> For the complete documentation index, see [llms.txt](https://mandober.gitbook.io/math-debrief/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://mandober.gitbook.io/math-debrief/300-logic/terms/deduction-theorem.md). # deduction-theorem ## Deduction theorem * [Natural deduction](https://en.wikipedia.org/wiki/Natural_deduction) * [Rule of inference](https://en.wikipedia.org/wiki/Rule_of_inference) * [System L](https://en.wikipedia.org/wiki/System_L) * [Curry-Howard correspondence](https://en.wikipedia.org/wiki/Curry-Howard_correspondence) * [Conditional proof](https://en.wikipedia.org/wiki/Conditional_proof) ## Deduction Theorem > The **deduction theorem** states that a sentence of the form `𝚽 -> 𝚿` is provable from a set of axioms `𝛁` iff the sentence `𝚽` is provable from the system whose axioms consist of `𝚿` and all the axioms of `𝛁`. * The deduction theorem (DT) is a metatheorem of propositional and FOL * DT is a formalization of the proof technique in which an implication `p -> q` is proved by assuming `p` and then deriving `q` from this assumption, conjoined with other, previously established, theorems. * DT explains why proofs of conditional sentences in math are logically correct * Though it seemed "obvious" for centuries that proving `q` from `p` conjoined with a set of theorems is tantamount to proving the implication `p -> q` based on those theorems alone, it was left to Herbrand and Tarski to show (independently) this was logically correct in the general case. The deduction theorem states that if a formula `B` is deducible from a set of assumptions `𝛁 ⋃ {A}`, where `A` is a closed formula, then the implication `A -> B` is deducible from `𝛁`. > `𝛁, {A} |- B` implies `𝛁 |- A -> B` In case `𝛁` is empty, DT shows that `{A} |- B` implies `|- A -> B`. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter: ``` GET https://mandober.gitbook.io/math-debrief/300-logic/terms/deduction-theorem.md?ask= ``` The question should be specific, self-contained, and written in natural language. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.