INTRODUCTION TO LOGIC Jennifer Wang Fall 2009 Midterm Review Quiz Game

ConceptsTrue/FalseTranslations Informal Proofs Formal Proofs

Final Question

(100) If A is a tautology, is it also a logical truth? Yes.

(200) How do you check if an argument is valid using truth tables? Check all the rows in which the premises are all true. If the conclusion is always also true, it’s valid. If there’s a row where the premises are all true but the conclusion is false, it’s invalid.

(300) Why is disjunction introduction a valid form of inference? By the truth table for , P  Q is true iff at least one of P or Q is true. If we already know that P is true, we know that P  Q is true.

(400) Name three of the four formal proof methods we’ve learned so far that use subproofs. Disjunction elimination (proof by cases) Negation introduction (proof by contradiction) Conditional introduction (conditional proof) Biconditional introduction (biconditional proof)

(500) What does it mean for a connective to be truth- functional? This means that we can determine the truth value of the connective from the truth values of its parts.

(100) What is the main connective in this sentence?:  (Small(a)  Small(b)  Small(c))  Small(d) Disjunction.

(200) How does this translate into FOL?: c is left of and to the back of b only if b is a cube.. (LeftOf(c,b)  BackOf(c,b))  Cube(b)

(300) How does this translate into FOL?: Homer is a Simpson but Marge is truly a Simpson also. Simpson(homer)  Simpson(marge)

(400) How does this translate into FOL? If d is a cube, then it is not in the same column as either a or b. Cube(d)   (SameCol(d,a)  SameCol(d,b))

(500) How does this sentence read in plain English?: Red(mars)   (Red(earth)  Red(jupiter)) Mars is red if and only if not both Earth and Jupiter are red.

(100) A valid argument can have false premises. True.

(200) Provided that Jane runs, Ron does not fall translates as:  Runs(jane)   Falls(ron) False.

(300) This equivalence holds:  (Q  P) iff Q   P True.

(400) This equivalence holds: P  Q iff (P  Q)  (  P   Q) False.

(500) This sentence is true in this world: SameCol(a,b)  (Cube(a)  Tet(b)) True.

(100) What is the informal correlate of this rule: = Elim? Indiscernibility of Identicals

(200) How many cases do you need in a proof by cases? How many ever disjuncts there are in the disjunction you’re using.

(300) Which ones of these informal rules do not correspond to a formal rule?: Disjunctive syllogism, modus ponens, modus tollens, reflexivity of identity Disjunctive syllogism, modus tollens

(400) Is this a valid argument? Explain why or why not. Yes. Premise 2 is equivalent to Small(a)   Small(b), so  Small(b) follows by modus ponens. By 3 and modus tollens, we get  Small(c), or Small(c). By this and premise 4, Small(d) follows from disjunctive syllogism, which is our conclusion.

(500) Give an informal proof of this argument, citing all rules and their justifications. Let’s do a proof by cases on premise 3. Case 1: Assume that A. From this and premise 1, B follows by modus ponens. Case 2: Assume that C. This is equivalent to  C by double negation. From this and premise 2,  B follows by modus tollens. But by double negation, this is just B. So either way B follows, which is the conclusion we want.

(100) What is the name of the method of inference used here? Disjunction elimination (proof by cases)

(200) What do you need to cite for a proof of P  Q? Two subproofs, one going from P to Q and one going from Q to P.

(300) There are two places to go after you derive the contradiction symbol. What are they? You can use contradiction elimination to derive anything you want, or if you’re in a subproof, you can finish the subproof you’re in and derive the negation of the assumption that led to the contradiction (negation introduction).

(400) Give a formal proof of this argument:

(500) Give a formal proof of this argument:

Final question Give an example of an argument where the conclusion is a logical but not tautological consequence of some premises. Give an informal or formal proof of the argument (your choice), and then construct a truth table to show this result. Make sure to write down your explanation of why it’s a logical but not tautological truth.