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Existential fallacy
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 | This article includes a list of general references, but it lacks sufficient corresponding inline citations. (November 2010) |
The existential fallacy, or existential instantiation, is a formal fallacy. In the existential fallacy, one presupposes that a class has members when one is not supposed to do so; i.e., when one should not assume existential import. Not to be confused with the 'Affirming the consequent', as in "If A, then B. B. Therefore A".
One example would be: "Every unicorn has a horn on its forehead". It does not imply that there are any unicorns at all in the world, and thus it cannot be assumed that, if the statement were true, somewhere there is a unicorn in the world (with a horn on its forehead). The statement, if assumed true, implies only that if there were any unicorns, each would definitely have a horn on its forehead.
Overview
[edit]An existential fallacy is committed in a medieval categorical syllogism because it has two universal premises and a particular conclusion with no assumption that at least one member of the class exists, an assumption which is not established by the premises.
In modern logic, the presupposition that a class has members is seen as unacceptable. In 1905, Bertrand Russell wrote an essay entitled "The Existential Import of Proposition", in which he called this Boolean approach "Peano's interpretation".
The fallacy does not occur in enthymemes, where hidden premises required to make the syllogism valid assume the existence of at least one member of the class.[citation needed]
Examples
[edit]- All trespassers will be prosecuted.
- Therefore, some of those prosecuted will have trespassed.[1]
This is a fallacy because the first statement does not require the existence of any actual trespassers (stating only what would happen if some do exist), and therefore does not prove the existence of any. Note that this is a fallacy whether or not anyone has trespassed.
See also
[edit]References
[edit]- ^ "Logical Fallacy: The Existential Fallacy". www.fallacyfiles.org.
External links
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Existential fallacy
View on Grokipediafrom Grokipedia
The existential fallacy is a formal fallacy in categorical logic that arises when a syllogism draws a particular conclusion—implying the existence of at least one member of a class—from two universal premises, which under the modern Boolean interpretation do not assert or imply the existence of that class.[1] This error stems from assuming "existential import" in universal statements like "All S are P," which in Boolean logic are treated as hypothetical and applicable even to empty classes (e.g., non-existent entities), whereas particular conclusions like "Some S are P" require actual instances to hold true.[1] For instance, the syllogism "All unicorns have horns; all horned creatures are mammals; therefore, some unicorns are mammals" commits the fallacy because the premises can be true even if no unicorns exist, rendering the conclusion false.[2]
The distinction traces back to interpretations of Aristotelian syllogistic logic, where universal propositions traditionally carried existential import—assuming the subject class has members—allowing certain inferences (e.g., moods like Darapti or Felapton) that modern logicians deem invalid.[3] This assumption led to extensive scholastic debates, such as whether the copula "is" in "All S are P" implies real existence, exemplified by puzzles like Noah's Ark containing pairs of all animals.[2] In the 19th century, George Boole's algebraic approach to logic rejected existential import for universals, treating them as conditional statements without presupposing existence, a view formalized by John Venn using diagrams and adopted in symbolic logic by figures like Charles Peirce and Bertrand Russell.[3] Under this Boolean standpoint, the existential fallacy invalidates any syllogism with two universal premises and a particular conclusion, enforcing a rule that prevents overreaching from abstract or hypothetical claims to existential assertions.[1]
The fallacy highlights broader tensions between traditional term logic and modern predicate logic, influencing fields like philosophy and mathematics by enabling rigorous reasoning about fictional, theoretical, or empty sets—such as frictionless planes or ideal gases—without unintended ontological commitments.[4] Critics like P.F. Strawson later argued that ordinary language often carries existential presuppositions, complicating strict Boolean applications, yet the fallacy remains a cornerstone for assessing syllogistic validity in formal contexts.[4]
This table summarizes the structures, emphasizing the universal-to-particular transition that triggers the fallacy in modern logic.[1]
Definition and Fundamentals
Core Definition
The existential fallacy is a formal fallacy in categorical syllogistic logic, occurring when an argument draws a particular conclusion from two universal premises, thereby asserting the existence of entities in the categories involved without justification from the premises.[1][5] This invalidates the inference because the universal premises alone cannot guarantee that the categories contain any members.[1] In deductive reasoning, the fallacy emerges when the conclusion presupposes the actual existence of subjects or predicates that the premises leave open to possibility, such as empty classes, leading to an overreach beyond what the premises logically support.[5] Universal premises, by their nature, make claims about all members of a category without requiring that the category be populated, allowing the premises to hold true vacuously in cases of non-existence.[1] A key distinction in modern logic underscores this issue: universal statements are considered true even for empty sets under the Boolean interpretation, whereas particular statements demand at least one existent instance to be true, rendering conclusions that bridge this gap deductively invalid.[1][5] The fallacy's occurrence relates to the assumption of existential import, which influences whether premises are interpreted as implying existence.[5]Existential Import in Logic
Existential import in logic refers to the assumption that certain categorical propositions, particularly universal statements, imply or presuppose the existence of entities in the subject class. In traditional Aristotelian logic, a universal affirmative proposition such as "All A are B" carries existential import, meaning it presupposes that at least one A exists, as the subject term must denote a non-empty class for the statement to be meaningful.[6] This presupposition extends to all categorical propositions in classical logic, ensuring that both subject and predicate terms refer to existing entities to maintain relations like those in the square of opposition.[7] In contrast, modern logic, influenced by developments following Gottlob Frege and Bertrand Russell, rejects existential import for universal statements under the Boolean interpretation. Here, "All A are B" is true even if no A exists, rendering the proposition vacuously true in cases of an empty subject class, without presupposing existence.[6] Particular propositions, such as "Some A are B," retain existential import in both traditions, entailing the existence of at least one A, but universals in the modern view do not assert or imply any such existence.[7] The lack of existential import in modern interpretations invalidates certain syllogisms that were deemed valid in the Aristotelian framework, particularly those drawing particular conclusions from two universal premises, as the premises provide no basis for asserting existence.[6] This discrepancy arises because the Aristotelian standpoint assumes existential commitment in universals, allowing inferences that affirm existence, whereas the Boolean approach treats validity independently of actual existence, exposing such inferences as fallacious.[7] The term "existential" in existential fallacy highlights this core issue: the erroneous importation of existence assumptions into logical reasoning where modern standards demand none.[6]Historical Context
Origins in Aristotelian Logic
The existential fallacy traces its roots to interpretations of Aristotle's syllogistic system, as outlined in his Prior Analytics, where universal propositions have traditionally been interpreted as carrying an implicit existential import. While some scholars argue that Aristotle himself did not hold a doctrine of existential import—proposing a "no-import" interpretation where categorical propositions, including affirmatives, do not presuppose existence—traditional views hold that in this framework, universal affirmative statements (A-propositions, such as "All S is P") presupposed the existence of at least one instance of the subject term S, rendering the proposition false if S referred to nothing in reality.[8][9][10] Similarly, universal negative statements (E-propositions, such as "No S is P") were understood to apply only to existing subjects, though their truth could hold vacuously in some interpretations; this assumption enabled deductions without requiring separate proofs of existence for the terms involved.[9][10] Aristotle's design of the syllogistic moods in the first figure, including the paradigmatic Barbara mood (two universal affirmatives leading to a universal affirmative conclusion), inherently avoided the existential fallacy under traditional interpretations by presupposing non-empty terms for the premises to yield valid inferences. For instance, in Barbara, premises like "All M is P" and "All S is M" imply "All S is P" under the assumption that S and M exist, ensuring the conclusion's existential commitment aligns with the premises without introducing invalid existential assumptions. This structure extended to other first-figure moods like Celarent (EAE), where the import facilitated perfect syllogisms that Aristotle deemed self-evident, as they directly mirrored natural reasoning patterns without needing existential qualifications. However, when Aristotle's system was applied to broader extensions beyond immediate inferences, the implicit reliance on existence could lead to conclusions implying unwarranted ontological commitments.[9][11] Early medieval interpreters, notably Boethius in his commentaries on Aristotle's On Interpretation and Prior Analytics, reinforced this existential import, solidifying it as a cornerstone of scholastic logic. Boethius adopted a copulative interpretation of categorical propositions, whereby affirmative universals required the actual existence of their subjects to be true, aligning with the square of opposition and subalternation principles that Aristotle had implied. This view, echoed in subsequent medieval traditions, set the historical stage for the fallacy's recognition by embedding the assumption deeply into logical discourse, where universal statements were routinely taken to affirm real-world existence without explicit verification.[12]Developments in Modern Logic
In the 19th century, Augustus De Morgan's Formal Logic (1847) marked a significant critique of existential assumptions in traditional syllogistic reasoning. De Morgan argued that affirmative propositions require the existence of both subject and predicate terms to be meaningfully true, emphasizing that such existence must be explicitly settled before evaluating the proposition's truth value. He identified issues with syllogisms that implicitly rely on non-empty classes, such as when universal statements lead to conclusions involving potentially empty terms like "unicorns," thereby challenging the unstated existential import in Aristotelian forms.[13] Building on this, George Boole's An Investigation of the Laws of Thought (1854) advanced an algebraic treatment of logic that explicitly rejected existential import for universal propositions. Boole interpreted universals (e.g., "All X is Y") as conditional relations between classes without presupposing the existence of instances in the subject class, which invalidated certain traditional syllogisms like those yielding particular conclusions from two universals when classes might be empty. This approach shifted focus to formal class inclusions, rendering syllogistic validity dependent on non-existential algebraic operations rather than ontological assumptions. John Venn further developed this by using diagrams to illustrate the Boolean interpretation, highlighting how universal premises do not imply existence. These 19th-century innovations set the stage for 20th-century formalizations in predicate logic, where analytic philosophers debated the fallacy's implications amid the transition from syllogistic to quantificational systems. Figures like Charles Peirce and Bertrand Russell adopted the rejection of existential import in symbolic logic. In the 1950s and 1960s, Peter Geach's Reference and Generality (1962) examined how modern predicate logic avoids the fallacy by requiring explicit existential quantifiers (∃) for particular claims, contrasting this with traditional syllogisms that infer existence illicitly from universals (∀). Discussions in analytic philosophy, including analyses of modal syllogisms, underscored the need for quantifiers to make existential import explicit, solidifying the fallacy's recognition as a key limitation of pre-modern logic.Logical Analysis
Syllogistic Moods Affected
The existential fallacy arises in categorical syllogisms where both premises are universal propositions (A or E type) and the conclusion is a particular proposition (I or O type), as these inferences assume the existence of entities in the classes involved, which is not guaranteed under modern Boolean interpretations of logic.[1] In traditional Aristotelian logic, universal premises carry existential import, meaning they presuppose the existence of at least one member in the subject class, allowing such syllogisms to be considered valid.[14] However, in modern logic, universal premises (e.g., "All S are P") are true even if the classes S or P are empty, rendering the particular conclusion (e.g., "Some S are P") potentially false because it asserts actual existence.[1] This discrepancy affects only specific moods that were deemed valid in the Aristotelian system but fail the Boolean test. The affected moods are limited to four: AAI-1 (known as Bamalip in the first figure), AAI-3 (Darapti in the third figure), EAO-3 (Felapton in the third figure), and AEO-4 (Fesapo in the fourth figure). No moods in the second figure commit this fallacy while maintaining validity in traditional logic, as valid second-figure syllogisms like Camestres (AEE-2) and Cesare (EAE-2) yield universal conclusions.[14] These four moods share the structure of two universal premises leading to a particular conclusion, relying on the existential import of the universal premises to bridge the gap to an existential assertion in the conclusion.[1] To illustrate the invalidity under modern interpretations, Venn diagrams can be used to visualize the potential for empty classes. For the AAI-3 mood (Darapti: All S are M; All P are M; therefore, some S are P), the three-circle Venn diagram shows the premises shading out areas outside the middle circle (M) for S and P, which holds true even if M is entirely empty—no members exist in M, making both universals vacuously true. However, the conclusion requires at least one "x" mark in the overlapping region of S and P within M, which cannot be guaranteed without assuming M is non-empty, thus invalidating the inference.[15] Similarly, for EAO-3 (Felapton: No S are M; All P are M; therefore, some P are not S), the diagram shades the S-M overlap (true if M empty) and includes P within M (true vacuously), but the conclusion demands an "x" in P outside S, which fails if no members exist at all.[1] For AEO-4 (Fesapo: All S are M; No P are M; therefore, some S are not P), the premises shade appropriately even with empty classes, but the particular negative conclusion assumes existent S members outside P, leading to the fallacy. These diagrams highlight how the Boolean approach rejects any inference implying existence from non-existential premises.[14]| Mood | Figure | Traditional Name | Premises | Conclusion |
|---|---|---|---|---|
| AAI-1 | 1 | Bamalip | All M are P; All S are M | Some S are P |
| AAI-3 | 3 | Darapti | All S are M; All P are M | Some S are P |
| EAO-3 | 3 | Felapton | No S are M; All P are M | Some P are not S |
| AEO-4 | 4 | Fesapo | All S are M; No P are M | Some S are not P |