The Socrates Example • We have the two premises: ○ “All men are mortal.” ○ “Socrates is a man.” • And the conclusion: ○ “Socrates is mortal.” • How do we get the conclusion from the premises? The Argument • We can express the premises (above the line) and the conclusion (below the line) in predicate logic as an argument: • We will see shortly that this is a valid argument Arguments in Propositional Logic • An argument in propositional logic is a sequence of propositions. • All but the final proposition are called premises. • The last statement is the conclusion. • The argument is valid if the premises imply the conclusion. • An argument form is an argument that is valid no matter what propositions are substituted into its propositional variables. • If the premises are p_1, p_2, …,p_n and the conclusion is q then ○ (p_1∧p_2∧…∧p_n)→q is a tautology. • Inference rules are all argument simple argument forms that will be used to construct more complex argument forms. Rules of Inference for Propositional Logic: • Modus Ponens ○ Equation ○ Corresponding Tautology: § (p∧(p→q))→q ○ Example: § Let p be “It is snowing.” § Let q be “I will study discrete math. § “If it is snowing, then I will study discrete math.” § “It is snowing.” § “Therefore , I will study discrete math.” • Modus Tollens ○ Equation ○ Corresponding Tautology: § (¬q∧(p→q))→¬p ○ Example: § Let p be “it is snowing.” § Let q be “I will study discrete math.” § “If it is snowing, then I will study discrete math.” § “I will not study discrete math.” § “Therefore , it is not snowing.” • Hypothetical Syllogism ○ Equation ○ Corresponding Tautology: § ((p→q)∧(q→r))→(p→r) ○ Example: § Let p be “it snows.” § Let q be “I will study discrete math.” § Let r be “I will get an A.” § “If it snows, then I will study discrete math.” § “If I study discrete math, I will get an A.” § “Therefore , If it snows, I will get an A.” • Disjunctive Syllogism ○ Equation ○ Corresponding Tautology: § (¬p∧(p ∨q))→q ○ Example: § Let p be “I will study discrete math.” § Let q be “I will study English literature.” § “I will study discrete math or I will study English literature.” § “I will not study discrete math.” § “Therefore , I will study English literature.” • Addition ○ Equation ○ Corresponding Tautology: § p→(p ∨q) ○ Example: § Let p be “I will study discrete math.” § Let q be “I will visit Las Vegas.” § “I will study discrete math.” § “Therefore, I will study discrete math or I will visit Las Vegas.” • Simplification ○ Equation ○ Corresponding Tautology: § (p∧q)→p ○ Example: § Let p be “I will study discrete math.” § Let q be “I will study English literature.” § “I will study discrete math and English literature” § “Therefore, I will study discrete math.” • Conjunction ○ Equation ○ Corresponding Tautology: § ((p)∧(q))→(p∧q) ○ Example: § Let p be “I will study discrete math.” § Let q be “I will study English literature.” § “I will study discrete math.” § “I will study English literature.” § “Therefore, I will study discrete math and I will study English literature.” • Resolution ○ Equation ○ Corresponding Tautology: § ((¬p∨r)∧(p∨q))→(q ∨ r) ○ Example: § Let p be “I will study discrete math.” § Let r be “I will study English literature.” § Let q be “I will study databases.” § “I will not study discrete math or I will study English literature.” § “I will study discrete math or I will study databases.” § “Therefore, I will study databases or I will study English literature.” Using the Rules of Inference to Build Valid Arguments • A valid argument is a sequence of statements. • Each statement is either a premise or follows from previous statements by rules of inference. • The last statement is called conclusion. Valid Arguments Example 1 ○ From the single proposition p∧(p→q) ○ Show that q is a conclusion. Example 2 ○ With 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.” ○ Using the inference rules, construct a valid argument for the conclusion: § “We will be home by sunset.” ○ Choose propositional variables: § p: “It is sunny this afternoon.” § r: “We will go swimming.” § t: “We will be home by sunset.” § q: “It is colder than yesterday.” § s: “We will take a canoe trip.” ○ Translation into propositional logic: § Hypotheses: ¬p∧q, r→p,¬r→s,s→t § Conclusion: t ○ Argument Handling Quantified Statements • Universal Instantiation (UI) ○ Example: § Our domain consists of all dogs and Fido is a dog. § “All dogs are cuddly.” § “Therefore, Fido is cuddly.” • Universal Generalization (UG) ○ Used often implicitly in Mathematical Proofs. • Existential Instantiation (EI) ○ Example: § “There is someone who got an A in the course.” § “Let’s call her a and say that a got an A” • Existential Generalization (EG) ○ Example: § “Michelle got an A in the class.” § “Therefore, someone got an A in the class.” Using Rules of Inference • Example 1 ○ Using the rules of inference, construct a valid argument to show that § “John Smith has two legs” ○ is a consequence of the premises § “Every man has two legs.” § “John Smith is a man.” ○ Notation and domain § Let M(x) denote “x is a man” § L(x) “x has two legs” § Let John Smith be a member of the domain. ○ Argument • Example 2 ○ Use the rules of inference to construct a valid argument showing that the conclusion § “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.” ○ Notation § 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.” ○ First we translate the premises and conclusion into symbolic form. ○ Argument Returning to the Socrates Example • Premises and conclusion • Argument The Barber Example • Show that from the statements • Every barber in Jonesville shaves those and only those who don t shave themselves. and There is a barber in Jonesville • We can derive a contradiction ○ ∀x(B(x)→∀y(S(x,y)⟷¬S(y,y))) ○ ∃x B(x) ○ ∀c B(c) ○ B(c)→∀y(S(c,y)⟷¬S(y,y)) ○ ∀y (S(c,y)⟷¬S(y,y)) ○ S(c,c)⟷¬S(c,c) ○ Thus, we have a contradiction Lewis Carroll • The first three are called premises and the third is called the conclusion ○ “All hummingbirds are richly colored.” ○ “No large birds live on honey.” ○ “Birds that do not live on honey are dull in color.” ○ “Hummingbirds are small.” • Notation ○ H(x)≔x is a hummingbird ○ C(x)≔x is richly colored ○ L(x)≔x is large ○ Ho(x)≔x lives on honey • Here is one way to translate these statements to predicate logic ○ ∀x (H(x)→C(x)) ○ ∀x (L(x)→¬Ho(x)) ○ ∀x (¬Ho(x)→¬C(x)) ○ ∀x (H(x)→¬L(x)) • Let c be an arbilirary element of the universe (1) ∀x (H(x)→C(x)) (2) ∀x (L(x)→¬Ho(x)) (3) ∀x (¬Ho(x)→¬C(x)) (4) H(c)→C(c)≡¬H(c)∨C(c) (5) L(c)→¬Ho(c)≡¬L(c)∨¬Ho(c) (6) ¬Ho(c)→¬C(c)≡Ho(c)∨¬C(c) (7) By resolution of (4) and (6), ¬H(c)∨Ho(c) (8) By resolution of (5) and (7), ¬H(c)∨¬L(c)≡H(c)→¬L(c) (9) By (8), ∀x(H(x)→¬L(x))
