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Logic
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Epistemology Branch Guide
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What are Syllogisms?
What are Syllogisms? keeps the same branch pressure in view but turns it from a different angle.
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Syllogistic Complexity
Syllogistic Complexity keeps the same branch pressure in view but turns it from a different angle.
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Many Logics?
Many Logics? keeps the same branch pressure in view but turns it from a different angle.
Prompt 1: Provide definitions of vicious and virtuous circularity, and discuss relevant attempts at delineation.
When circularity is fatal and when it is merely unavoidable
The prompt matters because it changes what the reader should investigate next about circularity. The section should turn a broad topic into a sharper line of inquiry.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
After this section, the reader should be able to restate circularity in plain language, identify an easy misuse of it, and say what would count as a better reason for or against the view.
Vicious Circularity Vicious circularity occurs when a definition, argument, or proof relies on itself in a way that renders it uninformative or invalid. It often leads to logical fallacies, paradoxes, or infinite regress. In essence, vicious circularity fails to provide a foundational basis for understanding because it presupposes the very thing it aims to establish.
Virtuous Circularity Virtuous circularity, on the other hand, refers to self-referential definitions or arguments that are coherent, informative, and logically sound. This type of circularity is often employed in recursive definitions, fixed-point theorems, and certain areas of semantics and mathematical logic where self-reference is handled in a controlled and meaningful way.
Vicious Circularity Defining a term by using the term itself without additional information. For instance, “Justice is the quality of being just.” This definition doesn’t advance understanding because it doesn’t break down the concept into more fundamental terms.
Virtuous Circularity The definition of natural numbers using the Peano axioms, where the successor function defines each natural number in terms of the previous one. This recursive definition is circular but provides a foundational structure for arithmetic.
Foundational Approaches Early logicians like Bertrand Russell tackled vicious circularity through the theory of types to resolve paradoxes like the Russell Paradox. By organizing sets into a hierarchy of types, self-referential sets that lead to paradoxes are avoided.
Fixed-Point Theorems Logicians such as Saul Kripke utilized fixed-point theorems to address semantic paradoxes like the Liar Paradox. In his seminal work on the theory of truth, Kripke demonstrated how a language could consistently contain its own truth predicate by allowing for partially defined truth values, thereby embracing a form of virtuous circularity.
Well-Foundedness The concept of well-founded sets avoids vicious circularity by ensuring there are no infinite descending membership chains. This approach is fundamental in set theory to prevent sets from containing themselves directly or indirectly in a problematic way.
Acyclic Definitions Logicians advocate for acyclic or non-circular definitions whenever possible. When circularity is necessary, they aim to construct definitions where the circularity does not prevent the assignment of meaning or truth. For example, in defining linguistic terms, mutual definitions can be acceptable if they form a finite, non-vicious circle.
Stratification and Hierarchical Models In formal semantics, stratified models are used to assign meanings at different levels, preventing vicious circularity by ensuring that a term at one level doesn’t depend on itself at the same level.
Alfred Tarski Tarski’s undefinability theorem showed that truth in formal languages cannot be consistently defined within the same language, highlighting issues of self-reference and vicious circularity. His work led to the development of Tarski’s hierarchy of languages, separating object languages from their meta-languages to avoid paradoxes.
Stephen Yablo Yablo introduced paradoxes that are self-referential but do not rely on direct self-reference, challenging traditional notions of circularity and prompting further examination of what constitutes vicious circularity.
Circular Proofs and Inductive Definitions Some logicians have explored circular proofs in the context of inductive definitions, where circular reasoning is controlled and contributes to a constructive foundation rather than leading to inconsistency.
- Attempts at Delineation: Academic logicians have proposed various methods to distinguish between vicious and virtuous circularity.
- Notable Contributions: The distinction between vicious and virtuous circularity is a nuanced and critical area of study in logic.
- Belief calibration: Circularity concerns how strongly the available evidence warrants belief, disbelief, or suspension of judgment.
- Evidence standard: Support, counterevidence, and merely persuasive appearances have to be kept distinct.
- Error pressure: Overconfidence, underconfidence, and ambiguous testimony each distort the conclusion in different ways.
Prompt 2: Provide examples in which apparently vicious circularity is christened virtuous without adequate justification.
What Explanation of Impredicative Definitions explains, and where it starts to strain
The point of this prompt is to make circularity answer to concrete cases. Once examples enter, the reader can see which distinctions are doing real explanatory work and which are ornamental.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
After this section, the reader should be able to restate circularity in plain language, identify an easy misuse of it, and say what would count as a better reason for or against the view.
Existence of Least Upper Bounds In real analysis, the completeness property states that every non-empty set of real numbers bounded above has a least upper bound (supremum). The supremum is defined by quantifying over all upper bounds, including itself.
Use in Type Theory Modern type theories, like Girard’s System F, allow impredicative constructions without fully addressing foundational issues raised by earlier logicians.
Dependence on Intuitive Notions The theory relies on intuitive concepts of truth and groundedness without fully formalizing them.
Semantic Complexity Using partially defined truth values complicates semantics without definitively resolving paradoxes.
Criticism from Other Logicians Some argue the theory shifts the problem rather than solves it, as it doesn’t provide a non-circular foundation for the truth predicate.
Lack of Foundational Clarity They do not provide the foundational explanations necessary for learners or precise linguistic analysis.
Potential for Misinterpretation Without clear, non-circular definitions, the risk of misunderstanding increases.
Assumption of Termination There’s sometimes an unwarranted assumption that recursion will naturally reach a base case without rigorous proof.
Overlooking Edge Cases Potential issues arising from certain inputs aren’t thoroughly analyzed, leading to software vulnerabilities or failures.
- Explanation of Impredicative Definitions: An impredicative definition defines an entity in terms of a totality that includes the entity itself.
- Christening as Virtuous Without Adequate Justification: Despite these concerns, impredicative definitions are widely used in mathematics without thorough justification of their legitimacy.
- Inadequate Justification: The acceptance of impredicative definitions often relies on the success and utility of the mathematical theories that use them rather than on rigorous foundational justification.
- Overview of Kripke’s Theory: Saul Kripke developed a semantic theory of truth to address the Liar Paradox by allowing sentences to be neither true nor false (truth-value gaps).
- Christening as Virtuous Without Adequate Justification: Proponents consider the circularity virtuous, claiming that the fixed-point construction provides a consistent semantics for self-referential sentences.
- Circular Definitions in Lexicography: In dictionaries, terms are sometimes defined using other terms that, in turn, are defined using the original term, creating a definitional circle.
Prompt 3: Presuppositional religious apologists often claim circularity in their ideology is virtuous. Critique these claims.
Clarifying Claim of Virtuous Circularity
The prompt matters because it changes what the reader should investigate next about circularity. The section should turn a broad topic into a sharper line of inquiry.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
After this section, the reader should be able to restate circularity in plain language, identify an easy misuse of it, and say what would count as a better reason for or against the view.
Cornelius Van Til Van Til argued that all reasoning is grounded in presuppositions, and the Christian worldview is the only one that can account for logic, science, and morality. He asserted that non-Christian worldviews are internally inconsistent and cannot provide a basis for rational thought.
Greg Bahnsen Bahnsen expanded on Van Til’s ideas, emphasizing the Transcendental Argument for God’s existence (TAG), which posits that the existence of God is the necessary precondition for the possibility of logic, science, and ethics.
Ultimate Authority An ultimate authority cannot be validated by any higher standard; thus, it must be self-authenticating.
Transcendental Necessity They argue that without presupposing the Christian God, one cannot make sense of the laws of logic, uniformity of nature, or objective moral values.
Vicious Circularity A logical fallacy where the conclusion is no more certain than the premises because the premises depend on the conclusion. It provides no external validation and thus is unpersuasive to those who do not already accept the conclusion.
Virtuous Circularity In specific contexts, such as certain mathematical proofs or definitions (e.g., recursive functions), circularity is acceptable when it is part of a well-founded system that leads to meaningful and non-contradictory results.
Self-Referential Justification Claiming that a text is true because it says it is true is a classic example of vicious circularity. It lacks external corroboration and does not provide an independent basis for validation.
Comparison with Other Ultimate Authorities Any religious or philosophical system could claim ultimate authority using the same reasoning (e.g., the Qur’an in Islam, the Vedas in Hinduism), leading to a plurality of mutually exclusive “ultimate authorities.”
Premise 1 Logical absolutes (laws of logic), uniformity of nature, and moral laws exist.
Premise 2 These phenomena require a transcendent, unchanging source.
Question-Begging TAG assumes that only the Christian God can account for these preconditions, which is the point under contention. This is an instance of begging the question.
False Dilemma The argument often presents a choice between Christianity and atheism, ignoring other theistic and non-theistic worldviews that may offer alternative explanations.
Non-Sequitur Even if the premises were accepted, the conclusion does not necessarily follow exclusively. The existence of logical absolutes does not inherently prove the Christian God’s existence over other possible explanations.
Foundationalism Epistemological view that certain basic beliefs (foundations) justify other beliefs but are themselves self-justified or infallible. Presuppositionalists attempt to align with this by positing God as the ultimate foundation. Critique: Foundational beliefs in foundationalism are typically self-evident or incorrigible, such as “I exist.” The existence of God is not self-evident in the same way and requires justification.
Critique Foundational beliefs in foundationalism are typically self-evident or incorrigible, such as “I exist.” The existence of God is not self-evident in the same way and requires justification.
Coherentism Beliefs are justified if they cohere with a system of interconnected beliefs. Critique: Presuppositionalists reject coherentism, but their reliance on a self-contained system of beliefs that justify each other resembles a coherentist approach, yet without addressing potential inconsistencies or alternative coherent systems.
Critique Presuppositionalists reject coherentism, but their reliance on a self-contained system of beliefs that justify each other resembles a coherentist approach, yet without addressing potential inconsistencies or alternative coherent systems.
Arbitrary Presuppositions By accepting any presupposed ultimate authority without independent justification, the selection becomes arbitrary.
- Claim of Virtuous Circularity: Presuppositionalists acknowledge that their argument is circular but contend that it is a “virtuous” circularity because.
- Definition of Circular Reasoning: In formal logic, circular reasoning (also known as begging the question or petitio principii ) occurs when an argument’s conclusion is assumed in its premises without independent support.
- Self-Authentication and Ultimate Authority: Presuppositionalists argue that the Bible is the ultimate authority and must authenticate itself.
- The Transcendental Argument for God’s Existence (TAG): Therefore, the Christian God exists as that necessary source.
- Lack of Falsifiability: While presuppositional apologists like Cornelius Van Til and Greg Bahnsen argue that their circular reasoning is virtuous due to the necessity of an ultimate authority, this claim faces significant logical challenges.
- References: Van Til, Cornelius. The Defense of the Faith. P&R Publishing, 1955.
Prompt 4: Provide the formal logical proof behind your claim that “ circularity in presuppositional apologetics remains vicious. “
Clarifying The Transcendental Argument for God’s Existence (TAG)
The prompt matters because it changes what the reader should investigate next about circularity. The section should turn a broad topic into a sharper line of inquiry.
A useful test case is an everyday disagreement where both sides have some evidence but not enough to claim certainty. The distinction only matters if it changes what each side should now infer, demand, or withhold.
After this section, the reader should be able to restate circularity in plain language, identify an easy misuse of it, and say what would count as a better reason for or against the view.
An argument is valid if, assuming the premises are true, the conclusion must be true.
A logical fallacy where the conclusion is assumed in one of the premises.
Circular reasoning that provides no independent support for the conclusion, making the argument unpersuasive to those who do not already accept the premises.
Logic, reasoning, morality, and science require the existence of the Christian God to be meaningful or intelligible.
Logic, reasoning, morality, and science are meaningful and intelligible.
If the Christian God exists, then is true.)
The Christian God exists.)
If is true, then the Christian God exists.)
The Christian God exists.)
The argument assumes in order to prove. The premise is only valid if is already accepted as true.
No external evidence or reasoning is provided to support without already assuming.
The argument’s validity depends on an implicit premise that is equivalent to the conclusion.
The existence of the Christian God ( ) is presupposed in the premises.
Specifically, it commits begging the question by assuming what it seeks to prove.
The argument lacks independent support for its conclusion and is unpersuasive to those who do not already accept its premises.
Readings and Analysis. P&R Publishing, 1998.
What is the central claim of presuppositional apologetics regarding the foundation of rational thought and logic?
Who are two prominent figures associated with the development and promotion of presuppositional apologetics?
- The Transcendental Argument for God’s Existence (TAG): The TAG is central to presuppositional apologetics. The epistemic pressure is how evidence, uncertainty, and responsible confidence interact before the reader accepts or rejects the claim.
- Formal Logical Proof Structure: Represent “Logic, reasoning, morality, and science are meaningful and intelligible.”
- Step 1: Identifying the Implicit Premise: The argument assumes that is the only possible explanation for.
- Step 2: Revealing the Circularity: However, this means that the conclusion is assumed in the premises because being true is predicated on being true.
- Step 3: Formal Proof of Circularity: Implicitly, (Assumed or required for the argument to work).
- Formal Fallacy Identification: The argument commits the fallacy of affirming the consequent if is taken alone.
What ties this page together.
The best route is to track how evidence changes credence, how justification differs from psychological comfort, and how skepticism can discipline thought without paralyzing it.
The recurring pressure is false certainty: treating a feeling of obviousness, a social consensus, or a useful assumption as if it had already earned the status of knowledge.
Read this page as part of the wider Epistemology branch: the prompts point inward to the topic, but they also point outward to neighboring questions that keep the topic honest.
For a companion resource on calibration, credence, and structured rational judgment, see Credencing.com.
- #1: What is the central claim of presuppositional apologetics regarding the foundation of rational thought and logic?
- #2: Who are two prominent figures associated with the development and promotion of presuppositional apologetics?
- #3: What is the main difference between vicious and virtuous circularity in logic?
- Which distinction inside Vicious & Virtuous Circularity is easiest to miss when the topic is explained too quickly?
- What is the strongest charitable reading of this topic, and what is the strongest criticism?
Deep Understanding Quiz Check your understanding of Vicious & Virtuous Circularity
This quiz checks whether the main distinctions and cautions on the page are clear. Choose an answer, read the feedback, and click the question text if you want to reset that item.
Future Branches
Where this page naturally expands
Nearby pages in the same branch include What are Syllogisms?, Syllogistic Complexity, and Many Logics?; those links are not decorative, but suggested continuations where the pressure of this page becomes sharper, stranger, or more usefully contested.