alpha _2}) = \mathsf {S}_{\beta _1}$ .

We will write:

if there is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ to $\{\mathsf {S}_\beta \}_{\beta \in B}$ with $\delta _2$ , and

if there is a strong surjective $\mathcal {L}_{\mathsf {P}}$ -homomorphism from $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ to $\{\mathsf { S}_\beta \}_{\beta \in B}$ with $\delta _2$ .Footnote ⁸

Given a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism f from $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ to $\{\mathsf {S}_\beta \}_{\beta \in B}$ with $\delta _2$ , any assignment $\sigma $ on $\{\mathsf {S}_\beta \}_{\beta \in B}$ determines a corresponding assignment $\sigma ^f$ on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ :

Definition 2.21. Given a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism f from $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ to $\{\mathsf {S}_\beta \}_{\beta \in A}$ with $\delta _2$ ,

and any assignment $\sigma $ , the assignment induced by f on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\sigma $ ) is $\sigma ^f$ where:

$$ \begin{align*} \sigma^f(\mathsf{S}_\alpha) & = ~\sigma(f(\mathsf{S}_\alpha)), \quad \text{if } \alpha \in A,\\ & = ~\top, \qquad\qquad \text{ otherwise}. \end{align*} $$

We now state (but do not yet prove) what we shall call the strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism theorem (or S $\mathcal {L}_{\mathsf {P}}$ H theorem):

Theorem 2.22. If

then (where f is any function witnessing the strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism), for any assignment $\sigma $ , if $\sigma $ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\beta \}_{\beta \in B}$ with $\delta _2$ then $\sigma $ , $\sigma ^f$ are $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ .

We can strengthen the conditional to a biconditional by requiring that the strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism be surjective:Footnote ⁹

Theorem 2.23. If

then (where f is any function witnessing the strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism), for any assignment $\sigma $ , $\sigma ^f$ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ if and only if $\sigma $ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\beta \}_{\beta \in B}$ with $\delta _2$ .

We could prove the S $\mathcal {L}_{\mathsf {P}}$ H theorem (and its strengthening Theorem 2.23) within the framework just described.Footnote ¹⁰ Here we shall take a somewhat different path. It turns out that these theorems can be easily reformulated and proven with very basic category-theoretic tools. Doing so is the task of the next section.

3 $\mathcal {L}_{\mathsf {P}}$ -homomorphisms and category theory

In this section we will reconstruct some of the machinery of the language of paradox $\mathcal {L}_{\mathsf {P}}$ within the category of sets Set, and use this reconstruction to prove the results given at the end of the previous section.

Before doing so, a historical note is in order. The results given in this section, and applied in the sections that follow, are not the first time that investigations of paradoxicality and related phenomenon have mobilized category-theoretic tools. Of particular note, in this regard, is [Reference Lawvere, Mitchell, Roos, Ulmer, Brinkmann, Chase, Dedecker, Douglas, Hilton, Sigrist, Ehresmann, Gruenberg, Knus, William Lawvere and Mac Lane6], which proves a number of fixed-point theorems (including versions of Tarski’s Theorem and Cantor’s Theorem) within a category-theoretic setting. Thus, the idea that category theory has something to teach us about paradoxes is not new. The particular methods deployed below, however, are novel, and no doubt merely scratch the surface of what can be learned about semantic paradoxes via adopting a category-theoretic perspective.Footnote ¹¹

Back to the mathematics! Understood category-theoretically, a denotation function $\delta $ is just a $\mathcal {P}$ -coalgebra from S to $\mathcal {P}(S)$ (where $\mathcal {P}$ is the (covariant) powerset functor $\mathcal {P}$ in Set) for some set S (understood as our set of sentence names).

Given this understanding of a denotation function, we can define acceptability as follows: Let $\Omega $ be the set of truth values (i.e., $\{\top , \bot \}$ ), and $\downarrow{\kern-1pt}: \mathcal {P}(\Omega ) \rightarrow \Omega $ be the following morphism from $\mathcal {P}(\Omega )$ to $\Omega $ :

$$ \begin{align*} \downarrow\!(\{\bot\}) & =\ \downarrow\!(\varnothing) = \top,\\ \downarrow\!(\{\top\}) &= \ \downarrow\!(\{\top, \bot\} = \Omega) = \bot, \end{align*} $$

$\downarrow $ is the category-theoretic analogue of the Pierce arrow expressing, in the present context (and put somewhat loosely), something like “all of the sentences are false.”Footnote ¹² Then, given a denotation function $\delta : S \rightarrow \mathcal {P}(S)$ , an assignment is just a morphism $\sigma : S \rightarrow \Omega $ , and an assignment $\sigma $ is $\mathcal {L}_{\mathsf {P}}$ -acceptable relative to a denotation function $\delta $ if and only if Figure 2 commutes.

Fig. 2

$\sigma $ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\delta $ .

Next, a strong homomorphism between denotation functions $\delta _1$ and $\delta _2$ is, in the category-theoretic setting, just a coalgebra morphism between the $\mathcal {P}$ -coalgebras $\delta _1: S_1 \rightarrow \mathcal {P}(S_1)$ and $\delta _2: S_2 \rightarrow \mathcal {P}(S_2)$ . $f : S_1 \rightarrow S_2$ is a coalgebra morphism from $\delta _1$ to $\delta _2$ if and only if Figure 3 commutes.

Fig. 3

f is a strong homomorphism from $\delta _1$ to $\delta _2$ .

Finally, the additional requirement that a strong homomorphism be surjective, needed for the strengthened version of the S $\mathcal {L}_{\mathsf { P}}$ H theorem (i.e., Theorem 2.23), becomes, in the present context, the requirement that the coalgebra morphism f is an epimorphism.

We can now provide a very simple category-theoretic proof of the S $\mathcal {L}_{\mathsf {P}}$ H theorem:

Proof. Assume $f: S_1 \rightarrow S_2$ is a coalgebra morphism from $\delta : S_1 \rightarrow \mathcal {P}(S_1)$ to $\delta _2 : S_2 \rightarrow \mathcal {P}(S_2)$ , and hence

commutes. Now, assume $\sigma $ is acceptable on $\langle S_2, \delta _2 \rangle $ , and hence

commutes. Then

commutes, and hence

commutes. Hence $\sigma \circ f$ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\langle S_1, \delta _1 \rangle $ .

Adding in surjectivity, we obtain the following category-theoretic version of Theorem 2.23:

Proof. Assume $f: S_1 \rightarrow S_2$ is a coalgebra morphism from $\delta : S_1 \rightarrow \mathcal {P}(S_1)$ to $\delta _2 : S_2 \rightarrow \mathcal {P}(S_2)$ , and hence

commutes, and assume further that f is an epimorphism, and hence, for any two $g, h : S_2 \rightarrow X$ , if $gf = hf$ , then $g = h$ . Assume $\sigma \circ f$ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\langle S_1, \delta _1 \rangle $ , and hence

commutes. The second diagram gives us

$$ \begin{align*} \downarrow \circ~ \mathcal{P}\sigma \circ \mathcal{P}f \circ \delta_1 = \sigma \circ f, \end{align*} $$

while the first diagram provides

$$ \begin{align*} \delta_2 \circ f = \mathcal{P}f \circ \delta_1. \end{align*} $$

A simple substitution then gives us

$$ \begin{align*} \downarrow \circ~ \mathcal{P}\sigma \circ \delta_2 \circ f = \sigma \circ f, \end{align*} $$

which, since f is an epimorphism, provides

$$ \begin{align*} \downarrow \circ~ \mathcal{P}\sigma \circ \delta_2 = \sigma. \end{align*} $$

Hence,

commutes, and $\sigma $ is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\langle S_2, \delta _2 \rangle $ .

The following corollary is straightforward:

Corollary 3.3. If then

$$ \begin{align*} &|\{\sigma : \sigma \text{ is } \mathcal{L}_{\mathsf{P}}\text{-acceptable on } \{\mathsf{S}_\alpha\}_{\alpha \in A} \text{ with } \delta_1\}| \\ \geq~& |\{\sigma : \sigma \text{ is } \mathcal{L}_{\mathsf{P}}\text{-acceptable on } \{\mathsf{S}_\beta\}_{\beta \in B} \text{ with } \delta_2\}|. \end{align*} $$

Proof. Assume that , and let

$$ \begin{align*} f : \{\mathsf{S}_\alpha\}_{\alpha \in A} \rightarrow \{\mathsf{S}_\beta\}_{\beta \in B} \end{align*} $$

be any $\mathcal {L}_{\mathsf {P}}$ -homomorphism witnessing this fact. Then, if $\sigma _1$ and $\sigma _2$ are distinct assignments that are $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\beta \}_{\beta \in B}$ , then there is an $\mathsf {S}_{\beta _1} \in \{\mathsf {S}_\beta \}_{\beta \in B}$ such that

$$ \begin{align*} \sigma_1(\mathsf{S}_{\beta_1}) \not= \sigma_2(\mathsf{S}_{\beta_1}). \end{align*} $$

But then, where $\mathsf {S}_{\alpha _1} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ is such that $f(\mathsf {S}_{\alpha _1}) = \mathsf {S}_{\beta _1}$ , we have:Footnote ¹³

$$ \begin{align*} \sigma_1^f(f(\mathsf{S}_{\alpha_1})) = \sigma_1(\mathsf{S}_{\beta_1}) \not= \sigma_2(\mathsf{S}_{\beta_1}) = \sigma_2^f(f(\mathsf{S}_{\alpha_1})). \end{align*} $$

Thus, the mapping $(\xi )^f$ (induced by the strong $\mathcal {L}_{\mathsf {P}}$ homomorphism f) from $\mathcal {L}_{\mathsf {P}}$ -acceptable assignments on $\{\mathsf {S}_\beta \}_{\beta \in B}$ with $\delta _2$ to $\mathcal {L}_{\mathsf {P}}$ -acceptable assignments on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta _1$ is one-to-one (but not necessarily onto).

As hinted in the final line of this proof, the inequality between the cardinalities of the classes of $\mathcal {L}_{\mathsf {P}}$ -acceptable assignments cannot be strengthened to an identity, as the following simple example shows. Select a denotation function $\delta $ that provides an instance of the Liar

$$ \begin{align*} \delta(\mathsf{L}) = \mathsf{F}(\mathsf{L}) \end{align*} $$

and an instance of the Open PairFootnote ¹⁴

$$ \begin{align*} \delta(\mathsf{OP_1}) &= \mathsf{F}(\mathsf{OP_2}),\\[-2pt] \delta(\mathsf{OP_2}) &= \mathsf{F}(\mathsf{OP_1}), \end{align*} $$

then $f : \{\mathsf {OP}_1, \mathsf {OP}_2 \} \rightarrow \{\mathsf {L}\}$ where

$$ \begin{align*} f(\mathsf{OP}_1) = f(\mathsf{OP}_2) = \mathsf{L}, \end{align*} $$

which is pictured in Figure 4, is a strong surjective $\mathcal {L}_{\mathsf {P}}$ -homomorphism from $\{\mathsf {OP}_1, \mathsf {OP}_2 \}$ with $\delta $ to $\{\mathsf {L}\}$ with $\delta $ . But the Liar is paradoxical, while the Open Pair is not (it is indeterminate, with two $\mathcal {L}_{\mathsf {P}}$ -acceptable truth value assignments, each of which assigns $\top $ to one sentence name and $\bot $ to the other).

Fig. 4

A simple strong (surjective) $\mathcal {L}_{\mathsf {P}}$ -homomorphism.

This example of a strong surjective $\mathcal {L}_{\mathsf {P}}$ -homomorphism from a non-paradoxical $\mathcal {L}_{\mathsf {P}}$ construction to a paradoxical one is an instance of the following fact:

Theorem 3.4. For any $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with denotation function $\delta $ ,

Proof. Let

$$ \begin{align*} f(\mathsf{S}_{\alpha_1}) = \mathsf{L} \end{align*} $$

for any $\mathsf {S}_{\alpha _1} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ .

If $\mathsf {Dep}_{\delta }(\mathsf {S}_{\alpha _1}, \mathsf {S}_{\alpha _2})$ , then, since $f(\mathsf {S}_{\alpha _1}) = f(\mathsf {S}_{\alpha _2}) = \mathsf { L}$ , and $\mathsf {Dep}_{\delta _{\mathsf {L}}}(\mathsf {L}, \mathsf {L})$ , Homo is satisfied.

For Strong, the only case is $\mathsf {Dep}_{\delta _{\mathsf {L}}}(f(\mathsf {S}_{\alpha _1}), \mathsf {L})$ for some $\mathsf {S}_{\alpha _1} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ . Let $\mathsf {S}_{\alpha _2} \in \{\mathsf {S}_\alpha \}_{\alpha \in A}$ be such that $\mathsf {Dep}_\delta (\mathsf { S}_{\alpha _1}, \mathsf {S}_{\alpha _2})$ (the existence of such is guaranteed by the seriality of $\mathsf {Dep}_\delta $ and the non-emptiness of $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ ). Then $f(\mathsf {S}_{\alpha _2}) = \mathsf {L}$ .

f is clearly surjective.

Perhaps the simplest and most useful corollary of the S $\mathcal {L}_{\mathsf {P}}$ H theorem is the following corollary:Footnote ¹⁵

Given this, we can use strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms to categorize $\mathcal {L}_{\mathsf {P}}$ constructions in two (ultimately equivalent) ways:

• • We can show that $\langle \{\mathsf {S}_\alpha \}_{\alpha \in A}, \delta _1 \rangle $ is paradoxical by identifying an $\langle \{\mathsf { S}_\beta \}_{\beta \in B}, \delta _2 \rangle $ such that

  
  where $\langle \{\mathsf {S}_\beta \}_{\beta \in B}, \delta _2 \rangle $ is known to be paradoxical.

• • We can show that $\langle \{\mathsf {S}_\alpha \}_{\alpha \in A}, \delta _1 \rangle $ is non-paradoxical by identifying an $\langle \{\mathsf { S}_\beta \}_{\beta \in B}, \delta _2 \rangle $ such that

  
  where $\langle \{\mathsf {S}_\beta \}_{\beta \in B}, \delta _2 \rangle $ is known to be non-paradoxical.Footnote ¹⁶

We can further demonstrate the usefulness of strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms by noting that we can re-construe part of the content of the unwinding theorem as a mere special case of the S $\mathcal {L}_{\mathsf {P}}$ H theorem. We begin with the following observation:

Theorem 3.6. Given any $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ with $\delta $ ,

Proof. The reader is invited to verify that

$$ \begin{align*} f : \{\mathsf{S}_\alpha\}_{\alpha \in A}^{\mathsf{U}} \rightarrow \{\mathsf{S}_\alpha\}_{\alpha \in A}, \end{align*} $$

where

$$ \begin{align*} f(\mathsf{S}_{\langle n, \alpha \rangle}) = \mathsf{S}_\alpha \end{align*} $$

is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from $\{\mathsf {S}_\alpha \}_{\alpha \in A}^{\mathsf {U}}$ with $\delta ^{\mathsf {U}}$ to $\{\mathsf { S}_\alpha \}_{\alpha \in A}$ with $\delta $ .

Thus, given any $\mathcal {L}_{\mathsf {P}}$ construction $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ), there is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from the unwinding of $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ) to $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ) itself. As a result, the right-to-left direction of Corollary 2.15 is also an immediate corollary of the S $\mathcal {L}_{\mathsf {P}}$ H theorem.Footnote ¹⁷

Before moving on to our examination of the McGee paradox (and its dual), we further strengthen the connections between the $\mathcal {L}_{\mathsf {P}}$ approach and category theory by noting that the Duality Theorem can also be given a simple category-theoretic proof. The proof is given in the Appendix.

4 The McGee paradox in $\mathcal {L}_{\mathsf {P}}$

The McGee paradox was originally formulated within the language of arithmetic with a truth predicate [Reference McGee7], and in that context the main point of interest in the construction is that it was the central component in a proof that certain axiomatic theories of truth (in particular, FS; see [Reference Friedman and Sheard4]) are consistent but $\omega $ -inconsistent. Here our interests will be quite different, since we will be looking at the McGee Paradox in the context of $\mathcal {L}_{\mathsf {P}}$ , and we will, as a result, be implicitly assuming that we are working in the standard model of arithmetic (and hence that the McGee construction is genuinely paradoxical). Nevertheless, a quick reminder of McGee’s original construction is in order.

Let $\mathcal {L}^{\mathsf {T}}$ be any language sufficient for Peano Arithmetic (PA) supplemented with a unary truth predicate $\mathsf {T}(\xi )$ . $\ulcorner \Phi \urcorner $ is the name (or Gödel code) of $\Phi $ . We abbreviate iterated applications of the truth predicate as follows:

$$ \begin{align*} \mathsf{T}^1(\xi) &=_{df} \mathsf{T}(\xi),\\ \mathsf{T}^{n+1}(\xi) &=_{df} \mathsf{T}(\ulcorner \mathsf{T}^n(\xi)\urcorner). \end{align*} $$

We define falsity in terms of truth and negation:

$$ \begin{align*} \mathsf{F}(\xi) =_{df} \neg \mathsf{T} ({\xi}). \end{align*} $$

And we abbreviate the iteration of falsity predicates along the same lines:

$$ \begin{align*} \mathsf{F}^1(\xi) &=_{df} \mathsf{F}(\xi),\\ \mathsf{F}^{n+1}(\xi) &=_{df} \mathsf{F}(\ulcorner \mathsf{F}^n(\xi)\urcorner). \end{align*} $$

We now obtain the McGee Paradox via an application of Gödel’s celebrated diagonalization theorem [Reference Gödel5] to the predicate

$$ \begin{align*} \neg (\forall n \geq1) \mathsf{T}^n(\xi) \end{align*} $$

obtaining the McGee sentence $\mathsf {M}$ , where

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M} \leftrightarrow \neg (\forall n \geq1) \mathsf{T}^n(\ulcorner \mathsf{M} \urcorner). \end{align*} $$

Loosely put, the McGee Sentence M is true if and only if it is not the case that all sentences of the form:

are true. Equivalently, M is true if and only if some sentence of that form is false—that is,

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M} \leftrightarrow (\exists n \geq1)\neg \mathsf{T}^n(\ulcorner \mathsf{M} \urcorner). \end{align*} $$

McGee’s construction has a dual version that will be more convenient for our purposes, since it can be reconstructed relatively straightforwardly within $\mathcal {L}_{\mathsf {P}}$ (McGee’s original construction is more naturally reconstructed in the $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}})$ . The Dual McGee Sentence is $\mathsf {M}^{\mathsf {D}}$ where (again via Gödelian diagonalization)

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M}^{\mathsf{D}} \leftrightarrow (\forall n \geq1) \neg \mathsf{T}^n(\ulcorner \mathsf{M}^{\mathsf{D}} \urcorner). \end{align*} $$

Loosely put, the Dual McGee Sentence $\mathsf {M}^{\mathsf {D}}$ is true if and only if all sentences of the form:

are false.

Calling this the “dual” of the original McGee construction is justified via noting that the above is equivalent to

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M}^{\mathsf{D}} \leftrightarrow \neg (\exists n \geq1) \mathsf{T}^n(\ulcorner \mathsf{M}^{\mathsf{D}} \urcorner). \end{align*} $$

In short, just as the move from $\mathcal {L}_{\mathsf {P}}$ to $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ involves switching conjunctions and disjunctions, the move from the original McGee construction to its dual involves replacing a universal quantifier (akin to an infinitary conjunction) with an existential quantifier (akin to an infinitary disjunction).

Given that we want to represent the McGee Paradox within $\mathcal {L}_{\mathsf {P}}$ , it will be somewhat more convenient to work with the Dual of the McGee Paradox formulated in terms of our (defined) falsity predicate, rather than the (primitive) truth predicate.Footnote ¹⁸ The following is equivalent to the formulation of the McGee Sentence given above:

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M}^{\mathsf{D}} \leftrightarrow \neg (\exists n) \mathsf{T}(\ulcorner\mathsf{T}^n(\ulcorner \mathsf{M}^{\mathsf{D}} \urcorner)\urcorner). \end{align*} $$

Replacing each occurrence of $\mathsf {T}(\xi )$ with $\mathsf {F}(\ulcorner \mathsf {F} (\xi )\urcorner )$ , we obtain

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M}^{\mathsf{D}} \leftrightarrow \neg (\exists n) \mathsf{F}(\ulcorner \mathsf{F}(\ulcorner\mathsf{F}^{2n}(\ulcorner \mathsf{ M}^{\mathsf{D}} \urcorner)\urcorner)\urcorner). \end{align*} $$

Commuting the initial negation and the quantifier, we obtain

$$ \begin{align*} \vdash_{\mathsf{PA}} \mathsf{M}^{\mathsf{D}} \leftrightarrow (\forall n) \neg \mathsf{F}(\ulcorner \mathsf{F}(\ulcorner\mathsf{F}^{2n}(\ulcorner \mathsf{ M}^{\mathsf{D}} \urcorner)\urcorner)\urcorner). \end{align*} $$

Finally, we can simplify this a bit, via “cancelling out” the initial negation-falsity prefix, obtaining the following falsity variant of the Dual McGee Sentence:Footnote ¹⁹

$$ \begin{align*} \mathsf{M}^{\mathsf{D}} \equiv (\forall n)\mathsf{F}(\ulcorner \mathsf{F}^{2n}(\ulcorner \mathsf{M}^{\mathsf{D}} \urcorner)\urcorner). \end{align*} $$

Reconstructing the Dual of the McGee Paradox in $\mathcal {L}_{\mathsf {P}}$ is now straightforward: Our sentence names are $\{\mathsf {M}^{\mathsf {D}}_n\}_{ n \in \mathbb {N}}$ , and our denotation function $\delta _{\mathsf {M^{\mathsf {D}}}}$ is such that, for all $n \in \mathbb {N}$ ,

$$ \begin{align*} \delta_{\mathsf{M^{\mathsf{D}}}}(\mathsf{M}^{\mathsf{D}}_0) &= \wedge\{\mathsf{F}(\mathsf{M}^{\mathsf{D}}_{2m}) : m \in \mathbb{N}\},\\ \delta_{\mathsf{M^{\mathsf{D}}}}(\mathsf{M^{\mathsf{D}}_{n+1}}) &= \mathsf{F}(\mathsf{M}^{\mathsf{D}}_n). \end{align*} $$

We can write this a bit more colloquially as

$$ \begin{align*} \mathsf{M}^{\mathsf{D}}_0 &= \mathsf{F}(\mathsf{M}^{\mathsf{D}}_0) \wedge \mathsf{F}(\mathsf{M}^{\mathsf{D}}_2) \wedge \mathsf{F}(\mathsf{M}^{\mathsf{D}}_4) \wedge \mathsf{F}(\mathsf{M}^{\mathsf{D}}_6) \wedge \cdots,\\ \mathsf{M}^{\mathsf{D}}_1 &= \mathsf{F}(\mathsf{M}^{\mathsf{D}}_0), \\ \mathsf{M}^{\mathsf{D}}_2 &= \mathsf{F}(\mathsf{M}^{\mathsf{D}}_1), \\ \mathsf{M}^{\mathsf{D}}_3 &= \mathsf{F}(\mathsf{M}^{\mathsf{D}}_2), \\ \text{etc.} \end{align*} $$

Loosely speaking, $\mathsf {M}^{\mathsf {D}}_0$ “is” $\mathsf {M}^{\mathsf {D}}$ in the informal construction from Section 1, and (again, loosely speaking) each $\mathsf {M}^{\mathsf {D}}_{n+1}$ “is” $\mathsf {F}(\mathsf {M}^{\mathsf {D}}_n)$ . Hence, for $n \geq 1$ ,

$$ \begin{align*} \mathsf{M}^{\mathsf{D}}_n \text{ "is" } \mathsf{F}^n(\mathsf{M^{\mathsf{D}}_0}), \end{align*} $$

and we can thus “rewrite” $\mathsf {M}^{\mathsf {D}}_0$ as

$$ \begin{align*} \mathsf{M^{\mathsf{D}}_0} \equiv \mathsf{F}(\mathsf{F}^0(\mathsf{M}^{\mathsf{D}}_0)) \wedge \mathsf{F}(\mathsf{F}^2(\mathsf{M}^{\mathsf{D}}_0)) \wedge \mathsf{F}(\mathsf{F}^4(\mathsf{M}^{\mathsf{D}}_0)) \wedge \mathsf{F}(\mathsf{F}^6(\mathsf{M}^{\mathsf{D}}_0)) \wedge \cdots \end{align*} $$

and hence as

$$ \begin{align*} \mathsf{M}^{\mathsf{D}}_0 \equiv \mathsf{F}(\mathsf{M}^{\mathsf{D}}_0) \wedge \mathsf{F}^3(\mathsf{M}^{\mathsf{D}}_0) \wedge \mathsf{F}^5(\mathsf{M}^{\mathsf{ D}}_0) \wedge \mathsf{F}^7(\mathsf{M}^{\mathsf{D}}_0) \wedge \cdots. \end{align*} $$

This is just (modulo some unimportant notational changes, and replacing an infinite conjunction with the corresponding universal quantification) the Dual of the McGee Sentence given above:

$$ \begin{align*} \mathsf{M} \equiv (\forall n)\mathsf{F}(\ulcorner \mathsf{F}^{2n}(\ulcorner \mathsf{M} \urcorner)\urcorner) \end{align*} $$

Figure 5 depicts (an initial segment of) the dependency relation for the $\mathcal {L}_{\mathsf {P}}$ version of the Dual of the McGee paradox (i.e., $\{\mathsf {M}^{\mathsf {D}}_n\}_{n \in \mathbb {N}}$ with $\delta _{\mathsf {M}^{\mathsf {D}}}$ ). Note that the dependency relation in the $\mathcal {L}_{\mathsf {P}}$ version of the Dual of the McGee paradox is not transitive. $\{\mathsf {M}^{\mathsf { D}}_n\}_{n \in \mathbb {N}}$ with $\delta _{\mathsf {M^{\mathsf {D}}}}$ is, however, paradoxical. In this section we give a direct proof of this fact—one that does not depend on the mobilization of strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms:

Proof. Assume for reductio that there is an assignment $\sigma $ that is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {M}^{\mathsf {D}}_n \}_{ n \in \mathbb {N}}$ with $\delta _{\mathsf {M}^{\mathsf {D}}}$ .

Fig. 5

An initial segment of $\mathsf {Dep}_{\delta _{\mathsf {M}^{\mathsf {D}}}}$ .

Assume that $\sigma (\mathsf {M}^{\mathsf {D}}_0) = \top $ . Then, for all $\mathsf {M}^{\mathsf {D}}_m$ such that $\mathsf {Dep}_{\delta _{\mathsf {M^{\mathsf { D}}}}}(\mathsf {M}^{\mathsf {D}}_0, \mathsf {M}^{\mathsf {D}}_m)$ , $\sigma (\mathsf {M}^{\mathsf {D}}_m) = \bot $ . In particular, $\sigma (\mathsf {M}^{\mathsf {D}}_0) = \bot $ . Contradiction.

Thus, $\sigma (\mathsf {M}^{\mathsf {D}}_0) = \bot $ . So there is an $m \in \mathbb {N}$ such that $\sigma (\mathsf {M}^{\mathsf {D}}_{2m}) = \top $ . This contradicts our lemma. Contradiction.

Hence there is no such assignment.

The paradoxicality of the Dual of the McGee paradox is, like the Yablo Paradox, a special case of a more general phenomenon. First, some additional definitions:Footnote ²⁰

Definition 4.3. Given a denotation function $\delta $ , an infinite sequence of ordinals

$$ \begin{align*} \langle \alpha_0, \alpha_1, \alpha_2, \alpha_3 \dots \alpha_n, \alpha_{n+1} \dots \rangle \end{align*} $$

is a $\delta $ -path if and only if, for all $m \in \mathbb {N}$ , $\mathsf {Dep}_\delta (\mathsf {S}_{\alpha _m}, \mathsf {S}_{\alpha _{m+1}})$ .

Definition 4.4. A $\delta $ -path

$$ \begin{align*} \langle \alpha_0, \alpha_1, \alpha_2, \alpha_3 \dots \alpha_n, \alpha_{n+1} \dots \rangle \end{align*} $$

is an n-repeater if and only if

$$ \begin{align*} \alpha_0 = \alpha_n \end{align*} $$

and there is no $m \in \mathbb {N}$ such that $0 < m < n$ and

$$ \begin{align*} \alpha_0 = \alpha_m. \end{align*} $$

Proof. Assume that $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ) contains an $\mathsf {S}_\beta $ such that every $\delta $ -path beginning with $\beta $ is an n-repeater for odd n. Assume for reductio $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ) is not paradoxical. Then there is an assignment $\sigma $ that is $\mathcal {L}_{\mathsf {P}}$ -acceptable on $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ). If $\sigma (\mathsf {S}_\beta ) = \top $ , then there is a $\delta $ -path

$$ \begin{align*} \beta, \alpha_1, \alpha_2, \alpha_3, \dots \end{align*} $$

such that

$$ \begin{align*} \sigma(\mathsf{S}_{\alpha_n}) & = \top, \quad \text{if } n \text{ is even,}\\ & = \bot, \quad \text{if } n \text{ is odd.} \end{align*} $$

Since every $\delta $ -path beginning with $\beta $ is an n-repeater for odd n, there is an m such that m is odd and $\mathsf {S}_\beta $ = $\mathsf {S}_{\alpha _m}$ . But then

$$ \begin{align*} \top = \sigma(\mathsf{S}_\beta) = \sigma(\mathsf{S}_{\alpha_m}) = \bot. \end{align*} $$

Contradiction. Hence $\sigma (\mathsf {S}_\beta ) \not = \top $ . A similar argument shows that $\sigma (\mathsf {S}_\beta ) \not = \bot $ . Hence, there is no such $\mathcal {L}_{\mathsf {P}}$ -acceptable assignment.

This result, like the theorem regarding transitivity from the previous section, is also unifying, in the sense that it also identifies a property shared by many (but again, not all!) paradoxes. Note that the $\mathcal {L}_{\mathsf {P}}$ version of the Liar paradox is also a construction involving a sentence where every $\delta $ -path beginning with that sentence is an n-repeater for odd n. But, just as the denotation relation on the Dual of the McGee paradox is not transitive, the dependency relation on the $\mathcal {L}_{\mathsf {P}}$ version of the Yablo paradox does not contain any such n-repeaters (since it contains no loops at all!).

Now that we have given a proof of the paradoxicality of (the Dual of) the McGee paradox using methods mobilized in extant work within the $\mathcal {L}_{\mathsf {P}}$ framework, it is time to see what can be done using our new tool—strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms.

5 The infinite flower and $\mathcal {L}_{\mathsf {P}}$ -homomorphisms

We will begin this section by constructing a novel paradox: The Infinite Flower. First, we use the following pairs as indexes for the sentence names:

$$ \begin{align*} A = \{\langle n, m \rangle : 0 < m < n \text{ and } n \text{ odd, or } n = 1 \text{ and } m = 0\}. \end{align*} $$

We now define the denotation function as follows (using IF rather than S merely for aesthetics—it’s a flower, after all):

$$ \begin{align*} \delta_{\mathsf{IF}}(\mathsf{IF}_{\langle n, m \rangle}) = \begin{cases} \{\mathsf{IF}_{\langle j, 1 \rangle}: j \text{ odd}\}\cup\{\mathsf{IF}_{\langle 1, 0 \rangle}\}, & \text{if } n = 1 \text{ and } m = 0, \\ \{\mathsf{IF}_{\langle n, m+1 \rangle}: n \in \mathbb{N}\}, &\text{if } n \text{ odd} \text{ and } 0 < m < n - 1, \\ \{\mathsf{IF}_{\langle 1,0 \rangle}\}, &\text{if } n \text{ odd} \text{ and } m = n-1. \end{cases} \end{align*} $$

The structure of the Infinite Flower can be seen a bit more clearly in Figure 6, which represents its first four “petals.”

Fig. 6

An initial segment of $\mathsf {Dep}_{\delta _{\mathsf {IF}}}$ .

Clearly, any path in the Infinite Flower beginning with $\mathsf {IF}_{\langle 0, 1 \rangle }$ is an n repeater for some odd n (the n is given by the first value of the pairs labeling the sentence names in the relevant “petal”). Hence the Infinite Flower is paradoxical, and is another instance of the sort of paradoxicality delineated by Theorem 4.5.

But we don’t need all of the fancy machinery involved in Theorem 4.5 in order to see the paradoxicality of the Infinite Flower. There is a much more straightforward, rather obvious route to this knowledge. First, we merely need to note the familiar fact that any “cyclic” $\mathcal {L}_{\mathsf {P}}$ construction of odd length is paradoxical:

Definition 5.1. Given a denotation function $\delta $ , an finite sequence of n-many ordinals

$$ \begin{align*} \langle \alpha_0, \alpha_1, \alpha_2, \alpha_3 \dots \alpha_{n-1} \rangle \end{align*} $$

is an n-cycle (with $\delta $ ) if and only if:

$$ \begin{align*} \mathsf{Dep}_\delta = \{\langle \mathsf{S}_{\alpha_m}, \mathsf{S}_{\alpha_{m+1}} \rangle : 0 \leq m < n-1\} \cup \{\langle \mathsf{S}_{\alpha_{n+1}}, \mathsf{S}_{\alpha_0} \rangle \}. \end{align*} $$

The Liar sentence L (with $\delta _{\mathsf {L}}$ ) is a 1-repeater (hence paradoxical), while the Open Pair $\mathsf {OP}_1, \mathsf {OP}_2$ (with $\delta _{\mathsf {OP}}$ ) is a 2-cycle, hence not covered by this result.Footnote ²¹ Examples of n-cycles for lengths 3, 5, and 7 are given by the respective “petals” in Figure 6.

Now, once we have noted this very basic fact about odd cycles, the reader might fairly ask what it has to do with the topic at hand: the McGee paradox and strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms. The answer is simple: We can give an alternative argument to show that (the dual of) the McGee paradox is indeed paradoxical by noting two facts:

1. 1. The Infinite Flower is paradoxical.

2. 2. There is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from the Infinite Flower to dual of the McGee paradox.

The first fact is easily verified once one notes that the Infinite Flower consists merely of a bunch of n-cycles pasted together (one for each odd n)—filling in the rather trivial details is left to the reader. For the second fact:Footnote ²²

Theorem 5.3.

Proof. The reader is invited to verify that

$$ \begin{align*} f(\mathsf{IF}_{\langle m, n\rangle}) = \mathsf{M}_n \end{align*} $$

is a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism from the Infinite Flower to the Dual of the McGee paradox.

This result, plus the observation just made regarding the paradoxicality of the Infinite Flower, provide a simple proof of the paradoxicality of (the dual of) the McGee paradox via a quick application of the S $\mathcal {L}_{\mathsf {P}}$ H theorem (and, in particular, of Corollary 3.5). Further, this proof is arguably a good bit simpler than the more direct proofs given in the previous section (via either Theorem 4.2 or an application of Theorem 4.5).

Thus, strong $\mathcal {L}_{\mathsf {P}}$ -homomorphisms are a powerful tool for showing that novel $\mathcal {L}_{\mathsf {P}}$ constructions are paradoxical, as advertised. In order to show that an $\mathcal {L}_{\mathsf {P}}$ construction $\{\mathsf {S}_\alpha \}_{\alpha \in A}$ (with $\delta $ ) is paradoxical, we need only show some familiar or simpler construction whose paradoxicality has already been (or can easily be) established can be mapped to it by a strong $\mathcal {L}_{\mathsf {P}}$ -homomorphism.

Appendix: Duality and category theory

Here, we give a novel category-theoretic proof of the Duality Theorem (Theorem 2.18). Recall that the only distinction between $\mathcal {L}_{\mathsf {P}}$ and $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ is that the former uses conjunctions whereas the latter uses disjunctions, and hence $\mathcal {L}_{\mathsf {L}}^{\mathsf {D}}$ mobilizes $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ -acceptability instead of $\mathcal {L}_{\mathsf { P}}$ -acceptability. Thus, our first task is to formulate the notion of $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ in the category-theoretic setting. Let $|: \mathcal {P}(\Omega ) \rightarrow \Omega $ be the morphism

$$ \begin{align*} |(\{\bot\}) &= |(\{\top, \bot\} = \Omega) = \top,\\ |(\{\top\}) &= |(\varnothing) = \bot. \end{align*} $$

$|$ is the category-theoretic analogue of the Sheffer stroke (expressing something like “at least one of the sentences is false”). Thus, given a denotation function $\delta $ (i.e., a coalgebra from S to $\mathcal {P}(S)$ ), an assignment $\sigma $ (i.e., a morphism from S to $\Omega $ ) is $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ acceptable on S relative to $\delta $ if and only if Figure 7 commutes.

Fig. 7

$\sigma ^{\mathsf {}}$ is $\mathcal {L}_{\mathsf {P}}^{\mathsf {D}}$ -acceptable on $\delta $ .

Now, let $\neg : \Omega \rightarrow \Omega $ be the mapping:

$$ \begin{align*} \neg(\top) = \bot,\\ \neg(\bot) = \top. \end{align*} $$

Note that $\neg \circ \neg = id_\Omega $ , and that $\neg $ is a monomorphism.Footnote ²³ Given an assignment $\sigma $ , the dual assignment $\sigma ^{\mathsf {D}}$ given in Definition 2.17 is just $\neg \circ \sigma $ . Recall that we defined the functor $\downarrow{\kern-1pt}: \mathcal {P}(\Omega ) \rightarrow \Omega $ as

$$ \begin{align*} \downarrow\!(\{\bot\}) & =\ \downarrow\!(\varnothing) = \top,\\ \downarrow\!(\{\top\}) &= \ \downarrow\!(\{\top, \bot\} = \Omega) = \bot. \end{align*} $$

The reader is invited to verify that

$$ \begin{align*} \neg \circ \downarrow \circ \mathcal{P}\neg & = |,\\ \neg \circ | \circ \mathcal{P}\neg & =\, \downarrow, \end{align*} $$

that is, the following diagrams commute:

Now, in order to prove the duality theorem, we need to merely prove

Which, in the present context, amounts to proving that the diagram on the left commutes if and only if the diagram on the right commutes:

But the diagram on the left commutes if and only if

commutes (since $\neg $ is a monomorphism), which commutes if and only if

commutes (since $\neg \circ | \circ \mathcal {P}\neg =\, \downarrow $ ), which commutes if and only if

commutes (since $\neg \circ \neg = id_\Omega $ ), which is just

Thus, the Duality Theorem is proven.