CS636 Final Term Notes

CS636 Notes · 1. Tautologies and Contradictions · Tautologies

A tautology is a statement that is always true, no matter what the truth values of its variables are.

Example:

P∨¬P (A statement is true, or its negation is true).

P

¬P

P∨¬P

T · F · T · F · T · T

o Result: Always true.

Contradictions

A contradiction is a statement that is always false, no matter what the truth values of its variables are.

Example:

P∧¬P (A statement cannot be true and false at the same time).

P

¬P

P∧¬P

T · F · F · F · T · F

o Result: Always false.

2. Arguments

An argument is a set of statements (called premises) that lead to a conclusion.

o

Example:

If it rains, the ground gets wet (premise). It is raining (premise). Therefore, the ground is wet (conclusion).

An argument is valid if the conclusion is always true whenever the premises are true.

Example:

Premises:

P: It is raining.

P→Q: If it is raining, the ground is wet. Conclusion: Q: The ground is wet.

P · Q · P→Q

Valid Argument?

T · T · T · Yes · F · T · T · Yes · T · F · F · No · F · F · T · Yes · Fallacy

A fallacy is an invalid argument, where the conclusion does not logically follow the premises.

Example (Fallacy):

Premises: P→Q , Q. Conclusion: P.

This is invalid because Q being true does not necessarily mean P is true.

3. Validity of Arguments

How to Check Validity

An argument is valid if the conclusion Q is true whenever all the premises are true.

Mathematically: (P1∧P2∧...∧Pn)→Q must be a tautology.

Example of Valid Argument

Argument: P→Q,¬Q⊢ ¬P

Explanation:

o

Premises:

1. If P, then Q (P→Q).

2. Q is false (¬Q).

o Conclusion: P must also be false (¬P).

P · Q · P→Q · ¬Q · ¬P

Valid?

T · T · T · F · F · Yes · T · F · F · T · F · No · 4. Three-Valued Logic

In classical logic, every statement is either true (T) or false (F).

In three-valued logic, a statement can also be undefined (U).

o Example: Division by zero (0/0) is undefined.

Operators in Three-Valued Logic

1. AND (∧): Both inputs must be true for the result to be true.

o

Example: · P · Q

P∧Q

T · T · T · T · F · F · T · U · U

2. OR (∨): At least one input must be true for the result to be true.

o

Example: · P · Q

P∨Q

T · F · T · F · U · U

3. Implication (→): If P is true, Q must be true for the result to be true.

o

Example: · P · Q · P →Q · T · T · T · T · F · F · U · T · T

4. Equivalence (↔): Both values must be the same for the result to be true.