2nd paper: Liar sentence mirroring our reasoning as Hegel's quasi-speculative sentence

 

Liar sentence mirroring our reasoning as Hegel's quasi-speculative sentence

 

ABSTRACT

 

This paper explores parallels between the liar paradox and various aspects of philosophical reasoning. It begins by analyzing the liar sentence, "This sentence is false," highlighting its self-referential nature and alternating truth values. The paper then draws connections between the Liar and Hegel's speculative sentence, proposing it as a "quasi-speculative sentence" that mirrors dialectical reasoning. Subsequent sections examine the logocentric predicament, Zeno's paradox, the realism vs. anti-realism debate, and determinism, illustrating how they embody similar self-negating structures. The analysis sheds light on the underlying structure of our philosophical reasoning.

 

Keywords: liar paradox, speculative sentence, logocentric predicament, determinism, dialectics

 

Introduction

 

Philosophers have long grappled with the intricacies of paradoxes. Among these, the liar paradox stands out for its persistent defiance of straightforward resolution. "This sentence is false" encapsulates a self-referential loop that oscillates between truth and falsehood. Several philosophers have treated it as a meaningless statement. For instance, Sobel (2008) argues that “it does not say anything at all” (emphasis in original, p. 136).[1] This argument does no injustice to the sentence. After all, it does not convey any substantive meaning.

 

Nevertheless, no one would deny its significance in the realm of philosophy and even mathematical logic. For instance, Gödel notes “a close relationship” between his proof strategy for the incompleteness theorems and the “‘liar’ antinomy” (Gödel, 1992, p. 40). However, instead of delving into its logical intricacies, this paper will focus on the similarities between the alternating appearances of truth and falsity within the Liar and some paradoxical features of our philosophical reasoning. Specifically, this paper will present an audacious claim that the liar paradox mirrors the structure of our reasoning.[2]

 

To that end, Section 1 analyzes the Liar by using two rules of classical logic: the principle of bivalence and the law of excluded middle. Based on this analysis, this paper will offer the following paraphrase of the Liar: “Affirmation of the falsity of the very affirmation.” It will serve as a useful phrase for illustrating paradoxical aspects of our reasoning.

 

Section 2 investigates Hegel’s speculative sentence. As Houlgate (1986) notes, however, “Hegel does not write much about the speculative sentence” (p. 145). Nevertheless, “it is clearly an important idea because it is the kernel of his theory of what constitutes non-metaphysical philosophical language, the mode of language appropriate to dialectical method.” This paper proposes that the liar sentence can partially achieve Hegel’s aim as a “quasi-speculative sentence.”

 

In Section 3, we will discuss the logocentric predicament, which was first mentioned by Sheffer (1926) in his review of Russell’s Principia Mathematica. Drawing on an extensive discussion in Chapter 3 of Hanna’s (2006) Rationality and Logic, the paper derives an analogue to the liar paradox: “logical affirmation of the groundlessness of logic.”

 

Section 4 reveals an apparent paradox in Zeno’s “dichotomy” argument. Two versions of this argument (the “progressive” and the “regressive”) in Bathfield (2018) are discussed. Then, it argues that Zeno’s argument can be described as “affirmation of immobility through mobility.”

 

Section 5 briefly touches upon the realism vs. anti-realism debate. Based on Loux’s (2006) and Dietrich’s (2020) discussions, the section illustrates how anti-realists’ criticism of realism can be considered to be self-defeating. Their position is described as “realist affirmation of anti-realism.”

 

Finally, in Section 6, we will discuss Lee’s (2024) formulation of a determinist’s assertion and how this can bear resemblance to the liar paradox. In parallel with the Liar, the paper derives: “Affirmation of the determinacy of the world events including the very affirmation.”

 

By considering the above paradoxical cases, this paper aims to reveal how the feature of the liar sentence is mirrored as a quasi-speculative sentence in our philosophical reasoning.

 

1. Liar sentence

 

In academia, the Liar has been conventionally discussed in the context of dialetheism[3] (i.e., the position that there are statements that are true and false at the same time). It also further inspired other liar-like[4] paradoxes. However, this section will focus on its appearance of alternating truth values.

 

Before we analyze the Liar, let us assume that the principle of bivalence holds for it. Specifically:

 

(1) There are only two truth values available for any proposition: T (true) and F (false). That is, there is no such thing as “only half true.” These two truth values are mutually exclusive.

 

We will also assume:

 

(2) The copula “is” in the propositional form “S is P” is used to indicate affirmation (assertion) of a particular property represented by P.

 

(3) S can exist as a lower-level sentence (e.g., “A is B”) within “S is P,” and a truth value of “S” can be alleged by plugging the truth value into P. For instance, we can state: “‘A is B’ is true.”[5]

 

Now, as an assumption, let us assert that the truth value of the liar sentence is F. Then, this assertion can be propositionally expressed as:

 

“This sentence is false” is false.

 

What does “this sentence” refer to? Naturally, it refers to “this sentence is false.” To simplify, let us simply call it K. Then:

 

K = K is false.

 

For simplicity, consider the following version of the statement on the right-hand side of the equation:

 

“K is false₂” is false₁.[6]

 

The first predicate “is false₁” negates the lower-level proposition (i.e., “K is false₂”). Therefore, based on the law of excluded middle, the only possibility is that “K is false₂” does not hold. This is satisfied in one of the two ways:

 

(i) K is [not false₂].

(ii) K [is not] false₂.

 

In the first instance, the property of falsity has been negated. In the second instance, the copula has been negated.[7] For our purposes, we will consider only the first instance. Now, what is the outcome of the negation of falsity? It should be undoubtedly “true,” based on the bivalence principle.

 

Thus, we find:

K is true₂.

 

This can be alternatively understood as “affirmation of the truth of K.” However, before affirming its truth, we find:

 

“K is false₃” is true₂.

 

The predicate “is true₂” now affirms the lower-level proposition “K is false₃.”

 

Thus, we find:

K is false₃.[8]

 

Following this reasoning, we observe:

 

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)

...

 

We see that truth values alternate in the following order:

 

F₁ -> T₂ ->F₃ -> T₄ -> ...[9]

 

F and T alternate as we move through the lower levels.[10] Note that a property (true/false) at a particular level is decided by affirmation of its antithesis at a higher level. For example, the property of falsity₃ at Level 3 is decided by the predicate “is true₂” at Level 2. Similarly, the property of truth₂ is decided by the predicate “is false₁.” Thus, F and T are inextricably linked. What is remarkable is that all these alternating Fs and Ts are contained within the single liar sentence, which can be indefinitely developed as follows:

 

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

 

We can paraphrase it as:

 

“Affirmation of the falsity of affirmation of the falsity of …”

 

How does the above phrase follow? In “This sentence is false,” the copula “is” serves to indicate affirmation, as previously noted in assumption (2). Thus, the sentence can be changed to “affirmation of the falsity of this sentence₁.” Regarding “this sentence₁,” we see:

 

“this sentence₁”

= This sentence₂ is false

= Affirmation of the falsity of this sentence₂.

 

Therefore, “Affirmation of the falsity of this sentence₁” can be extended to:

 

“Affirmation of the falsity of affirmation of the falsity of this sentence₂”

 

Or more briefly:

                                                                                    

“Affirmation of the falsity of the very affirmation.”

 

Note that “this sentence” has been omitted from the phrase. This omission is justified because the subject is only a placeholder. It merely serves to lengthen the phrase endlessly. Its additional function is for the copula “is” to be placed as a verb within the sentence.

 

The above key phrase 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. We will then see how the image of the alternating Ts and Fs in the liar paradox can place it within the context of Hegelian dialectics.

 

2. 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 predicate”[11] … “is destroyed by the speculative judgment.” Further, “[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).

 

However, Bowman (2013), in Hegel and the Metaphysics of Absolute Negativity, 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).

 

This paper argues that a single liar sentence can partially exhibit this dialectical feature. In Section 1, we observed that F and T continue to alternate through endless different levels as the sentence begins to unfold. These movements result from the recursive re-evaluation of the truth value of the subject. Although neither F nor T functions as a subject or predicate, the Liar resembles a speculative sentence in a significant way. To discuss this point in detail, we will consider a few more references below.

 

According to Phenomenology of Spirit, the philosophical (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). Building on this, Kolman (2023) notes that a speculative sentence is to “express, in a perspicuous and positive way, the reflective ascent associated with the negative nature of our experience” (p. 392). In addition, “this negativity must be applied to itself, not only in acknowledgment of the existence of the other side of any difference, but also of the other side of this differentiating itself” (emphasis added).

 

Let us examine the Liar by using both Hegel’s and Kolman’s expressions. After reading “this sentence is false,” the reader soon realizes that the subject within the sentence is equivalent to the whole sentence. Thus, she is repeatedly compelled “to come back to the proposition.” Then, she tries to “grasp it in some other way.” However, this sentence can never be fully grasped, because it refers to itself endlessly. The moment she believes that its truth value has been decided, she does not realize that it is negated by the predicate “is false” lurking in a lower-level sentence. Thus, any clear affirmation made “in a perspicuous and positive way” inevitably negates itself. By doing so, “this negativity” (“the falsity”) is affirmed “in acknowledgment of the existence of” “the other side of this differentiating itself” (“the very affirmation”). This recursive self-negation is succinctly illustrated in the paraphrased version of the Liar: “Affirmation of the falsity of the very affirmation.” This process closely resembles the incessant activity of the Absolute Idea.[12]

 

Admittedly, the liar sentence is not a speculative sentence per se. It does not fit neatly into the framework of a speculative sentence as Hegel defines it. However, both share a dynamic character that engages with contradiction. In particular, the Liar forces us to grapple with the coexistence of opposing truths -- an exercise that echoes Hegel’s exploration of the unity of opposites. Nevertheless, one may argue that its alternating structure of Ts and Fs seems to be a paradigm instance of what Hegel calls “bad infinity.” Bad infinity is described 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). It is well known in Hegelian scholarship that Hegel aimed to achieve unity of opposites rather than endlessly propagate them. So is the Liar an example of bad infinity or can it be called a quasi-speculative sentence?

 

In Lectures on the History of Philosophy (1892), Hegel discusses the Liar as one of the Megarian paradoxes (pp. 458-459). This issue is discussed in detail by d'Agostini, F., & Ficara, E. (2021). According to them, “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). Regarding this, d'Agostini, F., & Ficara, E. provide the following formulation of a 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 idea that the Liar resembles the dialectic motion of a speculative sentence is far from implausible. That is, it does not exemplify Hegel’s “bad infinity.” Rather, its truth values, T and F, are inextricably intertwined, constituting a dialectical whole. These observations justify the claim that the liar sentence is a quasi-speculative sentence. It will serve as a useful analogy for illustrating the paradoxical features of our reasoning. In the following section, we will first discuss the logocentric predicament.

 

3. Logocentric predicament

 

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.”[13] 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. 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). However, 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. Hence, we are led to make a radical conclusion that “logic is viciously circular, that is, groundless” (emphasis added, p. 65).[14]

 

Upon closer reflection, we notice an interesting issue. By indicating an infinite regress in modus ponens, Carroll led us to conclude that logic is groundless by appealing to logic itself. In other words, logic is groundless on the ground of logic. For convenience, consider the following version of this statement:

 

“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. Specifically, that logic₁ is groundless is based on the following argumentation:

 

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. Thus, the thesis that logic is groundless on the ground of logic holds.[15] This can be paraphrased as “logical affirmation of the groundlessness of logic.” This feature of our reasoning bears resemblance to the liar paradox. They are similar in that they are both self-defeating.

 

In Section 1, when discussing the Liar, 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. This process will go on indefinitely. Hence, we see the parallel between the Liar and the justification of groundless of logic.

 

While Carroll hinted at potential infinite intermediate steps missing in modus ponens, the ancient Greek philosopher Zeno suggested that there might be infinite segments within a physical distance. Let us examine this case more closely in the next section.

 

4. Zeno’s paradox

 

Zeno devised four paradoxes to defend his argument that motion is illusory. Among these, we will focus on his “dichotomy” argument. According to Bathfield (2018), there are two versions of this argument: the “progressive” and the “regressive” (p.12).

 

According to the progressive version, there is an infinite series of steps required to reach a destination, which suggests that motion cannot be completed because there is always some distance remaining. Specifically, to reach the destination, one must first traverse half the distance. After covering half the distance, there is still half the distance remaining, and this process continues indefinitely. This series of steps creates an infinite sequence where each step is half the remaining distance from the previous step. Even though each individual step becomes progressively smaller, the sequence never terminates. She can approach the destination but never actually reaches it. Therefore, she cannot travel the distance.[16] However, Bathfield mentions Maurice Caveing’s question: “[I]f a moving body can travel the first half-distance, what impedes it from running through the second one?” (p. 14). This question suggests that the argument relies on the assumption of motion to demonstrate the impossibility of motion. Adopting Caveing’s perspective, the argument can be expressed as “affirmation of immobility through mobility.”

 

Regarding the regressive version, Bathfield notes that “[it] seems much more convincing than the progressive one, and it leads to a true philosophical problem” (emphasis in original, p. 14). The regressive one argues that motion cannot even begin in the first place. To reach the destination, one must first traverse half the distance. Before covering half the distance, there is another half to cover, and this process continues indefinitely. That is, she cannot locate an immediate subsequent point with respect to the starting point. Therefore, no motion can begin. Since reaching a destination requires us to infinitely halve intermediate distances up to the destination beforehand, it is a priori impossible for us to begin moving.

 

It seems that the regressive version effectively rules out the progressive one’s presupposition of actual motion through intermediate stages. Nonetheless, it assumes that space is continuous and that a human or any physical object occupies this space. If a person is aware of space or can recognize an object as a finite entity within it, this manifestly disproves Zeno’s claim that no physical leap across a continuum is possible. If it is impossible to move due to the absence of an immediate subsequent point in space, then it should also be impossible to perceive spatial extension. Consequently, no perceptual information about the spatial arrangements should be available. When attempting to identify the shape of an object, the person would fall into an infinite regress trying to find a next particle from the edge of the object. However, the person can identify the object. It is also manifestly true that the person can move. For instance, in response to Zeno’s paradox, the Cynic philosopher Diogenes silently stood up and walked so that he could “point out that it is a matter of the most common experience that things in fact do move” (Huggett, 2024, 3.1). Of course, Zeno could respond that “it may appear that Diogenes is walking …, but appearances can be deceptive and surely we have a logical proof that [he is] in fact not moving at all.” Nevertheless, his realization that we cannot pinpoint the immediate subsequent point is made in light of the manifestation of continuous space.[17] His conclusion that motion is a priori impossible is based on a logical argument that accepts the a posteriori recognition of a discrete leap within continuous space. This ongoing conflict between his logical discovery of an infinite regress in motion and the physical manifestation of continuous space resembles the ceaseless interplay of opposite truth values within the Liar. This conflict is apparent in the phrase: “affirmation of immobility through mobility.”

 

Kant was also deeply puzzled by the question whether space is continuous or discrete.[18] In the Critique of Pure Reason, Kant contemplates which one of the following theses holds.

“Every composite substance in the world consists of simple parts, and nothing exists anywhere except the simple or what is composed of simples.”

 

“No composite thing in the world consists of simple parts, and nowhere in it does there exist anything simple” (Kant, 1998, pp. 476-477).

 

This is one of the four antinomies that led him to conclude that the ultimate truths of reality lie in the noumenal realm beyond human reason. Kant forever changed the landscape of philosophy by arguing that we perceive reality through a conceptual framework and cannot have direct knowledge of things-in-themselves. This principle is relevant to the following section, which deals with whether we can claim objective truths about the world as a mind-independent reality.

 

5. Realism vs. anti-realism

 

According to Loux (2006, p. 259), realists argue:

 

(1) There is a mind-independent world about which we form beliefs and make statements.

(2) The statements are true if they correctly correspond to the world they are about.

(3) The correspondence that is truth is a property that can transcend our ability to determine whether or not it obtains.

 

In other words, there are several true statements about the world that are linguistically identified and understood by us but cannot be proven to be true. But what verifies this claim? How could we know this claim to be true when the “true” statements are not verifiable? Aren’t the realists claiming a dogmatic view of the world?

 

Goldberg (2008) takes a step further and interprets the realist view to be such that “[s]ome empirical truths are not knowable through even ideal human inquiry” (p. 149).[19] This view is described by Dietrich (2020) as a “‘non-omniscience’ principle” (p. 134). Simply speaking, it means that “some truth is not known.” However, he concludes: “[H]owever plausible the non-omniscience thesis seems, it cannot be known to be true” (emphasis in original). He does not argue that realism must be false. Instead, he states that the realists’ stance is “awkward,” because they are essentially saying that “realism is itself a true but unknowable fact” (pp. 134-135).[20]

 

Can we address the issue through anti-realism?[21] According to anti-realists, reality is “constituted in part by our conceptual activities” (Loux, 2006, p. 287). This view is influenced by the Kantian thesis that the world as we understand it is necessarily mediated by our mental frameworks. This view appears to eliminate the necessity of positing true statements about a mind-independent world, as we only need to discuss the validity of particular statements according to whether they successfully cohere with our frameworks. However, one problem remains. The anti-realist view suggests that we would have no understanding of the world at all if these mental frameworks did not work the way they do. But “[h]ow is it possible to know that you can’t get outside the mind? If it’s impossible to know anything except as it is, say, mentally represented, then one of the things you can’t know is that it’s impossible to know anything except as it is represented” (Dietrich, 2020, p. 146).[22] In other words, under the anti-realist scheme, we are supposed to not know the fact that our knowledge is constricted within the realm of mental representations. 

 

Additionally, anti-realists argue that “word-world relations ... do not obtain” (Loux, 2006, p. 287). They argue that realists have no a priori basis to make references to objective truths. But “if the anti-Realist can make talk about reference intelligible by taking certain referring expressions at face value, why cannot the Realist do the same thing?” (p. 287). This question suggests that when the anti-realists attack the realists for taking referentiability for granted, they may not be aware they are complicit in making reference to an “objective” fact -- namely, that realists cannot make references to objective truths. Thus, the anti-realists’ position can be described as “realist affirmation of anti-realism.” The more legitimate their position is, the more paradoxical it becomes.[23]

 

Of the various metaphysical objects/statements that a realist or even an anti-realist can refer to, there is the thesis of determinism. In the following section, we will investigate how making a reference to determinism may create a paradoxical situation, akin to the Liar.

 

6. Determinism

 

Regarding a determinist’s assertion of determinism, Lee (2024) 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” (p. 20).

 

Lee argues that “[w]hile the referring₁ occurs dynamically, the referring₂ exists within a static realm.” This is one of the two reasons why he argues that there is a discrepancy between referring₁ and referring₂. The other reason is that the mind engaged in referring₁ differs from the mind engaged in referring₂, even though they should be identical. Specifically, “it is seen in hindsight” that the latter mind had been the former mind “all along without our knowing it.” His argument can be further explained as follows.

 

When a determinist asserts determinism, this suggests that her act of assertion has also been determined. From an external viewpoint, this seems totally reasonable. However, when she is involved in the act of assertion, there must be a brief time lapse where her mind separates itself 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. When she understands that this realization was also a predetermined event, the state of mind where this understanding takes place must also differ from the state where the realization took place.[24] Therefore, she can never achieve total identity between the mind engaged in referring₁ and the mind engaged in referring₂. Thus, contrary to Lee’s statement, in our sensible realm, there can be actually no case where the two minds had been the same all along.

 

However, if perfect identity could be achieved between the two minds, this would result in the following paradoxical formulation:

 

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

 

The subject engaged in the act of referring to such determinacy 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, similar to the liar paradox, from which there is no escape?[25]

 

Nonetheless, the above discussion does not necessarily lead to a conclusion that determinism must be false. Lee does not argue that determinism is false. Rather, he only reveals an inherent discrepancy in a situation where a determinist purports to claim self-identity. This characteristic resembles that of the liar paradox, where the liar perpetually self-negates her affirmation. 

 

Conclusion

 

This paper’s main ideas can be briefly illustrated as follows:

 

Liar sentence

l  Affirmation of the falsity of the very affirmation

 

Logocentric predicament

l  Logical affirmation of the groundlessness of logic

 

Zeno’s paradox

l  Affirmation of immobility through mobility

 

Anti-realism

l  Realist affirmation of anti-realism

 

Determinism

l  Affirmation of the determinacy of the world events including the very affirmation

 

As noted by Verene (2007), Hegel aimed to delineate through a speculative sentence “the dialectic motion necessary to present consciousness as alive” (p. 10). However, even if the speculative sentence could provide “the most concrete definition of reason as dialectical self-determination” (Houlgate, 1986, p. 150), it would still be mere textual representation unless read and examined through “consciousness.” The lone liar sentence, if read and understood by a philosophical reader, generates an infinite display of Fs and Ts traversing endless levels. Although this in itself is not an accurate representation of Hegelian dialectics, it closely mirrors the paradoxical features of our reasoning. Therefore, the author believes that the liar sentence merits the label “quasi-speculative sentence.”

 

REFERENCES

Bathfield, M. (2018). Why Zeno’s paradoxes of motion are actually about immobility. Foundations of Science, 23(4), 649-679. https://doi.org/10.1007/s10699-017-9544-9

Booij, E. J. (2023). Procedural semantics and its relevance to paradox. Logic and Logical Philosophy, 32(3), 291-314. https://doi.org/10.12775/LLP.2023.015

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[1] Żełaniec (2004) agrees that “the ‘Liar’ does not, contrary to appearances (if any …), express any proposition at all” (p. 105).

[2] Regarding the liar paradox, by d'Agostini, F., & Ficara, E. (2021) note that “what is interesting for Hegel is the structure of the puzzle” (emphasis in original, p. 7).

[3] See Field (2006); Bromand (2002, p. 741); and d'Agostini, F., & Ficara, E. (2021, pp. 10-12).

[4] See Ladstaetter (2013); Booij (2023, pp. 1-5); Tennant (2015, p. 585); and Clark (2003). For Tarski’s strategy for preventing liar-like sentences, see Leitgeb (2007, p. 284).

[5] In formal logic, we use the symbol “T(S)” to express a truth predicate (i.e., “is true”). However, for the purposes of discussion, we will stick to the form “‘A is B’ is true.”

[6] This analysis aligns with a “context-sensitive approach” (Juhl, 1997, p. 202). This approach “assigns ‘levels’ to occurrences of ‘true’ in particular sentence tokens.” For instance, “the ordinary liar, say, may be false₀ but true₁.” According to Juhl, these levels feature a “quasi-Tarskian hierarchy.”

[7] The second instance can be paraphrased as “negation of the truth” rather than “affirmation of the falsity.”

[8] Tarski’s Convention T defines: ‘Φ’ is true ↔ Φ. ‘Φ’ is a proposition, while Φ represents its actual corresponding case. As the truth predicate “is true₂” affirms “K is false₃,” we see: K is false₃. The quotation marks around “K is false₃” have been removed through Convention T. For details, see Horsten, L., & Leigh, G. E. (2017, p. 197).

[9] This ordering evokes the image of Hegel’s “negation of negation.”

[10] Level 1 is the highest level, and there is no limit to how low the levels can go.

[11] Regarding “S is P,” we say “is P” is a predicate in classical logic. Meanwhile, in Hegel’s speculative philosophy, only P is a predicate. For details, see Houlgate (1986, p. 146).

[12] In Less Than Nothing, Žižek (2012) describes “the mad self-referential play of the Absolute Idea” (p. 77). He further describes the absolute immanence of a criterion for the “Hegelian truth”, where “a statement is compared with itself, with its own process of enunciation.”

[13] As a Hegelian metaphysician, McNulty (2023) argues that “Subjective Logic” (traditional logic) depends on “Objective Logic” (ontology) and that this provides “Hegel’s resolution of the logocentric predicament” (p. xi). For the “presuppositionless” foundation of logic by Hegel, see Hentrup (2019).

[14] However, Hanna (2006) does not accept the conclusion that logic is “groundless.” He overcomes this issue through “the logic faculty thesis” (p. 3) He agrees with Wittgenstein that “[l]ogic is transcendental” (p. 59). This is evident in the fact that logic is embedded even in our denial of its ground.

[15] This external standpoint on logic is also based on logical reasoning. As such, our external standpoint on logic exists within the periphery of logic. This shows that we can take only a “perennially transient external standpoint” on logic. In the short moment when we decide that logic is groundless, we are not aware that this judgment is based on logical reasoning. When we transition into a new conscious state and conclude that the judgment was based on logical reasoning, the conclusion will again be revealed to be based on logical reasoning.

[16] While Zeno denies physical mobility, he seems to accept “logical mobility.” Unlike Carroll, he does not notice an infinite regress in logical reasoning. Accordingly, his argument can also be rephrased as “affirmation of physical immobility through logical mobility.” This logical mobility is not something to be taken for granted. According to Žižek (2012), “Hegel was the first to outline the contours of a logical temporality: even in the sphere of pure conceptual reasoning, the succession of moves does not work as an atemporal chain of consequences” (p. 629).

[17] Space may not be continuous, but even in that case, there is a contradiction. Why? Because our consciousness makes a “continuous” leap between two discrete spatial points. That is, discreteness requires that we be able to consider disparate objects at the same time despite no intermediate link between them.

[18] Assuming that space is discrete cannot easily resolve the issue either. In discrete space, a moving object such as an “arrow would have to skip some of the intervening space atoms entirely, never occupying them in the course of the trip” (Salmon, 1998, p. 145). For example, an object moving twice as fast as another object, which moves one space atom per one time atom, would skip one space atom per one time atom. Per Salmon, this creates an absurd situation where two objects can “pass one another traveling in opposite directions without ever being located next to one another.”

[19] Goldberg’s (2018) thesis that some truths are unknowable is subtly different from the thesis that the truths of some known statements about the world are unverifiable.

[20] He demonstrates this point through logical analysis. For details, see Dietrich (2020, p. 134).

[21] The term “anti-realism” was introduced by Michael Dummett, which is sometimes “imagined by many to be a form of old-fashioned idealism ... that reality runs no deeper than sense data, and so is, ... ‘mind-dependent’” (Dietrich, 2020, p. 123).

[22] Likewise, one may ask if Zeno’s argument could be held at all if no motion ever existed.

[23] Dietrich (2020) observes: “Anti-realism is the negation of realism, and since anti-realism is self-refuting, anti-realism is its own negation. Therefore, anti-realism is identical to realism” (p. 146). He thereby arrives at a radical conclusion that they are “in fact dialetheic, identical theses.” This echoes the idea that the “nameless” and the “nameable” are “the same but diverge in name as they issue forth” (Lao Tzu, 1996, Chapter 1).

[24] This process is similar to the truth value alternation that we saw regarding the liar paradox: F₁ -> T₂ ->F₃ -> T₄ -> ... For instance, Lee (2024) states, “Before the determinist decides on the determinacy of the events of the universe, these events must first be placed within her scope of thoughts” (p. 26). Then, as she declares determinism, she realizes that the entire process” “is also part of the deterministic scenario.” “Subsequently, she concludes from a transient God’s-eye perspective that every time she declares determinism, this would have also been predetermined.”

[25] This issue can be resolved using Lee’s (2024) “deterministic knowledge,” which is the “totality of facts associated with all the past, present, and future events in a deterministic world” (p. 21). He suggests that such facts could exist in “atomic-sentential form” (p. 22). Suppose that “deterministic knowledge” includes a statement that “Susan asserts determinism.” As this fact is actualized in the real world, Susan affirms₁ the determinacy of the world events including her very affirmation₂. Affirmation₁ is a proactive, agential declaration on her part. Affirmation₂ is the action scripted in the deterministic knowledge. However, affirmation₁ of affirmation₂ is an unintentional action; she unknowingly fulfills the prophesized action. This illustrates the dialectical transition between “I speak the truth” (e.g., the truth of determinism) and “the truth itself speaks (in/through) me” (Žižek, 2008, p. 2).