Propositional Logic - Rules of Inference

Conjunction Elimination

P∧Q
∴P

Figure 9

Conjunction Elimination says that given any two propositions P and Q, you are permitted to infer both P and Q individually from the conjunction P∧Q. (∧E)

Example of proof of deriving P from (P∧R)∧(Q∧S)

(P∧R)∧(Q∧S)

A
(P∧R) 1∧E
P 2∧E

Figure 10

Conjunction Introduction

P
Q
∴P∧Q

Figure 11

Conjunction Introduction says that given any two propositions P and Q, you are permitted to infer P∧Q if you already have both P and Q. (∧I)

Disjunction Introduction

P
∴P∨Q

Figure 12

Disjunction Introduction says that given any two propositions P, you are permitted to infer P∨Q for any proposition Q. (∨I) Although this rule produces weaker information, that is sometimes required to prove a conclusion.

Example of proof of (P∨R)∧(Q∨R) from P∧Q

P∧Q

A
P 1∧E
P∨R 2∨I
Q 1∧E
Q∨R 4∨I
(P∨R)∧(Q∨R) 3,5 ∧I

Figure 13

Disjunction Syllogism

P∨Q

¬P
∴Q

Figure 14

Disjunction Syllogism says that given P∨Q and ¬P, you can conclude Q. (DS)

Modus Ponens

P→Q

P
∴Q

Figure 15

Modus Ponens says that given P→Q and P, you can conclude Q. (MP) This could also be called “implication elimination”.

There is a fallacy called Affirming the Consequent where one of the premises are the consequent, you cannot validly infer from this the antecedent. Example: If I spill my cup, then the floor will be wet. I cannot deduce from a wet floor that I spilled my cup since it could have been wet due to other reasons.

Modus Tollens

P→Q

¬Q
∴¬P

Figure 16

Modus Tollens says that given P→Q and ¬Q, you can conclude ¬P. (MT)

There is a fallacy called Denying the Antecedent where one of the premises are the negation of the antecedent. Example: If I spill my cup, then the floor will be wet. I cannot deduce from the fact that the cup was not spilled that the floor is not wet since it could have been wet due to other reasons.

Double Negation

¬¬P
∴P

Figure 17

Double negation says that given P, then you can infer ¬¬P. Or you can infer P from ¬¬P. (DN)

Hypothetical Syllogism

P→Q

Q→R
∴P→R
Figure 18

Hypothetical syllogism states when you are given P→Q and Q→R, then you can infer P→R. (HS)

Constructive Dilemma

P→Q

R→S
P∨R
∴Q∨S

Figure 19

Constructive Dilemma states when you are given P→Q, R→S and P∨R, then you can infer Q∨S. (CD)


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