Liar Sentence Mirroring Our Reasoning as Hegel’s Quasi-Speculative Sentence

1. The Liar

In academia, the Liar has been conventionally discussed in the context of dialetheism3 (which posits statements that are simultaneously true and false). It also further inspired other liar-like4 paradoxes. However, we will focus on its appearance of alternating truth values.

1.1. Assumptions

Before analyzing the Liar, let us assume the following.

(1) NOT operator for T and F

The truth value T (true) is defined as the negation of F (false), and F (false) is defined as the negation of T (true).

T = not-F

F = not-T

(2) Copula and affirmation

The copula “is” in “S is true/false” functions to affirm a truth value for S.

(3) Quotation

S can exist as a lower-level sentence within: S is true/false.

For instance: “A is true/false” is true/false.5

(4) Disquotation

“A is B” is true   A is B.6

“A is B” is false   A is not-B.

1.2. The Liar’s Infinite Loop

Now, let us set:

K = This sentence is false.

Meanwhile, we note:

This sentence = This sentence is false.

Thus:

K = This sentence = This sentence is false.

From this, we derive:

K = K is false.

Accordingly, based on (3): K =K is false = “K is false” is false.

Assign parameters to two predicates:

“K is false₂” is false₁.7

Based on (4):

“K is false₂” is false₁ = K is not-false₂.

Based on (1):

K is not-false₂ = K is true₂.

However:

K is true₂. = “K is false₃” is true₂.

Thus:

K is false₃.

This line of reasoning shows:

K is false₁. (Level 1)

“K is false₂” is false₁.

K is true₂. (Level 2)

“K is false₃” is true₂.

K is false₃. (Level 3)

“K is false₄” is false₃.

K is true₄. (Level 4)

…

The predicates of Fs and Ts alternate as we move through the lower levels:8

F₁ -> T₂ ->F₃ -> T₄ -> …

A predicate (true/false) at a particular level is decided by affirmation of its antithesis at a higher level. For example, falsity₃ at Level 3 is decided by the predicate “is true₂” at Level 2. Similarly, truth₂ is decided by the predicate “is false₁.” Thus, Fs and Ts are inextricably linked. They are contained within a single sentence, which is indefinitely developed as follows:

[[[[…] is false₄] is false₃] is false₂] is false₁.

This can be paraphrased as:

affirmation of the falsity of affirmation of the falsity of …

How does the above phrase follow? In “This sentence₁ is₁ false₁,” the copula “is₁” serves to indicate affirmation.

Therefore, it can be changed to:

affirmation₁ of the falsity₁ of this sentence₁. …①

Meanwhile:

this sentence₁

= This sentence₂ is false₂.

= affirmation₂ of the falsity₂ of this sentence₂.

Thus:

① = affirmation₁ of the falsity₁ of affirmation₂ of the falsity₂ of this sentence₂.

This goes on indefinitely as follows:

affirmation₁ of the falsity₁ of affirmation₂ of the falsity₂ of … … ②

This can be succinctly put into our key phrase:

affirmation₁ of the falsity₁ of the very affirmation₂. …③

But how is ③ justified?

In ③:

the very affirmation₂ = affirmation₂ of the falsity₃ of the very affirmation₃.

Therefore:

③ = affirmation₁ of the falsity₁ of affirmation₂ of the falsity₃ of the very affirmation₃

Consequently, both ① and ③ equally represent ②. Admittedly, the very affirmation₂ in ③ cannot perfectly replace this sentence₁ in ①. Nevertheless, it serves the same purpose by allowing the phrase to develop infinitely. This is possible because affirmation_n+1 always shows up in this sentence_n. See below:

this sentence_n = affirmation_n+1 of the falsity_n+1 of this sentence_n+1.

this sentence_n+1 = affirmation_n+2 of the falsity_n+2 of this sentence_n+2.

…

Moreover, in the clausal form “This sentence_n is_n false_n,” this sentence_n is just a placeholder for the copula “is_n” to function. Since this sentence_n carries minimal intrinsic meaning, we can omit it from the key noun phrase.

The above discussion will illuminate an important aspect of our reasoning. Before exploring this further, we must first discuss its significance in the context of Hegel’s speculative sentence.

2. Speculative Sentence

2.1. What Is a Speculative Sentence?

The book Phenomenology of Spirit introduces speculative sentences such as “God is being” and “The actual is the universal” (Hegel, 2018, p. 39). In these sentences, the “difference between subject and predicate9 … is destroyed by the speculative judgment.” Furthermore, “[t]hinking loses its fixed objective basis which it had in the subject, when, in the predicate, it was thrown back to the subject, and when, in the predicate, it returns not into itself but into the subject of the content.” Hegel aims to demonstrate that a speculative sentence “has within itself the dialectical motion necessary to present consciousness as alive and self-developing through its determinate shapes to the organic whole of spirit as ‘absolute knowing’” (Verene, 2007, p. 10).

For example, in “God is being,” we would normally take the subject (God) and the predicate (being) as separate concepts. However, through a speculative judgment, these two come to reflect and transform into each other. In other words, God is not merely associated with being – She is being itself, and being itself is inseparable from Her. Accordingly, we can no longer think of one without the other. They are interdependent and establish a greater dialectical wholeness. Moreover, although Hegel said that the “difference between subject and predicate … is destroyed by the speculative judgment,” this does not mean that the distinction is completely erased. Rather, he likely meant that their traditional separation becomes invalidated. Otherwise, their relationship would reduce to a trivial identity (e.g., 1=1). This would not form any substantive dialectical pairing. Thus, they must retain their distinction.

Additionally, the speculative proposition “evokes the common opinion” that learns that “[the proposition] means something other than what it took itself to have meant, and this correction of its opinion compels knowing to come back to the proposition and now grasp it in some other way” (Hegel, 2018, p. 40). In other words, the speculative judgment forces us to rethink the speculative sentence in a way that supersedes our superficial understanding of it. In sum, we observe:

Speculative sentence: Its Four Features

(1) The existing separation between the concepts of the subject and predicate vanishes through a speculative judgment.

(2) The subject and predicate do not collapse into an empty identity but establish interdependence within a dialectical whole.

(3) This dialectical whole is not a static fusion but a process of further conceptual development.

(4) The speculative judgment compels us to rethink the sentence in a way that moves beyond our initial understanding.

However, Bowman (2013) quotes Rainer Schäfer’s argument that a single proposition is “not able to express the dialectical, internally self-reverting movement of the grammatical subject to the predicate and from the predicate … back to the grammatical subject” (p. 252). Hence, “there cannot actually be a speculative sentence as a sentence.” In a similar vein, Houlgate (1986) notes that “the Logic cannot be expressed by one speculative sentence alone, even if that sentence is the most concrete definition of reason as dialectical self-determination” (p. 150). Then, is the Liar a speculative sentence? Does it satisfy the four features?

2.2. Is the Liar a Speculative Sentence?

First, Feature (1) seems irrelevant to the Liar. Its subject “This sentence” (or “This sentence is false.”) is semantically subjugated to the predicate “false.” Since “This sentence” already contains the word “false,” there was no “rigid separation” at all between the subject and predicate. In this regard, the Liar falls short.

Moreover, unlike in (2), the Liar’s subject-predicate pair itself establishes no interdependence. Rather, the predicate “false” is merely encapsulated in the subject. Nevertheless, the false/true predicates that alternate through different levels within the Liar form a certain type of interdependency. In F₁ -> T₂ ->F₃ -> T₄ -> …, we saw that a predicate at a particular level would not exist without its preceding antithesis at a higher level. As such, the Liar reveals back-and-forth alternations along the different levels of true/false predicates whereas a speculative sentence focuses on a dialectical tension between the subject and predicate.

Then, do these alternating Fs and Ts achieve a dialectical whole where further “conceptual development” is apparent as stated in (3)? In fact, their continual oscillations raise a concern that they embody “bad infinity.” Hegel defines bad infinity as “the perpetual movement back and forth from one side of the persistent contradiction to the other, from the limit to its non-being, and from the latter back again to the other, the limit” (Hegel, 2010, p. 192). Let us look into what Hegel himself said on the Liar.

In Lectures on the History of Philosophy (1892), Hegel discusses the Liar as one of the Megarian paradoxes (pp. 458-459). This issue was studied in detail by d’Agostini, F., & Ficara, E. (2021). They state, “given the Liar’s sentence ‘µ’ that says ‘‘µ’ is false’, Hegel’s idea is that the conjunction ‘µ and not µ’ is true, while the two conjuncts ‘µ and ‘not µ’, separately taken, are untrue” (p. 1). Hegel “presents the paradox in the Megarian way: ‘if a man acknowledges that he lies, does he lie or speak the truth?” (p. 6). “In Hegel’s view, no other answer except ‘yes and no’ is possible” (p. 6). Specifically: “the separate ‘yes’ and ‘no’ are both untrue, as they correspond to partial and hence untrue accounts of the situation. This view is typical of Hegel’s theory of inconsistencies, normally interpreted as a form of epistemic dynamism, whereby ‘the consciousness’ first postulates that p, and then discovers that in fact not p, and so assumes that p and not p” (p. 9).

d’Agostini, F., & Ficara, E. provide the following dialectical triad: “First, the consciousness (the epistemic agent, the questioned person) realizes that ‘p’ (‘yes’) as such cannot be true. Then it realizes that ‘not p’ (‘no’) cannot be true either. Finally, it concludes that this is because what is true is the contradiction: ‘yes’ and ‘no’ is the true answer” (p. 9). These “three steps perfectly correspond to the phenomenology of the Liar’s case” (p. 9). Therefore, the Liar does not necessarily exemplify Hegel’s bad infinity. It exhibits bad infinity only when we attempt to ascertain its singular truth value through endless predication. If we embrace both F and T as a dialectical whole, which is regarded by Hegel to be true, the movement ceases to be bad infinity and instead becomes a productive dialectical progression, forcing a reevaluation of the nature of truth itself. For example, their interdependence teaches us that dialetheism is possible, thus generating new conceptual development. Therefore, the Liar satisfies Feature (3).

Finally, the Liar meets the criterion of (4). Obviously, we have to come back to the sentence again and again to seek its definitive truth value. Our initial judgment on its truth value is repeatedly betrayed by a predicate hidden at a deeper level.10 This feature reinforces our thesis that the Liar resembles a speculative sentence.

To summarize, the Liar does not satisfy (1), but partially matches (2) in an unconventional way. In addition, we saw how d’Agostini, F., & Ficara, E. helped to strengthen the case that the Liar satisfies (3). Lastly, it clearly exhibits (4) by compelling ongoing reinterpretation. While the Liar does not fully exemplify a speculative sentence in the strictest sense, it shows speculative tendencies, particularly in how it forces repeated engagement and redefines truth through dialectical contradiction. These observations justify the paper’s proposal to call the Liar a “quasi-speculative sentence.” The following details are pertinent to the quasi-speculative sentence.

(1) It presents a paradox forcing us to confront an infinite loop.

(2) By embracing an apparent dichotomy within it, we achieve conceptual development in a dialectical way.

This paradoxical structure, where each level of interpretation forces a reassessment of the previous one, is not unique to the Liar. A similar pattern arises in the foundational justification of logic itself.

3. Logocentric Predicament

3.1. Carroll Questions Modus Ponens

According to Sheffer (1926), “the attempt to formulate the foundations of logic is rendered arduous” by a “logocentric predicament” (p. 228). He explains, “In order to give an account of logic, we must presuppose and employ logic.”11 Hanna (2006) points out that Sheffer assumes that “epistemic noncircularity is a necessary condition of all legitimate explanations and justifications” (p. 55). However, Carroll (1895) questions whether logic can ever achieve non-circularity.12 Specifically, he believes that a modus ponens argument faces a regressive problem. The details of his discussion can be briefly reconstructed as follows.

P obtains.  Premise 1

P->Q obtains.  Premise 2

Thus, Q.  Conclusion

Carroll deems the above argumentation insufficient. He asserts that we need the following additional premise to reach the conclusion.

“If Premises 1 and 2 obtain, then Q.”

Essentially, he is suggesting that we must accept the entire existing argument process (Premises 1 and 2 leading to the Conclusion) as a premise in order to reach the conclusion. Upon adding Premise 3, the argument becomes:

P obtains.    Premise 1

P->Q obtains.    Premise 2

If Premises 1 and 2 obtain, then Q. Premise 3

Thus, Q.    Conclusion

However, Carroll still believes that the above process is incomplete and requires yet another premise – namely, that if Premises 1, 2, and 3 obtain, then Q. This leads to an infinite regress, where each step demands a further premise, and we never actually arrive at the conclusion.

His argument is based upon the following implicit assumptions:

“(1) Every valid deductive advance from the premises of an argument to its conclusion can be explained only by appeal to a principle of valid inference.

(2) That principle of valid inference must therefore itself be included as a true premise in the very same argument” (Hanna, 2006, p. 56).

Philosophers took issue with the second assumption. They held that “principles of valid deductive inference for a proof are not the same as true or logically true conditional premises in a proof” (emphasis in original, p. 57). That is, Carroll failed to see the distinction between the “object language” (logical premises) and the “metalanguage” (logical interference rules). Despite acknowledging the plausibility of this critique, Hanna argues that “it is not at all clear that the philosophical problem [Carroll] was trying” to get us to notice was solved (p. 58). Hanna asks, “by virtue of what logical resources are valid metalogical deductions to be explained or justified?” (emphasis in original, p. 58). If we justify them through meta-metalogical proofs, then these would require meta-meta-metalogical proofs. This again traps us in an infinite regress.

3.2. Wittgenstein: Logic Is Transcendental

This regress can be overcome by adopting Wittgenstein’s (1922) view that “logic is transcendental” (6.13). According to him, “[l]ogic is not a theory but a reflexion of the world” (6.13). Why? Because theory (consisting of “propositions”) “cannot represent what [it] must have in common with reality in order to be able to represent … the logical form” (4.12). He continues, “the logical form … mirrors itself in the propositions” (4.121). The propositions simply “exhibit” or “show” the logical form. This is revealed as we comprehend propositions. Alternatively, the logical form could be a kind of Kantian framework through which we build propositional knowledge about reality. Not surprisingly, Wittgenstein’s stance cannot be logically proved. It is an untestable metaphysical position appealing to the ontological status of logic as being embedded in our reasoning. We may recognize but never linguistically delineate the logical form.

But that would not discourage ambitious logicians from pursuing thorough theorization of logic. They would not be content merely enunciating Wittgensteinian quietism.13 Unfortunately, any attempt to build logic on an unassailable foundation would certainly be frustrated by Sheffer’s logocentric predicament. They would always end up concluding that logic is groundless.

Interestingly, its groundlessness cannot be proved either without falling into an infinite regress.

3.3. Analogue to the Liar

Observe the following argument:

We cannot resolve an infinite regress in modus ponens. (Premise 1)

If we cannot resolve an infinite regress in modus ponens, modus ponens is groundless. (Premise 2)

Thus, modus ponens is groundless. (Conclusion)

This argumentative process exactly employs modus ponens. In other words, modus ponens₂ was used to prove the groundlessness of modus ponens₁.

This can be further extended to the following thesis:

“Logic₁ is groundless on the ground of logic₂.”

When a reader first considers Carroll’s argument, she sees logic as logic₁. However, her reasoning process that led to the conclusion is based on logic₂. Thus, the thesis that logic is groundless on the ground of logic holds. This can be paraphrased as “logical affirmation of the groundlessness of logic.” This feature of our reasoning bears resemblance to the Liar. They are similar in that they are both self-defeating.

In Section 1, we saw a paradoxical situation where F₁ led to T₂, T₂ to F₃, F₃ to T₄, and so on. Similarly, when logic₂ is used to prove the groundlessness of logic₁, logic₂ will also be proved to be groundless by logic₃. Specifically:

Logic₁ is grounded. (Level 1)

Logic₁ is rendered groundless by the groundedness of logic₂. (Level 2)

Logic₂ is rendered groundless by the groundedness of logic₃. (Level 3)

…

In parallel with the Hegelian understanding that a conjunction of F and T is true for the Liar, it is reasonable to conclude that the alternation of the two antithetical features of logic -- its groundedness and groundlessness -- exemplifies Wittgenstein’s transcendental nature of logic in our epistemic realm. By embracing these two contrasting features, we can dialectically progress beyond the mere recurrence of an infinite regress. That is, logic is transcendental and cannot and need not be justified on its own ground within our epistemic domain.

However, this same structural pattern -- where an assertion of the groundlessness of logic collapses into its own negation -- reemerges in discussions of determinism. The following section explores how this self-negating structure manifests within the framework of determinism.

4. Determinism

4.1. The Determinist’s Dilemma

Regarding a determinist’s assertion of determinism,14 an article provides the following formulation:

“[T]he determinist refers₁ to:

The determinacy of all the events of the universe comprising the very event of referring₂ to the determinacy of all the events.”

The formulation carries two discrepancies:

(1) “While the referring₁ occurs dynamically, the referring₂ exists within a static realm.”

(2) The mind engaged in referring₁ differs from the mind engaged in referring₂.

(1) points to the tension between the dynamicity of an understanding that actively determines the rigid determinacy of all events and the staticity inherent in the “understanding” itself as one among those deterministic events. This naturally leads to (2). The mind engaged in referring₁ is actively targeting the mind engaged in referring₂. However, the dynamicity of the former is absent in the latter. In addition, the former as a subject has to separate from the latter as an object. But no subject can ever achieve full self-identity with itself as an object.

When one asserts determinism, this suggests that her act of assertion has also been determined. However, when she is involved in the act of assertion, there must be a moment where her mind separates from the objects of the world that are within her scope of determinism. When she realizes that the briefly separated mind was also part of the deterministic world, the state of mind where this realization takes place must be different from the briefly separated mind. Therefore, she cannot achieve total identity between the mind engaged in referring₁ and the mind engaged in referring₂. In parallel with the alternation of true/false predicates in the Liar, this endless momentary cognitive separation reveals:

The subject₁ declares determinism. (Level 1)

The subject₂ encapsulates the subject₁ within determinism. (Level 2)

The subject₃ encapsulates the subject₂ within determinism. (Level 3)

…

As in “F₁ -> T₂ ->F₃ -> T₄ -> …”, the determinist oscillates between her subjective viewpoint and objectification of her subjective agency. While the subject attempts to achieve full subject/object equivalency, she fails. This reveals an epistemic-level limitation in certifying a supposed ontic-level truth of determinism.

4.2. Is the Dilemma Unique to the Determinist Only?

One may argue that this issue of a subject/object discrepancy is not specific to determinism only.15 For example, the cited article asks whether a physicist asserting the influence of gravity faces a similar dilemma. The article posits:

“[A] physicist refers₃ to:

‘The omnipresent influence of gravity, which exerts its sway even on the neuronal activities of the physicist’s brain required for referring₄ to the omnipresent influence of gravity as well as on all the other events of the universe.’”

Let us apply two perspectives to the physicist: a 1st-person and 3rd-person perspective. The 3rd-person perspective on the physicist is assumed to sequentially unfold as follows:

… P1, P2, P3, …

These P’s refer to the respective states of her mind throughout temporal passage.16These mental states are supervened17 on by:

…B1, B2, B3, …

Meanwhile, the following unfolds from the 1st-person perspective at a particular time point, say, at t=2:

… P1’, X (dummy), P3’, …

These are mentally supervened on by:

…B1’, X (dummy), B3’, …

X (dummy) signifies that neither her mental nor neuronal state at t=2 is unavailable to her at t=2 from the 1st-person perspective. For instance, if she could identify her state of mind at t=2 as P2 (i.e., the 3rd-person perspective), she would no longer be the physicist herself. When a subject investigates an object, it is necessary that they be separated. When the subject objectifies the subject herself, there is necessarily a moment where the subject₁ engaged in the objectifying differs from the subject₂ being objectified. Therefore, the physicist as well as determinist fails to coherently apply an ontic principle to everything including herself. Nonetheless, this poses a greater problem to determinism. Why?

4.3. Determinism Is Stricter

In “… P1, P2, P3, …,” note that P1 and P3 differ from P1’ and P3’, respectively. From the 3rd-person perspective, the state of her mind at t=1 is immediately previous to P2. Meanwhile, from the 1st-person perspective, the state of her mind at t=1 is immediately previous to P3’. Likewise, from the 3rd-person perspective, the state of her mind at t=3 continues immediately from P2. From the 1st-person perspective, the state of her mind at t=3 continues immediately from P1’.

Of course, by appealing to the ontic nature of time, the physicist can refer to the objective future physical state of her brain. For example, if the physicist, from her 1st-person perspective at t=2, refers to her brain state to unfold at t=3, this should coincide with the brain state at t=3 identified from the 3rd-person perspective. However, when she is asked to refer to a particular physical/mental state by placing it relative to another mental/physical state (without relying on the ontic time), she errs.

The point is this. In her epistemic system that excludes time-reliant reference, even if she cannot help referring to, say, B3’ (instead of B3) as a physical substrate being supervened on by P3’, her argument that gravity applies to her brain is still valid. For example, in her real life, she might have once mistaken a tennis ball for a baseball. If she said that gravity applies to the “baseball,” would she have made any factual error regarding the principle of gravity? Probably not. Though the ball was a tennis ball, it does not change the fact that gravity applies to it regardless. In short, a small mislabel about physical objects does not threaten the universal applicability of gravity.

Yet, things are different with determinism. If the determinist cannot make, relative to one another, precise references to her mental states as well as brain states, how could she be so convinced in her justification of determinism on a subjective level? Determinism by definition suggests that every event is rigidly deterministic. Then, shouldn’t she exhibit utmost precision in referring to her subjective states -- whether mental or physical? If she could achieve perfect self-identity as in “P1, P2, P3,” her every state would be crystal-clear from her standpoint. Then there should be no tension at all between ontic and subjective levels when asserting determinism.18 But this is not possible.19

4.4. Analogue to the Liar

If the determinist could achieve perfect identity, this would result in the following paradoxical formulation:

“Reference to the determinacy of the world events including the reference itself.”

The subject is essentially arguing as follows:

“The world events, including the affirmation of this sentence, are deterministic.”

From this, we deduce:

“Affirmation₁ of the determinacy of the world events including the very affirmation₂.”

This phrase raises several questions. If “affirmation₁” was a predetermined event, why would it stand out compared to the other predetermined world events? Additionally, if affirmation₁ and affirmation₂ are indeed identical, wouldn’t this create an infinite loop from which there is no escape?20 Most importantly, what conceptual development do we dialectically achieve in light of this infinite loop?

The main lesson is not so much that determinism is incorrect or flawed as that a finite agent’s inability to justify determinism on a subjective level is a necessary condition for her to exist as a philosophical subject within a supposedly deterministic world as well as to legitimatize its determinacy on an ontic level. Without any subject to confirm/verify determinism in a deterministic system, a potential ontic truth of determinism would be ontologically insignificant. Besides, if she were truly capable of justifying determinism on an epistemic level, she would no longer be a subject imposing a perspective on the system. In this context, Žižek (2012)’s following remark is all the more apt:

“[O]ne cannot look ‘objectively’ at oneself and locate oneself in reality; and the task is to think this impossibility itself as an ontological fact, not only as an epistemological limitation. In other words, the task is to think this impossibility not as a limit, but as a positive fact -- and this, perhaps, is what at [the] most radical Hegel does” (emphasis in original, p. 239).