Rules of Inference Dr. Yasir Ali

Syllogism An argument form consisting of two premises and a conclusion is called a syllogism. The first and second premises are called the major premise and minor premise, respectively.

Modus Ponens The most famous form of valid syllogism in logic is called modus ponens. It has the following form: Following argument form is valid. If the sum of the digits of 371,487 is divisible by 3, then 371,487 is divisible by 3. The sum of the digits of 371,487 is divisible by 3. Therefore, 371,487 is divisible by 3. p q p →q T F

Modus Tollens Another valid argument form called modus tollens. It has the following form: If p then q. ￢ q ∴ ￢ p Following argument form is valid. If the watch-dog detects an intruder, the dog will bark. The dog did not bark. Therefore, no intruder was detected by the watch-dog. p q p →q ￢q ￢p T F

Additional Valid Argument Forms: Rules of Inference Generalization: The following argument forms are valid. Following argument form is valid. “It is below freezing now”, therefore, “It is below freezing now” or “It is raining now.” Or simply, It is below freezing now, therefore, it is below freezing or raining now. Also, It is raining now, therefore, it is below freezing or raining now.

Specialization: The following argument forms are valid. Following argument form is valid. “It is below freezing now,” and “It is raining now”, therefore, it is below freezing now. Or simply, It is below freezing and raining now, therefore it is below freezing now. Also, It is below freezing and raining now, therefore it is raining now.

Elimination: The following argument forms are valid. Following argument form is valid. It rains today or we will have a barbecue today. We won’t have barbecue today, therefore, it is raining today. Also, It rains today or we will have a barbecue today. It is not raining today, therefore, we won’t have barbecue today.

Transitivity: The following argument forms are valid. Following argument form is valid. If it rains today, then we will not have a barbecue today. If we do not have a barbecue today, then we will have a barbecue tomorrow. Therefore, if it rains today, then we will have a barbecue tomorrow.

Fallacies A fallacy is an error in reasoning that results in an invalid argument. Consider the following argument form: If Zeke is a cheater, then Zeke sits in back row. Zeke sits in the back row, therefore Zeke is a cheater. The above argument has the form: Which is not a valid argument form, this error in reasoning is known as converse error. p q p →q T F

Consider the following argument: If interest rates are going up, stock market prices will go down. Interest rates are not going up. Therefore, stock market prices will not go down. This is an invalid argument form, and the error in reasoning is known as inverse error. p q p →q ￢p ￢q T F

Rules of Inference Rule of Inference Tautology Name p p → q ∴ q Modus ponens ￢q p → q ∴ ￢p (￢q ∧ (p → q))→￢p Modus tollens p → q q → r ∴ p → r ((p → q) ∧ (q → r)) → (p → r ) Hypothetical syllogism p ∨ q ￢p ∴ q ((p ∨ q)∧￢p) → q Disjunctive syllogism ∴ p ∨ q p → (p ∨ q) Addition p ∧ q ∴ p (p ∧ q) → p Simplification q ∴ p ∧ q ((p) ∧ (q)) → (p ∧ q) Conjunction ￢p ∨ r ∴ q ∨ r ((p ∨ q) ∧ (￢p ∨ r)) → (q ∨ r) Resolution

Example: Show that the premises “If you send me an e-mail message, then I will finish writing the program,” “If you do not send me an e-mail message, then I will go to sleep early,” and “If I go to sleep early, then I will wake up feeling refreshed” lead to the conclusion “If I do not finish writing the program, then I will wake up feeling refreshed.” Let p :“You send me an e-mail message,” q :“I will finish writing the program,” r :“I will go to sleep early,” and s :“I will wake up feeling refreshed.” Then the argument takes the form p → q, ￢p → r, r → s Therefore ￢q → s.

For each of these sets of premises, what relevant conclusion or conclusions can be drawn? Explain the rules of inference used to obtain each conclusion from the premises. a) “If I play hockey, then I am sore the next day.” “I use the whirlpool if I am sore.” “I did not use the whirlpool.” b) “All insects have six legs.” “Dragonflies are insects.” “Spiders do not have six legs.” “Spiders eat dragonflies.” c) “Every student has an Internet account.” “Homer does not have an Internet account.” “Maggie has an Internet account.” d) “All foods that are healthy to eat do not taste good.” “Tofu is healthy to eat.” “You only eat what tastes good.” “You do not eat tofu.” “Cheeseburgers are not healthy to eat.” e ) “I am either dreaming or hallucinating.” “I am not dreaming.” “If I am hallucinating, I see elephants running down the road.”

A set of premises and a conclusion are given A set of premises and a conclusion are given. Use the valid argument forms (rules of inference) to deduce the conclusion from the premises, giving a reason for each step. Assume all variables are statement variables. p →q r ∨ s ￢ s → ￢ t ￢ q ∨ s ￢ s ￢ p ∧ r → u w ∨ t ∴ u ∧ w

Given the following information about a computer program, find the mistake in the program. There is an undeclared variable or there is a syntax error in the first five lines. If there is a syntax error in the first five lines, then there is a missing semicolon or a variable name is misspelled. There is not a missing semicolon. There is not a misspelled variable name.