Questions tagged [natural-deduction]

For questions concerning natural deduction, a formal proof system studied in proof theory. A natural deduction proof starts with a set of premises and applies introduction and elimination rules to arrive at the conclusion. This tag is not specific to any particular logic, classical or intuitionistic, propositional or allowing quantifiers.

This tag is for questions concerning natural deduction and proofs in natural deduction.

Natural deduction is a formalization in proof theory. A proof in natural deduction starts from a set of assumptions (formulae) and applies a series of introduction and elimination rules to arrive at the conclusion.

There are separate introduction rules and elimination rules for each logical connective (such as and ($\land$) or not ($\lnot$)). The introduction rules give some conditions under which we may assert the connective: for example, from $A$ and $B$ we may assert $A \land B$. The elimination rules give some conclusions we may assert from the connective: for example, from $A \land B$, we may conclude $A$. Thus in forming a proof, the introduction rules are used to introduce new logical connectives, and elimination rules are used to eliminate logical connectives.

The specific introduction and elimination rules used vary, depending on what logic (for example, classical, intuitionistic or minimal) we are working in.

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"Modus moron" rule of inference?

This is an exercise I got from the book "First Order Mathematical Logic" by Angelo Margaris (1967). I have never heard of this rule before, the question is whether what Margaris calls the modus moron rule of inference is correct or not and to…
user409521
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**Ended Competition:** What is the shortest proof of $\exists x \forall y (D(x) \to D(y)) $?

The competition has ended 6 june 2014 22:00 GMT The winner is Bryan Well done ! When I was rereading the proof of the drinkers paradox (see Proof of Drinker paradox I realised that $\exists x \forall y (D(x) \to D(y)) $ is also a theorem. I…
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Main differences and relations between Sequent Calculus and Natural Deduction

What are the main differences between the Sequent Calculus and the Natural Deduction (independently of if we're working with classical, intuitionistic or another logic) ? As far as I know : Differences : The sequent calculus is more suitable for…
Boris
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Difference between Logical Axioms and Rules of Inference

What's the difference between Logical Axioms and Rules of Inference? In my understanding, both are ordered pairs of formulas which are used to reach a conclusion through syllogisms. My questions Can both be formalized in a language? Are both…
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Motivation for natural deduction

I've been learning natural deduction recently. I've seen many problems and am starting to be able to solve problems more easily. For some reason I feel the need to ask what high school math students always ask about mathematics. What is the point…
John Smith
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Calculus of Natural Deduction That Works for Empty Structures

Currently, I am dealing with the calculus of natural deduction by Gentzen. This calculus gives us rules to manipulate so-called sequents. Definition. If $\Gamma$ is a set of formulas and $\phi$ a formula, then $\Gamma\vdash\phi$ is called a…
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What is the so-called eigenvariable or parameter in natural deduction?

I am reading the Wikipedia article on Natural Deduction. In section 6, the presentation of intr and elim rules for the universal and existential quantifiers, it mentions a concept called eigenvariable/parameter without further discussion. What is…
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Logic puzzle inspired by “Blue Eyes”

I learned of the “Blue Eyes” puzzle for the first time a couple of weeks ago, and I’ve been playing around with some of the logical concepts behind it since then. For anyone unfamiliar with this puzzle, I suggest you read and solve it before trying…
Franklin Pezzuti Dyer
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Does double negation distribute over disjunction intuitionistically?

Does the following equivalence $$\lnot \lnot (A \lor B) \leftrightarrow (\lnot \lnot A \lor \lnot \lnot B)$$ hold in propositional intuitionistic logic? And in propositional minimal logic? (In propositional classical logic this is obvious since $A…
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Relationship between sequent calculus and Hilbert systems, natural deduction, etc

I am trying to learn the basics of logic and I'm confused on how these proof systems work together. The big ones I see are Hilbert style, and then Gentzen style which includes natural deduction, and sequent calculus. I also see "intuitionistic…
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Proving De Morgan's Law with Natural Deduction

Here is my attempt, but I'm really not sure if I've done it right; as I'm just about getting the hang of Natural Deduction technique. Have I done it correctly? If not, where did I make errors and how should I do it? Thank you in advance! Sorry…
Haxify
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Why isn't Modus Ponens valid here

I have the following: $(\neg A \lor B) \rightarrow (\neg A \lor B) \\ (\neg A \lor B) \\ \vdash \neg A \lor B $ And in my mind this seems like a legitimate use of the Modus Ponens rule. But the textbook I'm using disagrees. Why is this…
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Simplifying propositional logic

Hi I asked a question a few hours ago which has been solved but I got stuck on another exercise so I thought I'd reach out for some help. I have the premise: $((A \to B) \land (\lnot A \to C))$ With the desired result at : $((A \land B) \lor (\lnot…
Kadana Kanz
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Show that $(p \to q) \lor (q \to p)$ is a tautology

I tried to prove that $(p \to q) \lor (q \to p)$ is a tautology. I used $p$ and $¬q$ as conditions. (Premises 1 and 5) I managed to get to a solution, but I'm not sure if it's right. Can you please check it? Thank you!!!:)
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Law of excluded middle. Do we need it in proofs?

Quite often when I am making a natural deduction proof, and I have no fixed idea on how to continue. I find myself thinking: "lets start with some form of the law of the excluded middle (LEM) and take it from there." But recently I more and more…
Willemien
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