The Foundations: Logic and Proofs

The Foundations: Logic and Proofs

Rules of Inference Valid Arguments An argument is a sequence of propositions. All but the final proposition are called premises. The last statement is the conclusion. The Socrates Example We have two premises: All men are mortal. Socrates is a man. The conclusion is: Socrates is mortal.

The argument is valid if the premises imply the conclusion. Arguments in Propositional Logic Formal notation of an argument: the premises are above the line the conclusion is below the line If it is raining then streets are wet It is raining Streets are wet How do we know that this argument is valid? Arguments in Propositional Logic

How do we know that this argument is valid? 1. Use truth tables T p q T p (pq) T

(p (pq)) q T T F F F T F

T T F T F F T

F T p q T Tedious: table size grows exponentially with number of variables 2. Establish rules to incrementally build argument

Arguments in Propositional Logic An argument form is an abstraction of an argument It contains propositional variables It is valid no matter what propositions are substituted into its variables, i.e.: If the premises are p1 ,p2, ,pn and the conclusion is q then (p1 p2 pn ) q is always T (a tautology) If an argument matches an argument form then it is valid Example: ( (p q ) p ) q is a tautology

Hence the following argument is valid: ( (if its raining streets are wet) its raining) streets are wet Inference rules are simple argument forms used to incrementally construct more complex argument forms Modus Ponens Corresponding Tautology: (p (p q)) q Example: Let p be It is snowing.

Let q be I will study math. If it is snowing then I will study math. It is snowing. Therefore I will study math. Modus Tollens Corresponding Tautology: (q (p q)) p Example: Let p be it is snowing. Let q be I will study math. If it is snowing, then I will study

math. I will not study math. Therefore, it is not snowing. Hypothetical Syllogism Corresponding Tautology: ((p q) (qr)) (pr) Example: Let p be It snows. Let q be I will study math. Let r be I will get an A. If it snows, then I will study math.

If I study math, then I will get an A. Disjunctive Syllogism Corresponding Tautology: (p (p q)) q Example: Let p be I will study math. Let q be I will study literature. I will study math or I will study literature. I will not study math. Therefore, I will study literature. Simplification

Corresponding Tautology: (pq) q Example: Let p be I will study math. Let q be I will study literature. I will study math and literature Therefore, I will study math. Addition Corresponding Tautology: p (p q) Example:

Let p be I will study math. Let q be I will visit Las Vegas. I will study math. Therefore, I will study math or I will visit Las Vegas. Conjunction Corresponding Tautology: ((p) (q)) (p q) Example: Let p be I will study math. Let q be I will study literature. I will study math. I will study literature. Therefore, I will study math and

literature. Resolution Corresponding Tautology: ((p r ) (p q)) (q Example: r) Let p be I will study math. Let r be I will study literature. Let q be I will study physics. I will not study math or I will study literature. I will study math or I will study physics.

Therefore, I will study physics or Valid Arguments Example: Given these hypotheses: It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip, then we will be home by sunset. Construct a valid argument for the conclusion: We will be home by sunset. Solution: 1. Choose propositional variables: p: It is sunny this afternoon.

p q q: It is colder than yesterday. r p r: We will go swimming. r s s: We will take a canoe trip. t: We will be home by sunset. s t 2. Translate into propositional logic t Valid Arguments 3. Construct the Valid Argument

p q r p r s s t t Common Fallacies: Affirming the conclusion (( p q) q) p This confuses necessary and sufficient conditions.

Example: If people have the flu, they cough. Alison is coughing. Therefore, Alison has the flu. This argument is not valid: Other things, such as asthma, can cause someone to cough. Having the flu is a sufficient condition for coughing, but it is not necessary Common Fallacies: Denying the hypothesis

(( p q ) p) q This also confuses necessary and sufficient conditions. Example: If it is raining outside, the sky is cloudy. It is not raining outside. Therefore, it is not cloudy. This argument is not valid: Skies can be cloudy without any rain. Rain is a sufficient condition of cloudiness, but it is not necessary.

Universal Instantiation (UI) If a predicate is true for all elements x in the domain then it is true for any specific element c Example: The domain consists of all dogs and Fido is a dog. All dogs are cuddly. Therefore, Fido is cuddly. This rule allows us to remove a quantifier Universal Generalization (UG) If a predicate is true for any element c in the

domain then it is true for all elements x Example: The domain consists of the dogs Fido, Spot, and Buddy. Fido is cuddly, Spot is cuddly, Buddy is cuddly. Therefore, all dogs in the domain are cuddly. This rule allows us to introduce a quantifier Existential Instantiation (EI) If a predicate is true some element in the domain then it is true for some specific element c

Example: There is someone who got an A in the course. Lets call her c and say that c got an A Existential Generalization (EG) If a predicate is true for a specific element c in the domain then there exists an element x for which it is true Example: Michelle got an A in the class. Therefore, there is someone who got an A in the class.

Returning to the Socrates Example 1 Using Rules of Inference Example: construct a valid argument showing that: Someone who passed the first exam has not read the book. follows from the premises A student in this class has not read the book. Everyone in this class passed the first exam. Solution: Let C(x) denote x is in this class

B(x) denote x has read the book P(x) denote x passed the first exam Using Rules of Inference Valid Argument:

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