# List of rules of inference

This is a list of rules of inference, logical laws that relate to mathematical formulae.

## Introduction

**Rules of inference** are syntactical **transform** rules which one can use to infer a conclusion from a premise to create an argument. A set of rules can be used to infer any valid conclusion if it is complete, while never inferring an invalid conclusion, if it is sound. A sound and complete set of rules need not include every rule in the following list, as many of the rules are redundant, and can be proven with the other rules.

*Discharge rules* permit inference from a subderivation based on a temporary assumption. Below, the notation

indicates such a subderivation from the temporary assumption to .

## Rules for classical sentential calculus

Sentential calculus is also known as propositional calculus.

### Rules for negations

- Reductio ad absurdum (or
*Negation Introduction*)

- Reductio ad absurdum (related to the law of excluded middle)

- Noncontradiction (or
*Negation Elimination*)

- Double negation elimination

- Double negation introduction

### Rules for conditionals

- Modus ponens (or
*Conditional Elimination*)

- Modus tollens

### Rules for conjunctions

- Adjunction (or
*Conjunction Introduction*)

- Simplification (or
*Conjunction Elimination*)

### Rules for disjunctions

- Addition (or
*Disjunction Introduction*)

- Case analysis (or
*Proof by Cases*or*Argument by Cases*)

- Constructive dilemma

### Rules for biconditionals

- Biconditional introduction

- Biconditional Elimination

## Rules of classical predicate calculus

In the following rules, is exactly like except for having the term everywhere has the free variable .

- Universal Generalization (or Universal Introduction)

Restriction 1:
is a variable which does not occur in
.

Restriction 2:
is not mentioned in any hypothesis or undischarged assumptions.

- Universal Instantiation (or Universal Elimination)

Restriction: No free occurrence of in falls within the scope of a quantifier quantifying a variable occurring in .

- Existential Generalization (or Existential Introduction)

Restriction: No free occurrence of in falls within the scope of a quantifier quantifying a variable occurring in .

- Existential Instantiation (or Existential Elimination)

Restriction 1:
is a variable which does not occur in
.

Restriction 2: There is no occurrence, free or bound, of
in
.

Restriction 3:
is not mentioned in any hypothesis or undischarged assumptions.

## Rules of substructural logic

The following are special cases of universal generalization and existential elimination; these occur in substructrual logics, such as linear logic.

- Rule of weakening (or monotonicity of entailment) (aka no-cloning theorem)

- Rule of contraction (or idempotency of entailment) (aka no-deleting theorem)

## Table: Rules of Inference

The rules above can be summed up in the following table.^{[1]} The "Tautology" column shows how to interpret the notation of a given rule.

Rules of inference | Tautology | Name |
---|---|---|

Modus ponens | ||

Modus tollens | ||

Associative | ||

Commutative | ||

Law of biconditional propositions | ||

Exportation | ||

Transposition or contraposition law | ||

Hypothetical syllogism | ||

Material implication | ||

Distributive | ||

Absorption | ||

Disjunctive syllogism | ||

Addition | ||

Simplification | ||

Conjunction | ||

Double negation | ||

Disjunctive simplification | ||

Resolution | ||

Disjunction Elimination |

All rules use the basic logic operators. A complete table of "logic operators" is shown by a truth table, giving definitions of all the possible (16) truth functions of 2 boolean variables (*p*, *q*):

p | q |
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | ||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

T | T | F | F | F | F | F | F | F | F | T | T | T | T | T | T | T | T | ||

T | F | F | F | F | F | T | T | T | T | F | F | F | F | T | T | T | T | ||

F | T | F | F | T | T | F | F | T | T | F | F | T | T | F | F | T | T | ||

F | F | F | T | F | T | F | T | F | T | F | T | F | T | F | T | F | T |

where T = true and F = false, and, the columns are the logical operators: **0**, false, Contradiction; **1**, NOR, Logical NOR; **2**, Converse nonimplication; **3**, **¬p**, Negation; **4**, Material nonimplication; **5**, **¬q**, Negation; **6**, XOR, Exclusive disjunction; **7**, NAND, Logical NAND; **8**, AND, Logical conjunction; **9**, XNOR, If and only if, Logical biconditional; **10**, **q**, Projection function; **11**, if/then, Logical implication; **12**, **p**, Projection function; **13**, then/if, Converse implication; **14**, OR, Logical disjunction; **15**, true, Tautology.

Each logic operator can be used in an assertion about variables and operations, showing a basic rule of inference. Examples:

- The column-14 operator (OR), shows
*Addition rule*: when*p*=T (the hypothesis selects the first two lines of the table), we see (at column-14) that*p*∨*q*=T.- We can see also that, with the same premise, another conclusions are valid: columns 12, 14 and 15 are T.

- The column-8 operator (AND), shows
*Simplification rule*: when*p*∧*q*=T (first line of the table), we see that*p*=T.- With this premise, we also conclude that
*q*=T,*p*∨*q*=T, etc. as showed by columns 9-15.

- With this premise, we also conclude that
- The column-11 operator (IF/THEN), shows
*Modus ponens rule*: when*p*→*q*=T and*p*=T only one line of the truth table (the first) satisfies these two conditions. On this line,*q*is also true. Therefore, whenever p → q is true and p is true, q must also be true.

Machines and well-trained people use this look at table approach to do basic inferences, and to check if other inferences (for the same premises) can be obtained.

### Example 1

Let us consider the following assumptions: "If it rains today, then we will not go on a canoe today. If we do not go on a canoe trip today, then we will go on a canoe trip tomorrow. Therefore (Mathematical symbol for "therefore" is ), if it rains today, we will go on a canoe trip tomorrow". To make use of the rules of inference in the above table we let be the proposition "If it rains today", be "We will not go on a canoe today" and let be "We will go on a canoe trip tomorrow". Then this argument is of the form:

### Example 2

Let us consider a more complex set of assumptions: "It is not sunny today and it is colder than yesterday". "We will go swimming only if it is sunny", "If we do not go swimming, then we will have a barbecue", and "If we will have a barbecue, then we will be home by sunset" lead to the conclusion "We will be home by sunset." Proof by rules of inference: Let be the proposition "It is sunny today", the proposition "It is colder than yesterday", the proposition "We will go swimming", the proposition "We will have a barbecue", and the proposition "We will be home by sunset". Then the hypotheses become and . Using our intuition we conjecture that the conclusion might be . Using the Rules of Inference table we can proof the conjecture easily:

Step | Reason |
---|---|

1. | Hypothesis |

2. | Simplification using Step 1 |

3. | Hypothesis |

4. | Modus tollens using Step 2 and 3 |

5. | Hypothesis |

6. | Modus ponens using Step 4 and 5 |

7. | Hypothesis |

8. | Modus ponens using Step 6 and 7 |

## References

- ↑ Kenneth H. Rosen:
*Discrete Mathematics and its Applications*, Fifth Edition, p. 58.