Definition:Natural Deduction/Elementary Valid Argument Forms

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Definition

In most treatments of PropLog various subsets of the following rules are treated as the axioms. Some of them are obvious. Others are more subtle.

These rules are not all independent, in that it is possible to prove some of them using sequents constructed from combinations of others. However, when a set of proof rules is selected as the axioms for any particular treatment of this subject, those rules are usually selected carefully so that they are independent.

Rule of Assumption

An assumption 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 $\neg \left({\phi \land \psi}\right)$, and we can also conclude $\phi$, then we may infer $\neg \psi$.
$(2): \quad$ If we can conclude $\neg \left({\phi \land \psi}\right)$, 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 $\phi \implies \psi$.


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 does not depend upon the assumption $\phi$.


Rule of Explosion

If a contradiction can be concluded, it is possible to infer any statement.


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 does not depend upon the assumption $\neg \phi$.


Also known as

Some sources refer to elementary valid argument forms as axioms of natural deduction.

They are also seen referred to as rules of inference.


Sources

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