# Definition:Natural Deduction/Rules of Inference

## Definition

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 $\psi$.

$(1): \quad$ 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 $\psi \implies \phi$.

If we can conclude both $\phi$ and $\neg \phi$, we may infer a 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 $\phi$.

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 $\neg \phi$.

### Remark

These rules are not all independent, in that it is possible to prove some of them using combinations of others. This owes to the intended practical applicability of natural deduction.

## Also known as

Some sources refer to the rules of inference of natural deduction as elementary valid argument forms or as axioms of natural deduction.