# Gemini Answer In formal logic, particularly [[intuitionistic logic]], "[[entailment]]" (also known as [[logical consequence]]) is a fundamental concept that describes the relationship between propositions or sentences. It signifies that one proposition (or set of propositions) logically follows from another. In other words, if the premises are true, then the conclusion must also be true. **Formal Definition** Entailment is often represented symbolically as: ``` P ⊨ Q ``` This means "P entails Q" or "Q is a logical consequence of P." **Intuitionistic Logic** Intuitionistic logic is a type of logic that differs from classical logic in its interpretation of truth and proof. In intuitionistic logic, a proposition is considered true only if there is a constructive proof for it. This leads to a different understanding of entailment. **Entailment in Intuitionistic Logic** In intuitionistic logic, entailment means that there exists a constructive proof that demonstrates the truth of the conclusion based on the truth of the premises. This is a stricter requirement than in classical logic, where a proposition can be considered true even without a constructive proof (e.g., by contradiction). **Examples** - **Classical Logic:** In classical logic, the law of excluded middle (P ∨ ¬P) holds, meaning that for any proposition P, either P is true or its negation ¬P is true. Therefore, in classical logic, P ∨ ¬P entails any proposition Q. - **Intuitionistic Logic:** In intuitionistic logic, the law of excluded middle does not hold. Therefore, P ∨ ¬P does not necessarily entail any proposition Q. There must be a constructive proof connecting P ∨ ¬P to Q for entailment to hold. **Importance of Entailment** Entailment is a crucial concept in [[logic]] as it underpins the validity of arguments and the soundness of reasoning. Understanding entailment in different logical systems allows us to analyze and evaluate the logical relationships between propositions. **Additional Notes** - Different logical systems may have different definitions of entailment. - Entailment is closely related to the concept of logical implication, but they are not always interchangeable. - Intuitionistic logic has applications in [[constructivism|constructive mathematics]], computer science, and philosophy. I hope this explanation clarifies the meaning of "entailment" in formal logic, particularly intuitionistic logic. Please let me know if you have any further questions. # ChatGPT In formal logic, "[[entailment]]" refers to a relationship between statements or sets of statements where one statement or set logically follows from another. This concept is fundamental across different systems of logic, including classical and intuitionistic logic. ### Entailment in General In a broad sense, entailment occurs when the truth of one or more premises necessitates the truth of a conclusion. If the premises are true, then the conclusion must also be true; the truth of the premises entails the truth of the conclusion. This is formally expressed as: If $A_1,A_2,…,A_n$ are true, then $B$ is true. Here, the statement $B$ is said to be entailed by the statements $A_1, A_2, \ldots, A_n$. ### Entailment in Intuitionistic Logic Intuitionistic logic, which differs from classical logic in its rejection of the law of excluded middle (i.e., not all propositions are either true or false), has a distinctive approach to entailment: - **Constructive Reasoning**: In intuitionistic logic, entailment is not merely about the truth values of statements but involves the construction of a proof. To say that $A$ entails $B$ ($A \vdash B$) means there is a constructive proof of $B$ assuming $A$ as a premise. This is in contrast to classical logic where entailment often involves showing that it is impossible for the premises to be true and the conclusion false without necessarily constructing a proof of the conclusion. - **Context Dependence**: Entailment in intuitionistic logic is context-sensitive. The validity of an inference depends on the assumptions in the prevailing context, and these assumptions must be explicitly maintained and managed throughout a proof. - **Models and Semantics**: The semantics of intuitionistic logic typically use models such as Kripke models or topological models where the truth of statements can vary with context (e.g., across different points in a [[Kripke model]] or open sets in a topology). In these models, $A$ entails $B$ means that in every context where $A$ holds, $B$ must also hold. Entailment in [[intuitionistic logic]] emphasizes the ability to [[constructivism|constructively prove]] the conclusion from the premises, reflecting a more dynamic and proof-oriented understanding of truth compared to the static truth values of classical logic. This perspective aligns with the philosophy underlying intuitionism, which sees mathematical truths not as abstract entities existing independently of human knowledge but as mental constructions within the framework of logical thought. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Entailment") sort title, authors, modified, desc ```