Post Truth is how many commentators discuss our present situation. Post truth says that these days everyone is bullshitting, lying or both. Bullshitting's when you don't care whether a claim is true or false, a lie is when you do but you knowingly assert something false. So I thought I'd pivot and look at the Liar Paradox via arguments and insights from the philosopher Joel David Hamkins.
Hamkins’ argument about the Liar Paradox and especially his claim that it is an illusion rests on a particular interpretation of the formal structure of semantic paradoxes. His view is not so much that the Liar Paradox is trivial or irrelevant, but that it misleads us into thinking that a contradiction must be real when it is, in fact, a kind of junk artifact.
In this piece I’ll first sketch his argument and then apply his approach to a some key philosophical puzzles in Kant to show that what may seem merely a technical formalism carries philosophical punch.
The Liar Paradox, dating back to Epimenides 2500 years ago, is typically formulated as: “This sentence is not true.” We seem to have a sentence that, if it’s true, then what it says must be the case i.e., it’s not true, which is a contradiction. If it’s not true, then that’s exactly what it says, so it must be true. So, the sentence is both true and not true, violating classical logic, which cannot accommodate such contradictions. Historically, most logicians, including medieval ones, assumed something was wrong with the sentence, or with the idea of self-reference, and tried to diagnose and eliminate the problem. But there's still no consensus even after millennia. In classical logic, the principle of explosion (ex contradictione quodlibet) says that from a contradiction, anything follows, and logicians call this triviality. Thus, accepting contradictions like the Liar would lead to triviality.
Paraconsistent logic, advocated by Graham Priest, rejects this principle and permits contradictions without allowing them to collapse the entire logical system. Graham Priest’s intuition is that if you want to take the argument seriously (i.e., not dismiss it as meaningless or flawed), and yet avoid explosion, paraconsistent logic becomes a necessity. The decision to "change the logic" arises from an earnest attempt to understand and explain the paradox without handwaving it away. Hamkins has a different intuition. He wonders whether the paraconsistent solution really explains anything at all. His thought is that rather than offering an explanation it merely licences the contradiction. He thinks that by examining the phenomenon more closely it is possible to explain the puzzling nature of the sentence without needing to take the step of revising logic.
A formal and structured account of his argument goes like this:
The Liar Sentence (L): L = “This sentence is false.”
Formalisation (with truth predicate T):
Let L denote a sentence such that: L ↔ ¬T(⌜L⌝)
That is, L is true if and only if L is not true.
Classical Derivation of Contradiction: Suppose T(⌜L⌝) is true → then L is true → ¬T(⌜L⌝) is true → contradiction.
Suppose T(⌜L⌝) is false → then L is false → ¬T(⌜L⌝) is false → T(⌜L⌝) is true → contradiction.
Hamkins' Diagnosis: Hamkins contends that the Liar paradox appears to generate a contradiction, but this is an illusion due to illegitimate semantic assumptions.
The three key components of his argument are:
1. The T-schema[i] is not universally valid. The T-schema says: T(⌜A⌝) ↔ A for all sentences A. Hamkins notes that this schema is not valid for all sentences in natural language (or formalized language with self-reference), because it leads to paradoxes like the Liar. Thus, in any sufficiently rich language (e.g., one containing its own truth predicate), the unrestricted T-schema must be limited. Therefore, the Liar Paradox relies on assuming the T-schema holds universally, when in fact, it should not.
2. The Liar sentence does not denote a meaningful proposition. For a sentence to be true or false, it must express a coherent proposition. Hamkins suggests that the Liar sentence does not express a coherent proposition because it is semantically defective. The sentence “This sentence is false” is not well-founded in its reference: its truth depends on its own ungrounded negation. This ill-foundedness means it fails to denote a classical proposition, and hence its apparent contradiction arises from treating a non-proposition as a proposition.
3. The contradiction is an illusion of illegitimate reasoning. The paradox is generated by treating the Liar sentence as if it behaves like a normal sentence with a definite truth value and satisfying the T-schema. But if L does not denote a proposition, and if the T-schema does not apply to L, then the inference from L ↔ ¬T(⌜L⌝) to contradiction is invalid. The contradiction is not real, but rather a result of applying rules outside their domain of validity.
Therefore, the Liar Paradox does not require us to revise our logic (e.g., adopt paraconsistency or any other alternative to classical logic), because no true contradiction arises under proper semantic restrictions. Hamkins draws an analogy to naive set theory. Russell’s Paradox (e.g., the set of all sets that do not contain themselves) seems to generate a contradiction. But axiomatic set theory (e.g., ZFC[ii]) blocks the paradox by restricting set formation. The contradiction disappears not because logic changed, but because the rules of the system were clarified. Similarly, the Liar Paradox is the result of naively assuming the full T-schema and the meaningfulness of all self-referential sentences. With proper semantic restrictions (analogous to axioms in set theory), the contradiction is avoided.
Hamkins therefore argues that the Liar Paradox does not reveal a real contradiction in truth or logic. Rather, it exposes flaws in our naïve semantic assumptions, especially the universal application of the T-schema and the assumption that all self-referential sentences are meaningful. When those assumptions are corrected, the paradox dissolves. Therefore, no need arises to change logic; instead, we must better understand and constrain the semantics of truth. Hamkins insists that logical revision should serve explanatory clarity, and the Liar Paradox, when properly diagnosed, does not warrant abandoning classical logic. He notes that any attempt tends to lead to some form of “revenge paradox,” where efforts to resolve the issue spawn further problems. Rather than addressing paradoxical self-reference head-on, Hamkins prefers to understand the concept of truth through well-founded uses of truth predicates as typically found in mathematics and everyday discourse, where statements are not self-referential or paradoxical. Just as Russell’s Paradox mistakenly assumes we can always form a set of all sets with a property, the Liar Paradox assumes we can always assert the truth value of any statement, even ones already involving truth. Hamkins sees both as violating foundational hierarchies.
Hamkins therefore treats the Liar as an illusion generated by misuse or misinterpretation of truth, a linguistic or conceptual trap. The contradiction it yields proves its flawed nature and solutions like Priest’s paraconsistent logic (accepting true contradictions) fall into the trap and accept the illusion without explaining it. Hamkins wants a solution to show why the paradox arises in the first place and how truth should be modelled and he thinks merely defusing the paradox by licensing it doesn’t do either of these. Hamkins argues that we by reconsidering other assumptions, such as the unrestricted validity of the T-schema (T(⌜A⌝) ↔ A), or the assumption that every meaningful sentence has a determinate truth value, we get the required explanatory solution. Examining the applicability scope of the T-schema (e.g. ‘Snow is white’ if and only if snow is white) explains why what appears as a meaningful sentence with a determinate truth value (e.g. ‘This sentence is false’) is an illusion.
For Hamkins the notion of hierarchies of language embedded in Tarski’s T-schema is crucial to this approach to the paradoxes. When we talk about a “lower-level language”, we are referring to a fundamental idea in Tarski’s semantic theory, which distinguishes between the Object language, the language that is being talked about or analysed and the Metalanguage, a different, usually more expressive language used to talk about or describe the object language. The metalanguage is “higher-level” because it contains terms and concepts that the object language does not have, such as a truth predicate that can apply to sentences in the object language. For example, suppose English is the object language. Then a metalanguage might be a language like English plus the predicate “is true” that can be used to say things like “The sentence ‘Snow is white’ is true.” The metalanguage can talk about the sentences of the object language, while the object language cannot talk about its own sentences’ truth in a fully unrestricted way. The Liar paradox arises because sentences like “This sentence is false” attempt to talk about their own truth status. They effectively combine the object language and metalanguage levels into one, creating a vicious self-reference.
Tarski’s key move is, in order to avoid such self-reference, to insist that the truth predicate is only defined in the metalanguage. The object language itself does not have a truth predicate that can be applied to its own sentences. This creates a hierarchy of languages. At the base, you have the object language, which does not contain a truth predicate. At the next level, the metalanguage can talk about the truth of object language sentences. If needed, you can have a meta-metalanguage talking about the truth of metalanguage sentences, and so on. Because the truth predicate only applies “downward” (from metalanguage to object language), a sentence cannot make statements about its own truth within the same language level. The Liar sentence fails to be a well-formed proposition within this framework because to say “This sentence is false” requires the sentence to talk about its own truth or falsity. But this kind of self-reference is disallowed by the language hierarchy because the truth predicate only applies to sentences at a strictly lower level. Hence the Liar sentence is not expressible as a meaningful proposition in the object language. It is “semantically defective”, meaning it simply cannot be assigned a truth value (true or false) without violating the hierarchy. Therefore, the apparent paradox does not arise because the setup that would create it is ruled out.
Hamkins uses this idea formally, employing set theory (ZFC) to model the languages and their truth predicates and rigorously show that no global truth predicate (one applying to all sentences at the same language level) can exist without contradiction. In so doing he confirms that sentences like the Liar sentence are not legitimate meaningful propositions in a classical semantic framework respecting Tarski’s hierarchy. Thus, the hierarchy of languages and the restriction of truth predicates to apply only to lower-level languages is what fundamentally prevents the Liar paradox from arising in classical logic.[iii] So Hamkins shows that truth in classical logic requires a hierarchy of languages, the lower object language and the metalanguage which can apply truth predicates to the lower language. Analagously, in set theory, there is a distinction between sets and classes which, if overlooked (like the Liar sentence overlooks the hierarchical distinction of languages) also leads to illusions. For example, when people say things like “The set of all sets exists,” they often mean something intuitive and straightforward: that there is a totality of everything, or that all things can be gathered into a single “whole.” However, the paradoxical or contradictory feeling arises not because the content, the idea of ‘everything’ or ‘the world’, is incoherent or impossible, but because the statement is expressed or interpreted in a way that causes logical confusion. This confusion usually involves self-reference or circularity.
For example, treating “the world” as a set that contains all sets, including itself, leads to classical paradoxes like Russell’s paradox. Hamkins would say the feeling of contradiction is an illusion caused by mixing levels of language or misunderstanding the precise formal meaning of these concepts. In formal set theory, especially the commonly used Zermelo-Fraenkel set theory with Choice (ZFC)[iv], the universe of all mathematical sets is not itself a set, but something called a proper class. A set is a collection of objects that itself can be an object of study within set theory. A proper class is a collection too big to be a set and so cannot be treated as an element of any set or class. Why is this distinction important? If “the set of all sets” existed as a set, it would contain itself as a member (since it contains all sets). This self-membership leads directly to paradoxes, like Russell’s paradox (the set of all sets not containing itself), which undermines the foundations of naive set theory. Modern set theory avoids this by saying that the “collection of all sets” is a proper class, not a set. Proper classes cannot be elements of other classes or sets, so self-membership is ruled out by definition. This avoids the paradox without denying the “existence” of the universe of all sets in some sense.
Several of the great German Idealists argued that the world as a totality didn’t exist because they conceived of ‘the world’ in terms of it being the set of all sets. They concluded that it was impossible because contradictory. But had they been able to draw on the distinction between sets and classes then they would not have been so compelled to draw that conclusion and would not have been motivated to conceive of the world as something other than a fixed totally determined object. The fact that the “set of all sets” cannot exist as a set does not mean the concept “the world exists” or “everything exists” is logically impossible. It just means that to formalise the idea correctly, one must work at the right level of abstraction and language. For example: “The world” can be interpreted as the universe of all sets, which is a proper class, not a set. By moving from the naive idea of “set of all sets” to the more precise idea of a proper class, the paradox is avoided. So, the content or concept “the world exists” is coherent and meaningful. But the naive phrasing, as if it were a set that contains everything including itself, is incorrect and leads to paradox. This all points to the importance of the right level of language and conceptual framework. Many philosophical paradoxes arise when one confuses informal intuitive language with formal mathematical language.
Hamkins emphasises that many paradoxical feelings, such as those caused by the Liar paradox or by totality claims like “everything exists,” come from imprecise or naive formulations. Having resources such as the set/class distinction and, in semantics, a hierarchical stratification of truth application, dissolves apparent contradictions or “illusions” by respecting the correct levels and kinds of objects. Collections that are “too large” are not forced into the structure of sets. Truth is not treated as a simple predicate within the same language, but carefully handled via meta-languages. Thus, technical distinctions in modern logic and set theory, between sets and classes, between object language and metalanguage, and the use of the Tarski truth schema, allow us to avoid classical paradoxes like the Burali-Forti and Liar. These tools show that many concepts that appear impossible appear so only because they are imprecisely formulated.
Kant’s antinomies, presented in his Critique of Pure Reason, were understood by Kant as revealing the limits of human reason when it tries to grasp the totality of the world. Each antinomy is a pair of conflicting propositions, thesis and antithesis, both of which Kant argues can be supported by apparently sound reasoning, leading to a paradox. I think had they been more precisely formulated Kant would have drawn different conclusions.
The first antinomy concerns the finitude or infinitude of the universe in space and time; the second concerns the divisibility or indivisibility of matter; the third concerns the freedom or necessity of causal determinism; and the fourth concerns the existence or non-existence of a necessary being (God). Kant takes these contradictions as evidence that reason, when applied beyond possible experience, falls into transcendental illusions that cannot be resolved by pure speculative means. This leads him to his critical conclusion that metaphysical knowledge of the world as a whole is fundamentally limited, and that reason must be disciplined by the boundaries of possible experience. However, from the vantage point of modern formal logic and set theory, the appearance of contradiction in the antinomies can be understood differently, as arising from conflations of conceptual and linguistic levels, rather than from inherent limitations of reason or the world itself. Two crucial insights provide this alternative framework: the distinction between sets and classes in Zermelo-Fraenkel set theory with Choice (ZFC), and Tarski’s semantic hierarchy formalised in his truth schema.
The first antinomy debates whether the universe is finite or infinite in space and time. Kant’s paradox here mirrors classical set-theoretic paradoxes involving “the totality of all things,” much like the “set of all sets.” In ZFC, we cannot form the set of all sets, as this leads to contradictions such as Russell’s paradox. Instead, such totalities are regarded as proper classes , collections too large to be sets, which exist at a meta-level rather than as members of the universe itself. The analogy here is that the “world as a whole” should be understood not as an object or set within the universe, but as a proper class, a meta-level totality outside the framework of ordinary objects. Kant’s original paradox arises because he treats the world as an object within the same conceptual universe it is supposed to encompass. This leads to contradictory conclusions when reason tries to infer properties of this “totality” by treating it like an ordinary object. By recognising the distinction between sets and proper classes, the paradox dissolves: the “world as a totality” is not a set with properties susceptible to finite/infinite dichotomy but a proper class that escapes such self-referential treatment. Hence, the contradiction reflects a failure to respect the proper ontological level, not a fundamental metaphysical problem.
The second antinomy concerns the divisibility of matter, whether matter is infinitely divisible or composed of simple, indivisible atoms. Here, the paradox arises when we attempt to apply infinite regress or atomic discreteness to matter conceived as a whole. This again reflects a confusion about levels: infinite divisibility can be understood in terms of a hierarchy of parts and subparts modelled by sets of increasing size or complexity. But the notion of “matter as a whole” treated as a single indivisible object falls prey to the same kind of set/class conflation. If “matter” is seen as a proper class (an all-encompassing collection), then trying to treat it as a finite or atomic set is conceptually flawed. The infinite divisibility thesis applies when we talk about parts within the universe of sets, while the atomicity thesis tries to ascribe a fundamentally different ontological status to matter itself. Both claims become intelligible and non-contradictory once we appreciate that these statements are about different levels of description: the part-whole hierarchy (sets) versus the totality (proper class).
The third antinomy debates whether causality is free or necessarily determined. Kant is grappling here with the relation between empirical causality within the phenomenal world and the notion of freedom or spontaneity in the noumenal realm. From the modern viewpoint, this paradox can be reframed as a confusion between object-level causation and meta-level agency or self-reference. Freedom is often treated as a property that must be assigned within the same logical or conceptual framework as causal determinism, leading to an apparent contradiction. But by recognizing different levels, for example, viewing freedom as a meta-level property of agents who exist outside deterministic chains described at the object-level, the contradiction is dissolved. Similar to the Tarski hierarchy preventing a sentence from referring to its own truth within the same level, here the causal determinism applies at the physical level, while freedom is a property attributed at a higher, meta-level, preventing direct logical conflict.
Finally, the fourth antinomy concerns the existence or non-existence of a necessary being, often interpreted as a proof for or against God’s existence. Kant argues that both the thesis “there is a necessary being” and the antithesis “there is not” can be rationally defended, leading to contradiction. From the perspective of set/class theory and semantic hierarchies, this paradox can again be seen as a confusion of levels. The concept of a “necessary being” or “first cause” is often treated as an object within the universe, subject to the same logical operations as ordinary entities. But if we treat such a being as a proper class or meta-level entity, a concept that transcends ordinary ontology, then assigning existence or non-existence to it in ordinary object-level terms is a category error. The paradox arises when we apply object-level existence predicates to meta-level or class-level entities, which is prohibited or ill-defined in formal frameworks. Moreover, Tarski’s truth schema reminds us that semantic notions like “existence” or “necessity” require precise linguistic framing. When statements about the necessary being’s existence are treated without distinguishing object language and metalanguage, contradictions emerge. But if we respect these levels, the contradictions vanish as ill-formed statements.
In summary, Kant’s antinomies largely arise from failing to recognise that totalities, infinite regressions, freedom, and necessity involve moving between conceptual and linguistic levels. By conflating these levels, Kant’s reasoning falls into paradox. Modern set theory’s distinction between sets and proper classes clarifies the ontological status of “totalities” such as the universe or matter as a whole, preventing self-membership paradoxes like Burali-Forti or Russell’s. Tarski’s semantic hierarchy, encoded in the T-schema, prevents self-referential semantic paradoxes by insisting that truth and existence statements about object-level sentences must be made in a higher-level metalanguage. If Kant’s antinomies are reframed with these tools, the paradoxes cease to be metaphysical dead-ends signalling the limits of reason. Instead, they become signs that conceptual rigor is needed: a recognition of levels in ontology and language.
This does not trivialise Kant’s insights but complements them by providing formal clarity. The metaphysical enterprise, properly disciplined, can avoid contradictions not by retreating from the question of totality or freedom, but by carefully structuring concepts to respect levels. Thus, modern logic invites a reevaluation of Kantian philosophy. It suggests that the contradictions Kant found do not necessitate a fundamental skepticism about metaphysics. Instead, they reveal that metaphysical language must be precise about the status of totalities, the hierarchy of concepts, and the language used to describe truth and existence. By doing so, philosophical inquiry may overcome the antinomies and move toward a coherent metaphysics that embraces the totality of the world, the nature of causality, and the possibility of necessity without falling prey to paradox.
[i] For readers not familiar with the Tarski T-schema I’ll sketch here what it is. The Tarski T-schema is a principle about the meaning of the truth predicate. It states that for any sentence A: T(“A”) if and only if A. Here, T(“A”) means “the sentence A is true.” The schema says: the sentence ‘A is true’ exactly corresponds to the sentence A itself. For example: T(“Snow is white”) ↔ Snow is white. This principle captures the idea that truth predicates should reflect what they claim: if a sentence says something true, then it is true, and vice versa. However, Alfred Tarski, a logician, showed that you cannot have a single, universal truth predicate that applies to all sentences in a language without running into contradictions like the Liar Paradox. This is known as Tarski’s undefinability theorem. In other words: if a language is rich enough to talk about its own sentences (self-reference), then any attempt to define ‘truth’ for all sentences in that language leads to paradox. So, the T-schema cannot hold unrestrictedly for all sentences in a sufficiently expressive language. Hamkins’ key insight is that Tarski’s hierarchy of languages or semantic stratification avoids such self-reference by not allowing the truth predicate to apply to sentences in the same language. That is, the truth predicate only applies to sentences in a lower-level language, preventing sentences like a Liar sentence L from expressing meaningful propositions. In formal terms, we separate languages into object language and meta-language. The truth predicate is defined only in the meta-language about the object language sentences. This avoids the self-referential loop causing Liar paradox. Thus, any attempt to define a global truth predicate satisfying the T-schema for all sentences, including self-referential ones, is blocked. Hence, L is semantically defective or ill-formed in the classical framework. The T-schema thus clarifies the illusion: the Liar sentence seems paradoxical only if we treat it as a well-formed, meaningful sentence with a global truth predicate. When the hierarchy is respected, the contradiction disappears.
[ii] Hamkins uses Zermelo-Fraenkel set theory with Choice (ZFC) to model languages and truth predicates formally. Within this formalism, he shows that no global truth predicate satisfying the unrestricted T-schema can exist for languages capable of expressing self-reference. The Liar sentence is not a well-defined proposition in this setting. Instead, such sentences are semantically defective, lacking a truth value. In other words, Hamkins rigorously proves that the Liar paradox arises from an improper semantic setup one that ignores the limitations formalized by Tarski and ZFC.
[iii] Object Language vs Metalanguage -
1. Concrete Example: Suppose we have a very simple object language LLL: Its sentences are just simple statements like: S1:S_1:S1: “Snow is white.” S2:S_2:S2: “Grass is green.” Now, LLL does not contain any truth predicate. It only has basic vocabulary and syntax to talk about the world, but no way to say “S1S_1S1 is true” within LLL.
The Metalanguage MMM We introduce a metalanguage MMM, which is richer: It contains all the sentences of LLL, plus a truth predicate TTT. TTT can be applied to the names (or codes) of sentences from LLL, allowing us to say things like: T(‘Snow is white’)T(\text{‘Snow is white’})T(‘Snow is white’) , which means: “The sentence ‘Snow is white’ is true.” The key: MMM talks about the sentences of LLL, but no sentence in LLL can talk about its own truth.
2. The Tarski T-schema and Its Role .The Tarski T-schema formalizes the connection between truth and satisfaction: T(⌈ϕ⌉)↔ϕT(\lceil \phi \rceil) \leftrightarrow \phiT(⌈ϕ⌉)↔ϕ Here, ⌈ϕ⌉\lceil \phi \rceil⌈ϕ⌉ is the name (or code) of the sentence ϕ\phiϕ in the object language. The schema says: “The sentence named ϕ\phiϕ is true if and only if ϕ\phiϕ itself holds.”
For example: T(⌈‘Snow is white’⌉)↔Snow is whiteT(\lceil \text{‘Snow is white’} \rceil) \leftrightarrow \text{Snow is white}T(⌈‘Snow is white’⌉)↔Snow is white. This schema is only stated in the metalanguage MMM, not in LLL.
3. Why the Liar Sentence is Blocked. The Liar sentence tries to say: L:“This sentence is false.”L: \text{“This sentence is false.”}L:“This sentence is false.” But what is “this sentence”? To say this properly, the sentence must refer to its own name or code. But the object language LLL cannot talk about its own sentences’ truth or falsity because it lacks the truth predicate. To express self-reference on truth, you need to be in a metalanguage MMM that talks about sentences of LLL, but LLL itself cannot do this. Thus: The Liar sentence is not well-formed in LLL. Trying to write it down violates the strict separation: a sentence in LLL cannot quantify over or mention its own truth.
Hence, no paradox arises within the formal system respecting this hierarchy. Hamkins formalizes these ideas using standard set theory, ZFC (Zermelo-Fraenkel set theory with Choice), which allows: Coding sentences of LLL as sets or numbers. Defining the truth predicate TTT for the object language within the metalanguage. Proving that if you try to define a global truth predicate TTT that applies to all sentences at the same level (self-applicative), you get contradictions. Thus, TTT must be defined in a hierarchical way to avoid contradictions. This rigorous approach shows that the hierarchy and restriction on the truth predicate are not just informal rules but necessary to maintain consistency.
[iv] You might ask: why can't a class just be a very big set? If every class were a set, some classes would have to contain themselves or be “too big,” causing classical paradoxes such as Russell’s paradox where the set of all sets that do not contain themselves cannot exist as a set, because it leads to contradiction. Similarly, the Burali-Forti paradox arises when we consider the “set of all ordinals.”
Ordinals are well-ordered sets that represent types of orderings extending infinitely. If such a set of all ordinals existed as a set, it would have to be larger than every ordinal (including itself), leading to a contradiction. Hence, the “set of all ordinals” cannot exist as a set in ZFC. Because of these paradoxes, ZFC restricts what can be a set to avoid these contradictions. A proper class is a collection too large to be a set, it cannot be an element of any set or class. For example, the collection of all sets is a proper class, not a set. Proper classes avoid paradox by not being objects within the universe of sets, you cannot talk about them as members of other collections inside ZFC. So a class is not just a “very big set.” It’s a different kind of “collection” that lies outside the universe of sets.
