# Definition:Natural Deduction System/Copi

## Definition

Source: Irving M. Copi: Symbolic Logic, 4th ed. (MacMillan, $1973$)

### Rules of Inference

#### Modus Ponendo Ponens

 $\ds p$ $\implies$ $\ds q$ $\ds p$  $\ds$ $\ds \vdash \ \$ $\ds q$  $\ds$

#### Modus Tollendo Tollens

The Modus Tollendo Tollens can be symbolised by the sequent:

 $\ds p$ $\implies$ $\ds q$ $\ds \neg q$  $\ds$ $\ds \vdash \ \$ $\ds \neg p$  $\ds$

#### Hypothetical Syllogism

 $\ds p$ $\implies$ $\ds q$ $\ds q$ $\implies$ $\ds r$ $\ds \vdash \ \$ $\ds p$ $\implies$ $\ds r$

#### Disjunctive Syllogism

 $\ds p \lor q$  $\ds$ $\ds \neg p$  $\ds$ $\ds \vdash \ \$ $\ds q$  $\ds$

#### Constructive Dilemma

 $\ds \paren {p \implies q} \land \paren {r \implies s}$  $\ds$ $\ds p \lor r$  $\ds$ $\ds \vdash \ \$ $\ds q \lor s$  $\ds$

#### Destructive Dilemma

 $\ds \paren {p \implies q} \land \paren {r \implies s}$  $\ds$ $\ds \neg q \lor \neg s$  $\ds$ $\ds \vdash \ \$ $\ds \neg p \lor \neg r$  $\ds$

#### Rule of Simplification

 $\ds p \land q$  $\ds$ $\ds \vdash \ \$ $\ds p$  $\ds$

#### Rule of Conjunction

 $\ds p$  $\ds$ $\ds q$  $\ds$ $\ds \vdash \ \$ $\ds p \land q$  $\ds$

 $\ds p$  $\ds$ $\ds \vdash \ \$ $\ds p \lor q$  $\ds$

### Rule of Replacement

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.

The Rule of Replacement can then subsequently be used on the following:

#### De Morgan's Theorems: 1

$\vdash \paren {\neg p \lor \neg q} \iff \paren {\neg \paren {p \land q} }$

#### De Morgan's Theorems: 2

$\vdash \paren {\neg p \land \neg q} \iff \paren {\neg \paren {p \lor q} }$

#### Rule of Commutation: 1

$\vdash \paren {p \lor q} \iff \paren {q \lor p}$

#### Rule of Commutation: 2

$\vdash \paren {p \land q} \iff \paren {q \land p}$

#### Rule of Association: 1

$\vdash \paren {p \lor \paren {q \lor r} } \iff \paren {\paren {p \lor q} \lor r}$

#### Rule of Association: 2

$\vdash \paren {p \land \paren {q \land r} } \iff \paren {\paren {p \land q} \land r}$

#### Rule of Distribution: 1

$\vdash \paren {p \land \paren {q \lor r} } \iff \paren {\paren {p \land q} \lor \paren {p \land r} }$

#### Rule of Distribution: 2

$\vdash \paren {p \lor \paren {q \land r} } \iff \paren {\paren {p \lor q} \land \paren {p \lor r} }$

#### Double Negation

$\vdash p \iff \neg \neg p$

#### Transposition

$\vdash \paren {p \implies q} \iff \paren {\neg q \implies \neg p}$

#### Material Implication

$\vdash \paren {p \implies q} \iff \paren {\neg p \lor q}$

#### Material Equivalence: 1

$\vdash \paren {p \iff q} \iff \paren {\paren {p \implies q} \land \paren {q \implies p} }$

#### Material Equivalence: 2

$\vdash \paren {p \iff q} \iff \paren {\paren {p \land q} \lor \paren {\neg p \land \neg q} }$

#### Exportation

$\vdash \paren {\paren {p \land q} \implies r} \iff \paren {p \implies \paren {q \implies r} }$

#### Tautology: 1

$\vdash p \iff \paren {p \lor p}$

#### Tautology: 2

$\vdash p \iff \paren {p \land p}$