Are Logically Alien Beings Possible? Descartes, Kant, Frege, and the Tractus

02 Jun 2025

This post was machine translated and has not yet been proofread. It may contain minor errors or unnatural expressions. Proofreading will be done in the near future.

This article is a summary of James Conant, The Search for Logically Alien Thought: Descartes, Kant, Frege, and the Tractatus (1991). THe post includes the author’s supplementary remarks.

Abstract

One of the principal debates in the history of Western philosophy concerns the necessity of logical laws. Are logical laws necessarily necessary, or are they contingently necessary? That is, is a world with logical laws different from ours, or aliens who think according to different logical laws, possible?

The responses of philosophers to this question may be summarised as follows:

• Aquinas argued that logical laws are necessarily necessary, on the grounds that even God operates according to logical laws.

• However, Descartes contended that if God is truly omnipotent, He must be able to alter logical laws as well, and accordingly maintained that logical laws are contingently necessary.

• Kant overcame Descartes’s position by presenting logical laws as transcendental¹ conditions of rational thought.

• However, Descartes’s position survived as psychologism_Psychologism, which held that since logic is also part of human thought, logic should be reduced to psychology.

• Frege, borrowing from Kant’s position, launched a fierce attack on psychologism, but his stance of viewing logical laws simultaneously as transcendental conditions and as ‘the most universal laws of nature’ created internal tension.

• Wittgenstein argued that logical laws are senseless_Sinnlos. Furthermore, he diagnosed that the cause of Frege’s internal tension was that the question “Are logical laws necessarily necessary?” lacks sense_Unsinn.

Through this philosophical-historical trajectory, the author defends the interpretation of the “New Wittgenstein_{New Wittgenstein}” school, which contends against the traditional interpretation that the Tractus is a work about ineffable truths, arguing instead that the Tractus contains no philosophical claims whatsoever, and that making one realise this fact is the ultimate goal of the Tractus.

1. Aquinas and Descartes: Is God Subject to Logic?

For scholastic philosophers, the debate over the necessity of logical laws was a serious problem, as God’s omnipotence and the necessity of logical laws appeared incompatible. To overcome this dilemma, Aquinas presented a distinction that might be called absolute possibility_{possible absolutely} and absolute impossibility_{impossible absolutely}. Absolute possibility refers to states of affairs that an intellect can understand, whilst absolute impossibility refers to states of affairs that are incomprehensible. Aquinas’s claim is that God’s omnipotence means that everything absolutely possible—such as making the Earth flat—is possible for God, whilst absolutely impossible things—such as violating the law of excluded middle—are irrelevant to discussions of God’s omnipotence.

However, Descartes objects that Aquinas’s claim is blasphemous. When Aquinas distinguishes between absolute possibility and absolute impossibility, the “intellect” upon which he bases this distinction is implicitly presupposed to be human intellect. That is, Aquinas conflates the limitations of human intellect—we cannot imagine an apple that is simultaneously red and not red—with the limitations of God’s omnipotence—therefore God cannot create an apple that is simultaneously red and not red. Descartes criticises this as an overreach of human intellect. God can act in ways that humans cannot understand. For such a God, logical laws are contingent products that can be altered at will, according to Descartes.

Then why do humans perceive logical laws as necessary? Descartes’s answer is that the reason God gave to humans is designed to understand logical laws as necessary. If God created this world according to specific logical laws, a benevolent God would give humans dwelling in this world an intellect that perceives those logical laws as necessary (contrast this passage with Descartes’s famous evil demon argument). That is, according to Descartes, the apparent necessity of logical laws originates in the design of reason (given by God).² If we merely replace theological expressions with neurophysiological ones in Descartes’s claim, we obtain the psychologist claim that the apparent necessity of logical laws originates in the workings of the brain.

However, Cartesianism appears to contain an internal contradiction. Descartes wishes to assert the following three propositions simultaneously:

① Humans cannot understand logically impossible states of affairs.

② God can create logically impossible states of affairs.

③ Humans can understand the statement in ② and furthermore judge it to be true.

Here ① and ③ seem to conflict. If ‘logically impossible states of affairs’ is a concept incomprehensible to humans, would not the statement that God can accomplish such things be equally incomprehensible? Just as the statement ‘a mathematical system in which the continuum hypothesis is true is possible’ is equally incomprehensible to someone who does not understand the continuum hypothesis. Feeling the need to address this problem, Descartes presented a subtle distinction between comprehension_comprehend and apprehension_apprehend. If comprehension is the complete grasping of something within reason, apprehension is reason’s touching of something.

I know that God is the creator of all things, that there are immutable truths, and that God is their creator. I say that I know this, not that I perceive or grasp it. For although our mind is finite and thus cannot perceive or grasp God, we can know that God is infinite and omnipotent. This is like how our hands can touch a mountain but we cannot embrace a mountain with our two arms.

Therefore, Descartes would modify ③ as follows:

③ Humans can apprehend the statement in ② and furthermore judge it to be true.

Later, we shall encounter once again in the traditional interpretation of Wittgenstein’s Tractus such attempts to justify through subtle distinctions of words—dare one say, wordplay—the existence of propositions that are, strictly speaking, senseless but nevertheless pass for valid truths.

2. Leibniz and Kant: Freedom, Reason, Logic

Leibniz opposed Descartes’s claim that God arbitrarily created logical laws. To refute Descartes, Leibniz resurrected the scholastic philosophical debate: “Is something good because God does it, or does God do it because it is good?” Leibniz argues for the latter. God acts because something is good. However, according to Leibniz, this does not suggest that God is bound by the concept of ‘good’. Rather, this fact shows that God understands that something is good. That is, the concept of ‘good’ precedes God’s will but is subsumed under God’s reason.

Leibniz emphasises that the concept of ‘good’ that precedes God’s will, far from being contrary to God’s freedom, is a condition for God’s freedom to be established. If there were no principles preceding God’s will, God would have no grounds for performing any particular action. (If God nonetheless acted, it could only be seen as random choice. However, we do not call random behaviour a choice by free will. Just as we do not say that electrons passing through a double slit choose which slit to pass through by free will. Therefore, free will requires principles to distinguish it from randomness.)

Similarly, Leibniz argues that logical laws are principles subsumed under God’s reason whilst preceding God’s will, and are conditions for God’s freedom to be established. This Leibnizian philosophy is inherited by Kant. Where Descartes viewed logical laws as products of reason, Kant reverses Descartes’s claim through the so-called ‘Copernican revolution’. According to Kant, logical laws are not products of reason but constitutive conditions of reason. And in a Leibnizian sense, free reason is reason that follows logical laws. Reason that does not follow logical laws cannot, in the strict sense, be called reason. (This is like how chess that does not follow the rules of chess cannot, in the strict sense, be called chess.) Putnam writes the following about Kant’s philosophy of logic:

Kant’s Lectures on Logic presents a position that stands in extreme opposition to what we today call ‘psychologism’—a pioneering, perhaps the first, work. […] What interests me is [Kant’s] repeated assertion that illogical thought is not, strictly speaking, thought at all.

Certainly logic contains no metaphysical assumptions. That thought in the normative sense, i.e., judgement that can be proven true [or false], follows logic is not something that metaphysics must explain. To explain something presupposes logic. For Kant, logic precedes all rational activity.

Kant particularly emphasises that logic must be strictly distinguished from psychology. As a science of empirical facts, psychology is a theory reached through rational judgement activity, whilst logic is a condition that must be presupposed for such rational activity to be possible. Therefore, attempts to reduce logic to psychology constitute a categorical error. Note here that the object of Kant’s discussion is not ‘human reason’ but reason in general.

However, Kant draws stricter boundaries of logic than Leibniz. According to Kant, logic concerns only the pure form of sentences. (Therefore, the law of excluded middle, modus ponens_{modus ponens}, contraposition, etc., belong to the category of logic, but what we today call set theory, model theory, and modal logic do not.) What this implies is that logic states nothing about the world. (Unlike set theory, which includes ontological statements such as the axiom of infinity—the axiom that at least one infinite set exists—or the empty set axiom—the axiom that the empty set exists—Kantian logic makes no ontological claims.)

Since logic is thus divorced from worldly affairs, attempts to justify the necessity of logic or to gain metaphysical insights from logic become subjects of transcendental critique. This is as much beyond the limits of reason as questions about the immortality of eternity or the existence of God. Later we shall see Wittgenstein accepting Kant’s claim whilst transposing it to a more fundamental level. In short, where Kant diagnosed the cause of philosophical confusion about logic as reason overstepping its limits, Wittgenstein diagnoses the cause of confusion as the illusion that there are limits that reason inevitably encounters.

3. Frege: The Logically Alien Thought Experiment

Like Kant, Frege emphasises that logic is not a product of reason but a constitutive condition of reason—not human reason but reason in general. Frege particularly argues that for judgement to be established, it must be a proposition whose truth or falsity can be assessed. Thoughts that do not satisfy the conditions for judgement are pseudo-thoughts_{Scheingedanke}. This is a requirement of the law of excluded middle, which Frege—and Kant as the sole logical law—puts forward as a logical law.

Based on this position, Frege strongly criticised the psychologism that was fashionable in his time. Frege criticises psychologism for confusing questions of causation with questions of justification. Psychological and physiological research on the human brain can provide explanations for why we accept logical laws as absolutely true. However, such explanations do not justify why logical laws are absolutely true. The validity of logical laws precedes psychology.

Frege’s argument successfully refutes what we might call weak psychologism—the position that recognises the absolute status of logical laws whilst attempting to reduce logic to psychology. However, it is still insufficient to refute strong psychologism—the position that does not recognise the absolute status of logical laws and attempts to reduce logic solely to psychology. Strong psychologists can dismiss questions about the absoluteness of logical laws as meaningless questions, like questions about the existence of God.

Frege’s logically alien thought experiment points out that there is a fundamental error in strong psychologism’s thinking. According to strong psychologism, aliens with different brain structures who think of logical laws contradicting ours as absolutely true are possible. Let us suppose that such an alien actually appeared before us. In this situation, Frege poses the following question: Which is correct: the alien’s logical laws or our logical laws?

At this point, the thought experiment divides into two paths. The first path is when the psychologist acknowledges Frege’s question as meaningful. In this case, the psychologist comes to acknowledge that there exists a logical system that precedes both the alien’s logical laws and our logical laws, and can judge which of the two laws is correct. The question “Which is correct, the alien or us?” belongs neither to alien-psychology nor to human-psychology. It transcends psychology, and acknowledging the meaningfulness of this question itself acknowledges logic that transcends psychology. Compared to this higher logic, the alien’s logic is revealed to be not different logic but wrong logic—that is, not logic at all.

(Here someone might ask whether our logic might be wrong. However, in this case, the alien thought experiment itself, and indeed all our reason, falls into a pit of nonsense. Scepticism about logic makes the arguments that led to such scepticism illogical, thereby completely destroying the possibility of philosophical enquiry. We have no choice but to regard our logic as correct. This becomes the cause of Frege’s internal tension, to be discussed below.)

The second path is when the psychologist dismisses even Frege’s question as meaningless. In this case, the psychologist would claim that no one’s logical laws are absolutely correct. The alien’s logical laws are correct for the alien, human logical laws are correct for humans, and there is nothing more to claim. However, Frege points out that such a psychologist’s attitude makes the position the psychologist wishes to maintain—namely, the possibility of aliens who think according to logical laws different from ours—incomprehensible.

What is crucial in Frege’s point is that for the proposition “A and B are logically different” to be meaningful, there must be presupposed a logic that can determine the identity of A and B. (For ease of understanding, let us use an example from topology rather than logic. The reason we can say that a sphere and a torus³ are topologically different is that there exist topological concepts—such as the number of holes⁴—that are commonly valid for both spheres and tori, and spheres and tori show differences with respect to those concepts.)

Similarly, the psychologist’s claim that aliens who are logically different from us can exist presupposes by itself a background logic that establishes the difference between our logic and the alien’s logic, and in this case we return to the first path. On the other hand, if what the psychologist claims is the existence of aliens who merely make different utterances from us, the difference here is nothing more or less than the difference between the sounds of two cows. The psychologist’s claim that aliens who think according to different logic from ours exist merely reveals that such aliens do not think in the first place. (Cows and humans both make sounds, but only the latter thinks. Bingo machines and calculators both output numbers, but only the latter calculates.)

4. Internal Tension in Frege’s Philosophy of Logic

From the arguments thus far, Frege’s conclusion appears to be “logically alien beings cannot exist”. If this were truly the conclusion of the argument, our thinking would have followed this process:

1. We posited the existence of logically alien beings.
2. We considered the characteristics such beings would have.
3. We realised that these characteristics are incompatible with the nature of logic.
4. Therefore, we reject the possibility of logically alien beings.

However—at least according to the “conditions for judgement” that Frege himself presents—is what we experience here not an illusion of thought? If we examine Frege’s argument closely, we can see that what he means to say is not “logically alien beings cannot exist for such and such reasons” but “the very positing of logically alien beings is incomprehensible”. The positing of logically alien beings makes our thinking non-thinking by placing our thought outside the scope of logic. Metaphorically speaking, it is a Trojan horse.

What is ultimately revealed here is that the thought experiment about logically alien beings is itself a kind of pseudo-thought_{Scheingedanke}. The psychologist’s claim that “logically alien beings are possible” is a sentence that lacks sense. It does not satisfy the condition for judgement that it should be “a proposition whose truth or falsity can be assessed”. And what this implies is that our claim that “logically alien beings are impossible” also lacks sense.⁵

However, in some sense, the sentence “logically alien beings are impossible” seems to contain something that is a true statement. Does it not reveal some truth about the nature of logic? It is merely that the logical structure of our language makes the utterance of that truth impossible. The Tractus has traditionally been interpreted in this context. Scholars such as Hacker_Hacker and Geach_Geach have interpreted the Tractus as a work that “shows” truths that “cannot be said in the strict sense” through counter-syntactic propositions_{countersyntactic proposition}. Here, counter-syntactic propositions are propositions that follow the natural syntax of language—for instance, the subject must precede the predicate—but at some point violate logical syntax—for instance, quantifying over type-theoretic objects at different levels with the same predicate. For example, the distinction between object_object and concept_concept cannot be expressed in strict logical language because they are type-theoretically different. The distinction between object and concept generates problems of the same type as Russell’s paradox by objectifying ‘concept’. Nevertheless, this distinction seems to bear important characteristics of logic. According to traditional interpretation, the Tractus is also composed of such counter-syntactic propositions—propositions that are strictly speaking senseless but nevertheless bear meaningful truths—and the final message of the Tractus to “throw away the ladder” is the passage that exposes the counter-syntactic characteristics of those propositions.

However, the author of the paper strongly criticises such traditional interpretation. Traditional interpretation has merely pretended to resolve the internal tension of the Tractus through spurious distinctions—”strictly speaking thought” and “thought”, “merely senseless” and “deeply senseless”, “fact” and “fact“—that are no more than wordplay, like Descartes’s distinction between ‘understanding’ and ‘grasping’, but has completely failed to actually resolve that tension. Traditional interpretation is completely dependent on these distinctions whilst failing to clearly explain how sentences that violate logical rules can be “senseless” but not “merely senseless”, how “thought can never be illogical (§3.03)” can be acknowledged whilst illogical “thought” remains possible. Therefore, the author presents a new interpretation that breaks with traditional interpretation. This is part of the interpretation called the “New Wittgenstein”.

Continued in Part 2

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1. Note that in Kantian philosophy, ‘transcendent’ means “beyond the limits of experience,” while ‘transcendental’ means “related to the preconditions of rational judgment.”

2. Descartes’s claim is strangely Kantian. Indeed, Margaret Wilson evaluates Descartes’s position on logical laws as the progenitor of Kant’s transcendental philosophy and modern naturalism. However, as will become apparent later, there is an unbridgeable gap between Descartes’s position on logical laws and Kant’s position.

3. Strictly speaking, a torus. A torus is the surface of a doughnut.

4. Strictly speaking, the first homology group. For a sphere this is $\mathbb{Z}$ and for a torus this is $\mathbb{Z} \times \mathbb{Z}$.

5. The original paper states:

If we take the sentences “illogical thought is impossible” or “we cannot think illogically” to indeed present us with thoughts (with senses which we can affirm the truth of), then we concede what a moment ago we wished to deny (namely, that the negation of these sentences present us with a genuine content, one which is able to stand up to the demand for judgment). But if we conclude that these words (which we want to utter in response to the psychologistic logician) do not express a thought with a sense, then aren’t we, if we judge psychologism to be false, equally victims of an illusion of judgment? This is the problem at the heart of the onion.

That is, the author of the paper is pointing out that the following sentence is not actually a thought:

1. Illogical thought is impossible.

Translating this into logical notation:

2. $\not\exists x (I(x) \land T(x))$

Now contrast this with the following sentence:

3. Green thoughts are impossible.
4. $\not\exists x (G(x) \land T(x))$

(1) and (3) are morphologically identical sentences. Yet (3) appears to be a sentence with meaning. That meaning is that there exist no objects in this world that are both green and thoughts. An empiricist might say that (3) also passes the falsifiability principle. In short, if any green object amongst all green objects in the world is discovered to correspond to a thought, then (3) is rejected. Moreover, (4), the logical expression of (3), appears to violate no logical rules.

However, in my view, the idea that (1) and (2) are meaningful is, to borrow Frege’s expression, captivated by an “illusion of thought”. The most efficient way to argue this position is to emphasise the impropriety of the unrestricted quantifier $\exists x$ appearing in (4). What is the range of $x$? If the apple in front of me can be quantified by $x$, can the stem of the apple also be quantified by $x$? What is the criterion by which something—such as a leaf—should be considered an independent object rather than part of an object? Conversely, what is the criterion by which something—such as a tree—should be considered an independent object rather than a simple combination of objects? Should we acknowledge mathematical Platonism and consider ‘the set of natural numbers’ also as an object of quantification? Is ‘God’ also an object of quantification?

Thus unrestricted quantifiers require users to make strong ontological commitments. One way to avoid this awkward situation is to quantify $x$ to the truth set of $T$ instead of using unrestricted existential quantification. In short, modifying (4) as (5):

5. $\exists x \in \mathrm{Th}\; G(x)$

Here $\mathrm{Th}$ is the set of thoughts. But having written this, (5) becomes ill-formed. The problem lies in defining $G$. If we wish to avoid unrestricted quantification, we must also avoid unrestricted quantification in defining the domain of $G$. That is, a) is possible but b) is not:

    a) G: physical objects → {T, F} (O)

    b) G: all objects → {T, F} (X)

Therefore (5) is ill-formed. (5) violates logical rules by applying a predicate to an object that does not belong to its domain. In other words, (5) is a sentence that lacks sense.

This can be understood through the following mathematical example. You will have learnt that $0 < i$ is an incorrect inequality because in that context $<$ is a relation defined only over real numbers. And the reason $<$ can only be defined over real numbers relates to real numbers, imaginary numbers, and the logical rules required by $<$. According to the rules of $<$, multiplying both sides by a number greater than 0 maintains the direction of the inequality, whilst multiplying by a number less than 0 reverses the direction of the inequality. But if $0 < i$, then $0 = 0 \cdot i < i \cdot i = -1$, which is contradictory, and if $i < 0$, then $-1 = i \cdot i > 0 \cdot i = 0$, which is contradictory. Therefore $0 < i$ is an inequality that lacks sense. This does not mean that $0 < i$ is false, because if $0 < i$ were false, then $0 > i$ would have to be true.

See also the discussion in: https://forum.owlofsogang.com/t/topic/6465