Definition:Natural Deduction
Definition
Natural deduction is a technique for deducing valid sequents from other valid sequents by applying precisely defined proof rules, by a technique called logical inference.
As such, natural deduction forms a proof system, which is focused on practical applicability.
Motivation
In its practical applicability, natural deduction differs from most proof systems in literature, which are more pedantic and formalistic, but therefore also more rigorous.
One may interpret natural deduction as the full-fledged proof system to actually use, once a formalistic alternative has been proved to satisfy all the rules of inference of natural deduction.
In doing so, it no longer matters which exact formalism was employed, and one can focus on the mathematical content itself.
On $\mathsf{Pr} \infty \mathsf{fWiki}$, we use natural deduction for both propositional logic and predicate logic.
Additionally, natural deduction may be applied not only for classical propositional logic, but also for more limited forms such as intuitionistic propositional logic, by employing the appropriate restrictions.
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Rules of Inference
The proof system called natural deduction deals exclusively with the notion of provable consequence.
As such, it does not contain any axioms.
Practically, this means that any proof of natural deduction will start with premises or the Rule of Assumption.
The complete list of rules of inference of natural deduction is as follows:
Rule of Assumption
- An assumption $\phi$ may be introduced at any stage of an argument.
Rule of Conjunction
- If we can conclude both $\phi$ and $\psi$, we may infer the compound statement $\phi \land \psi$.
Rule of Simplification
- $(1): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\phi$.
- $(2): \quad$ If we can conclude $\phi \land \psi$, then we may infer $\psi$.
Rule of Addition
- $(1): \quad$ If we can conclude $\phi$, then we may infer $\phi \lor \psi$.
- $(2): \quad$ If we can conclude $\psi$, then we may infer $\phi \lor \psi$.
Proof by Cases
- If we can conclude $\phi \lor \psi$, and:
- $(1): \quad$ By making the assumption $\phi$, we can conclude $\chi$
- $(2): \quad$ By making the assumption $\psi$, we can conclude $\chi$
- then we may infer $\chi$.
Modus Ponendo Ponens
- If we can conclude $\phi \implies \psi$, and we can also conclude $\phi$, then we may infer $\psi$.
Modus Tollendo Tollens
- If we can conclude $\phi \implies \psi$, and we can also conclude $\neg \psi$, then we may infer $\neg \phi$.
Modus Tollendo Ponens
- $(1): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \phi$, then we may infer $\psi$.
- $(2): \quad$ If we can conclude $\phi \lor \psi$, and we can also conclude $\neg \psi$, then we may infer $\phi$.
Modus Ponendo Tollens
- $(1): \quad$ If we can conclude $\map \neg {\phi \land \psi}$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
- $(2): \quad$ If we can conclude $\map \neg {\phi \land \psi}$, and we can also conclude $\psi$, then we may infer $\neg \phi$.
Rule of Implication
- If, by making an assumption $\phi$, we can conclude $\psi$ as a consequence, we may infer $\phi \implies \psi$.
- The conclusion $\phi \implies \psi$ does not depend on the assumption $\phi$, which is thus discharged.
Double Negation Introduction
- If we can conclude $\phi$, then we may infer $\neg \neg \phi$.
Double Negation Elimination
- If we can conclude $\neg \neg \phi$, then we may infer $\phi$.
Biconditional Introduction
- If we can conclude both $\phi \implies \psi$ and $\psi \implies \phi$, then we may infer $\phi \iff \psi$.
Biconditional Elimination
- $(1): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\phi \implies \psi$.
- $(2): \quad$ If we can conclude $\phi \iff \psi$, then we may infer $\psi \implies \phi$.
Principle of Non-Contradiction
- If we can conclude both $\phi$ and $\neg \phi$, we may infer a contradiction.
Proof by Contradiction
- If, by making an assumption $\phi$, we can infer a contradiction as a consequence, then we may infer $\neg \phi$.
- The conclusion $\neg \phi$ does not depend upon the assumption $\phi$.
Rule of Explosion
- If a contradiction can be concluded, it is possible to infer any statement $\phi$.
Law of Excluded Middle
- $\phi \lor \neg \phi$ for all statements $\phi$.
Reductio ad Absurdum
- If, by making an assumption $\neg \phi$, we can infer a contradiction as a consequence, then we may infer $\phi$.
- The conclusion $\phi$ does not depend upon the assumption $\neg \phi$.
Derived Rules
In practically working with natural deduction, the following derived rules are useful.
Rule of Substitution
Let $S$ be a sequent of propositional logic that has been proved.
Then we may infer any sequent $S'$ resulting from $S$ by substitutions for letters.
Rule of Sequent Introduction
Let the statements $P_1, P_2, \ldots, P_n$ be conclusions in a proof, on various assumptions.
Let $P_1, P_2, \ldots, P_n \vdash Q$ be a sequent for which we already have a proof.
Then we may infer, at any stage of a proof (citing SI), the conclusion $Q$ of the sequent already proved.
This conclusion depends upon the pool of assumptions upon which $P_1, P_2, \ldots, P_n$ rest.
Rule of Theorem Introduction
We may infer, at any stage of a proof (citing $\text {TI}$), a theorem already proved, together with a reference to the theorem that is being cited.
Also known as
Natural deduction is seen under various less precise names, for example decision procedure or decision method.
Some sources call it the axiomatic method.
Also see
Certain schools of logic have investigated the situation of what happens when certain of the above proof rules (and their equivalents) are disallowed:
- Johansson's Minimal Logic disallows the Rule of Explosion and the Law of Excluded Middle.
- Intuitionistic Propositional Logic disallows the Law of Excluded Middle.
- Classical Propositional Logic is the school of propositional logic which allows all the above rules.
- Results about natural deduction can be found here.
Historical Note
The first system of rules for natural deduction was devised by Gerhard Gentzen in $1934$.
Technical Note: Templates
In order to make the use of the proof rules of natural deduction in a tableau proof on $\mathsf{Pr} \infty \mathsf{fWiki}$, the following templates have been developed:
For convenience, other templates are also available, for the following derived rules:
Template:Commutation | to invoke | the Rule of Commutation | |||||||
Template:DeMorgan | to invoke | an instance of De Morgan's Laws | |||||||
Template:Idempotence | to invoke | the Rule of Idempotence | |||||||
Template:IdentityLaw | to invoke | the Law of Identity |
For the other general proof rules, there exist the following templates:
Template:SequentIntro | to invoke | the Rule of Sequent Introduction | |||||||
Template:TheoremIntro | to invoke | the Rule of Theorem Introduction | |||||||
Template:Substitution | to invoke | the Rule of Substitution |
Sources
- 1959: A.H. Basson and D.J. O'Connor: Introduction to Symbolic Logic (3rd ed.) ... (previous) ... (next): $\S 4.1$: The Purpose of the Axiomatic Method
- 1980: D.J. O'Connor and Betty Powell: Elementary Logic ... (previous) ... (next): $\S \text{II}$: The Logic of Statements $(2): \ 1$: Decision procedures and proofs
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): natural deduction
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2$: Natural Deduction
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): natural deduction