# Definition:Natural Deduction/Rules of Inference

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## 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 $\psi$.

### 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 $\neg \phi$.

### 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**.

## Sources

- 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $3$: The Method of Deduction: $3.1$: Formal Proof of Validity - 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 - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.2.3$: Natural Deduction in summary