Transcript Intro to Logic Quiz Game Midterm

Quiz Game
INTRODUCTION TO LOGIC
Jennifer Wang
Fall 2009
Midterm Review
Concepts
True/False
Translations
Informal
Proofs
Formal
Proofs
100
100
100
100
100
200
200
200
200
200
300
300
300
300
300
400
400
400
400
400
500
500
500
500
500
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 truthfunctional?
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.