Rules of Inference are the building blocks of logical reasoning in propositional logic. They help us draw valid conclusions from given premises, allowing us to construct sound arguments and proofs.

These rules, like Modus Ponens and Disjunctive Syllogism, are essential tools in Natural Deduction. They enable us to manipulate logical statements and derive new truths, forming the foundation for more complex logical reasoning.

Inference Rules for Conditional Statements

Modus Ponens

• States that if a conditional statement is true and the antecedent is true, then the consequent must also be true
• Symbolically represented as: $(p \to q) \land p \vdash q$
• The premise $p \to q$ is a conditional statement
• The premise $p$ is the antecedent of the conditional statement
• From these two premises, we can conclude $q$, which is the consequent of the conditional statement
• Allows us to draw conclusions based on the truth of a conditional statement and its antecedent (if it is raining, then the ground is wet)

Modus Tollens

• States that if a conditional statement is true and the consequent is false, then the antecedent must also be false
• Symbolically represented as: $(p \to q) \land \neg q \vdash \neg p$
• The premise $p \to q$ is a conditional statement
• The premise $\neg q$ is the negation of the consequent of the conditional statement
• From these two premises, we can conclude $\neg p$, which is the negation of the antecedent of the conditional statement
• Allows us to draw conclusions based on the truth of a conditional statement and the falsity of its consequent (if the ground is not wet, then it is not raining)

Hypothetical Syllogism

• States that if two conditional statements are true, and the consequent of the first is the antecedent of the second, then we can conclude a new conditional statement
• Symbolically represented as: $(p \to q) \land (q \to r) \vdash (p \to r)$
• The premise $p \to q$ is a conditional statement
• The premise $q \to r$ is another conditional statement, where the antecedent $q$ matches the consequent of the first statement
• From these two premises, we can conclude a new conditional statement $p \to r$, where the antecedent is from the first statement and the consequent is from the second statement
• Allows us to chain together conditional statements to form a new conditional statement (if it is raining, then the ground is wet; if the ground is wet, then it is slippery; therefore, if it is raining, then the ground is slippery)

Inference Rules for Disjunctions

Disjunctive Syllogism

• States that if a disjunction is true, and one of the disjuncts is false, then the other disjunct must be true
• Symbolically represented as: $(p \lor q) \land \neg p \vdash q$
• The premise $p \lor q$ is a disjunction
• The premise $\neg p$ is the negation of one of the disjuncts
• From these two premises, we can conclude $q$, which is the other disjunct
• Allows us to conclude the truth of one disjunct based on the falsity of the other (either the car is red or blue; the car is not red; therefore, the car is blue)

Addition

• States that if a statement is true, then any disjunction involving that statement is also true
• Symbolically represented as: $p \vdash (p \lor q)$
• The premise $p$ is a statement
• From this premise, we can conclude $p \lor q$, which is a disjunction involving the original statement and any other statement
• Allows us to introduce a disjunction based on the truth of one of its disjuncts (the sky is blue; therefore, the sky is blue or the grass is green)

Inference Rules for Conjunctions

Conjunction

• States that if two statements are true, then their conjunction is also true
• Symbolically represented as: $p, q \vdash (p \land q)$
• The premises $p$ and $q$ are individual statements
• From these premises, we can conclude $p \land q$, which is the conjunction of the two statements
• Allows us to combine two true statements into a single conjunctive statement (the sun is shining; the birds are singing; therefore, the sun is shining and the birds are singing)

Simplification

• States that if a conjunction is true, then each of its conjuncts is also true
• Symbolically represented as: $(p \land q) \vdash p$ and $(p \land q) \vdash q$
• The premise $p \land q$ is a conjunction
• From this premise, we can conclude $p$, which is one of the conjuncts
• Similarly, we can also conclude $q$, which is the other conjunct
• Allows us to break down a conjunctive statement into its individual components (the apple is red and sweet; therefore, the apple is red; therefore, the apple is sweet)

Other Inference Rules

Double Negation

• States that a statement is logically equivalent to its double negation
• Symbolically represented as: $p \equiv \neg(\neg p)$
• The statement $p$ is equivalent to the negation of its negation, $\neg(\neg p)$
• Double negation can be used to affirm a statement by negating its opposite (it is not the case that the door is not open; therefore, the door is open)
• This rule allows for the simplification of statements containing multiple negations (it is not not raining; therefore, it is raining)