Logical closure

USAGE SUMMARY

The phrase "Logical closure" is correct and usable in written English.
It can be used in contexts related to philosophy, mathematics, or logic, referring to the property of a set or system being closed under certain operations or conditions. Example: "In formal logic, the principle of logical closure ensures that if a statement is true, all its logical consequences are also true."

✓ Grammatically correct

Science

News & Media

Human-verified examples from authoritative sources

Exact Expressions

6 human-written examples

Heyting arithmetic, after all, has exactly the same axioms as Peano arithmetic, and is the logical closure of those axioms under intuitionistic, rather than classical, logic.

SEP

However, due to logical closure the belief set contains many elements that are not really worth to be taken seriously.

SEP

Due to logical closure it then also contains the sentence p ∨ q, "Either Shakespeare wrote Hamlet or Charles Dickens wrote Hamlet".

SEP

Truth and membership are inductively generated by iterating rules that correspond to natural logical closure conditions and can be formalized by means of positive (i.e. negation- and implication-free) clauses.

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As we have seen above, the Rantala frames approach is in a sense more radical: by allowing anarchic impossible worlds of any kind to be accessible, it destroys all forms of logical closure for the relevant cognitive states.

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The possibility of multiple extensions different possible coherent, inferentially complete conclusion sets that can be drawn from a single set of premises means that we have to think of logical consequence not as a function taking a set of axioms into its logical closure, but as a relation between a set of axioms and alternative logical closures.

SEP

Human-verified similar examples from authoritative sources

Similar Expressions

54 human-written examples

The reason for this has to do with questions concerning the logical (or Boolean) closure of the set of physical properties — if P, Q and R are physical properties, which of the various logical permutations of P, Q and R are likewise physical properties?

SEP

Tarski notwithstanding, since 1975 the hierarchical approach has been somewhat superseded by new ideas that have rendered the ideal of logical and semantical closure in many respect accessible (especially by means of the fixed point methods used by Kripke 1975 and Martin-Woodruff; see Martin 1984).

SEP

Behind such closures, "the logical thinking is that you are dealing with clients whose track record is hard to establish and who are in the front line in terms of job vulnerability," said Emilio Ontiveros, chairman of Analistas Financieros Internacionales, a research consulting firm in Madrid.

The New York Times

More explicitly, one requires closure under the logical operations of negation, conjunction, existential quantification and suitable combinatorial operations of permutation and expansion.

SEP

In a BBC interview in 2002 Reid said the occupation was "the only logical effective form" of opposition to closure.

The Guardian

Expert writing Tips

Best practice

In mathematical or philosophical writing, always define the logical system being used (e.g., classical logic, intuitionistic logic) to provide a precise understanding of what "Logical closure" entails.

Common error

Avoid assuming that logical closure implies logical equivalence. Logical closure means including all logical consequences, whereas logical equivalence refers to statements that are mutually entailing. Ensure your use of "Logical closure" accurately reflects its meaning in the specific context.

Linguistic Context

The primary grammatical function of "Logical closure" is as a noun phrase. It refers to the property of a set of statements or a system of beliefs being closed under logical consequence, meaning that it contains all statements that logically follow from it. As illustrated by Ludwig, this phrase is most commonly found in academic or philosophical contexts.

Expression frequency: Rare

Frequent in

Science

66%

News & Media

33%

Formal & Business

0%

Less common in

Encyclopedias

0%

Wiki

0%

Reference

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Ludwig's WRAP-UP

In summary, "Logical closure" is a noun phrase primarily used in academic and philosophical contexts to describe the property of a system or set of beliefs containing all its logical consequences. Ludwig AI indicates it's grammatically correct, though rare in general usage. When writing, ensure clear context to avoid ambiguity and consider alternative phrases like "inferential completeness" or "deductive closure" based on nuance. Avoid confusing it with logical equivalence. Remember, understanding and applying "Logical closure" correctly can enhance precision in logical and philosophical discussions.

More alternative expressions

Phrases that express similar concepts, ordered by semantic similarity:

Deductive closure

Highlights the deductive aspect of the closure, emphasizing that conclusions are reached through deduction.

Inferential closure

Focuses on the property of a system where all valid inferences are included.

Inferential completeness

Focuses on the system's ability to derive all possible conclusions, rather than just the property of being closed under logical operations.

Entailment closure

Emphasizes that all entailed statements are included in the closure.

Theory closure

Refers to the closure of a theory under logical operations.

Consistent closure

Specifies that the closure maintains consistency, avoiding contradictions.

Logical consequence

Refers to what can be validly inferred or derived from a set of premises, a result of applying logical closure.

Complete inference

Highlights achieving a state where all possible valid inferences have been made.

Semantic closure

Focuses on closure related to meaning and truth conditions.

Boolean closure

Specifies that closure is under Boolean operations.

FAQs

How is "Logical closure" used in philosophy?

In philosophy, "Logical closure" often refers to the principle that if someone believes a set of propositions, they also believe all logical consequences of those propositions. This is related to discussions about rationality and ideal agents.

What is an example of "Logical closure" in mathematics?

In mathematics, a set is said to be closed under an operation if performing that operation on members of the set always results in another member of the set. For example, the set of integers is closed under addition because adding two integers always results in an integer.

How does "Logical closure" relate to artificial intelligence?

In AI, the concept of "Logical closure" is relevant to creating reasoning systems. Ideally, an AI system should be able to infer all logical consequences from its knowledge base, but this can be computationally challenging.

What are some alternatives to "Logical closure"?

Depending on the context, you could use alternatives like "inferential completeness", "deductive closure", or "logical consequence" to express similar ideas.