Rules of Inference 2

Valid Arguments 3 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.” Solution: 1. 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.” 2. Translation into propositional logic: Hypotheses: p  q, r  p, r  s, s  t Conclusion: t Continued on next slide  Valid Arguments 4 3. Construct the Valid Argument Step 1. p  q Reason Premise 2. p Simplification using 1 3. r  p Premise 4. r Modus tollens using  2 and  3 5. r  s Premise 6. s Modus ponens using  4 and  5 7. s  t Premise 8. t Modus ponens using  6 and  7 Handling Quantified Statements Valid arguments for quantified statements are a sequence of statements. Each statement is either a premise or follows from previous statements by rules of inference which include: • Rules of Inference for Propositional Logic • Rules of Inference for Quantified Statements The rules of inference for quantified statements are introduced in the next several slides. Universal Instantiation (UI) xP  x  P  c Example: Our domain consists of all dogs and Fido is a dog. “All dogs are cuddly.” “Therefore, Fido is cuddly.” Universal Generalization (UG) P  c  for an arbitrary c xP  x Used often implicitly in Mathematical Proofs. Existential Instantiation (EI) xP  x  P  c for some element c 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) P  c for some element c xP  x Example: “Michelle got an A in the class.” “Therefore, someone got an A in the class.” Using Rules of Inference 1 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.” Solution: Let M(x) denote “x is a man” and L(x) “ x has two legs” and let John Smith be a member of the domain. Valid Argument: Step Reason 1. x  M  x  L  x   Premise 2. M  J   L  J  UI from 1 3. M  J  Premise 4. L  J  Modus Ponens using  2 and  3 Using Rules of Inference 2 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.” Solution: Let C(x) denote “x is in this class,” B(x) denote “ x has read the book,” and P(x) denote “x passed the first exam.” First we translate the premises and conclusion into symbolic form. x  C  x   B  x   x  C  x   P  x   x  P  x  B  x  Continued on next slide  Using Rules of Inference Valid Argument: Step Reason 1. x  C  x   B  x  Premise 2. C  a   B  a  EI from 1 3. C  a  Simplification from  2 4. x  C  x   P  x   Premise 5. C  a   P  a  UI from  4 6. P  a  MP from  3 and  5 7. B  a  Simplification from  2 8. P  a   B  a  Conj from  6 and  7 9. x  P  x  B  x  EG from  8 3 Returning to the Socrates Example x  Man  x   Mortal  x  Man  Socrates   Mortal  Socrates Solution for Socrates Example Valid Argument Step Reason 1. x  Man  x  Mortal  x  Premise 2. Mam  Socrates  Mortal  Socrates UI from 1 3. Mam  Socrates  Premise 4. Mortal  Socrates MP from  2 and  3 Universal Modus Ponens Universal Modus Ponens combines universal instantiation and modus ponens into one rule. x  P  x   Q  x   P  a  , where a is a particular element in the domain  Q  a This rule could be used in the Socrates example.