PHIL12A
Section answers, 28 March 2011
Julian Jonker
1
How much do you know?
1. (9.10) Evaluate the following sentences in Edgars World (see opposite):
(1) x Tet(x) : False
(2) x (Tet(x) Large(x) : False
(3) x (Tet(x) Large(x) : True
(4) x (Tet(x) Tet(x)
PHIL12A
Section answers, 14 March 2011
Julian Jonker
1
How much do you know?
1. Use truth tables to check whether the following equivalences hold:
(a) P Q
Q P
P
Q
P
Q
Q
P
T
T
T
T
T
F
T
T
F
T
T
F
T
F
F
T
F
F
F
T
F
T
F
T
T
F
T
T
T
F
F
F
F
T
F
T
F
T
T
F
(b)
PHIL12A
Section questions, 2 March 2011
Julian Jonker
1
How much do you know?
1. Show that the sentences in each pair are tautologically equivalent.
(a) (Ex 7.1) A B and A B
A
B
A
B
A
B
T
T
T
T
T
F
T
T
T
T
F
T
F
F
F
T
F
F
F
T
F
T
T
T
F
T
T
F
F
F
T
F
T
F
T
PHIL12A
Section answers, 28 Feb 2011
Julian Jonker
1
How much do you know?
Give formal proofs for the following arguments.
1. (Ex 6.18)
1
AB
2
A B
Proof:
1
AB
2
A
3
A B
4
B
Elim: 2
5
B
6
Intro: 4,5
7
B
Intro: 5-6
8
A B
Intro: 7
9
A B
Elim: 1, 2-3, 4-8
1
2
PHIL12A
Section answers, 18 April 2011
Julian Jonker
FOL translation exercises
1. Look carefully at how the following sentences have been translated into FOL. (You should try to do them
on your own rst.)
(a) Each cube is to the left of a tetrahedron.
Actu
PHIL12A
Section answers, 30 March 2011
Julian Jonker
1
How much do you know?
1. Translate the following sentences (these are the four Aristotelian forms). For each sentence, come up
with a translation that uses the quantier and one that uses the quantier:
PHIL12A
Section answers, 16 March 2011
Julian Jonker
1
How much do you know?
Construct formal proofs for the following arguments:
1. (Ex 8.19)
1
AB
2
B
3
A
Proof:
1
AB
2
B
3
A
4
B
Intro: 1, 3
5
Intro: 2, 4
6
A
Intro: 3-5
2. (Ex 8.24)
1
AB
2
AC
3
BD
4
C D
PHIL12A
Section answers, 16 February 2011
Julian Jonker
1
How much do you know?
1. Show that the following sentences are equivalent.
(a) (Ex 4.16) A B A and A B
A
B
(A
B)
A
A
B
T
T
T
T
T
T
T
T
T
T
T
F
T
T
F
T
T
T
T
F
F
T
F
T
T
T
F
F
T
T
F
F
F
F
F
F
F
F
F
PHIL12A
Section answers, 11 April 2011
Julian Jonker
1
How much do you know?
1. Translate the following pairs of sentences, and show by a chain of equivalences that they are equivalent:
(a)
i. It is not the case that all Ps are Qs.
x(P(x) Q(x)
ii. Some Ps
PHIL12A
Section questions, 25 April 2011
Julian Jonker
1
Formal proofs
Decide whether the following arguments are valid or not. If they are, give formal proofs for them; otherwise
give counterexamples.
1. (Ex 13.2)
1
x(Cube(x) Small(x)
2
xCube(x)
3
xSmall
PHIL12A
Section answers, 23 February 2011
Julian Jonker
1
How much do you know?
1. The following questions are adapted form exercises 5.1-5.6. Decide whether each pattern of inference is
valid. If it is, show that it is using truth tables. If it is not, g
PHIL12A
Section questions, 13 April 2011
Julian Jonker
1
How much do you know?
1. Translate the following sentences into FOL twice, using universal quantiers for one translation and
existential quantiers for the other.
(a) Every cube is left of every tetr
PHIL12A
Section answers, 4 April 2011
Julian Jonker
1
How much do you know?
1. (Ex 9.25) Assume you are working in an extension of the rst-order language of arithmetic with the
additional predicates Even(x) and Prime(x). Express the following in this lang
PHIL 12A / Answers, 25 January 2011
Julian Jonker
1
How much do you know?
Consider a rst-order language that has the names Jesse, Kendall, Lou, Maria
and Nate, and the function symbols leftof(x) referring to the person to the left of
x (from your perspect
Philosophy 12A. Introduction to Logic.
Spring 2015
Prof. Paolo Mancosu.
Office: 230 Moses Hall.
Office hours: W 11-12; F 12.00-13.00.
Email: [email protected]
Lectures: M, W, F, 10-11, 160 Kroeber
Graduate Instructors:
Russell Buehler: [email protected]
Handout on Unless
Brian van den Broek
20150307
Unless is one particle of natural language that many students find difficult
to translate into natural language. This handout walks through one example
in detail, to motivate the claim that the translation of
Worksheet 1 Solutions
xkcd.com
Identify the premises and conclusions in each argument; indicate whether each is valid or sound.
(1) If Socrates is a man, Socrates is mortal. Socrates is mortal. Therefore, Socrates is a man.
1
If Socrates is a man, Socrate
PHIL12A
Answers, 31 January 2011
Julian Jonker
1
How much do you know?
1. The following questions are based on Ex 2.3 of the textbook. For each argument, identify the premises and
conclusion, and state whether the argument is valid. (Be prepared to defend
PHIL12A
Section answers, 2 February 2011
Julian Jonker
1
How much do you know?
The following arguments are given in the blocks language of Tarskis World. Decide whether the argument is
valid. If it is, nd a way to persuade others that it is. (In other wor
PHIL12A
Section answers, 7 February 2011
Julian Jonker
1
How much do you know?
1. (Ex 2.22) Is the following argument valid? Sound? If it is valid, give an informal proof of it. If it is not,
give an informal counterexample to it.
All computer scientists
PHIL12A
Section answers, 9 February 2011
Julian Jonker
1
How much do you know?
1. I have constructed a world in Tarskis World using objects named a through f, but Im not going
to show it to you. Now consider the sentences below, and decide whether you can
PHIL12A
Section answers, 14 February 2011
Julian Jonker
1
How much do you know?
1. You should understand why a truth table is constructed the way it is: why are the truth values listed in
the order they are? In principle, it doesnt matter how they are lis
PHIL12A
Section answers, 7 March 2011
Julian Jonker
1
Revision problems
1. Translate the following sentences into FOL, devising your own names and predicates where necessary:
(a) Block c is a tetrahedron just in case block a is in between block c and bloc
PHIL12A
Section answers, 6 April 2011
Julian Jonker
1
How much do you know?
1. Here are two questions from last time. Assess whether the following arguments are (a) tautologically
valid, (b) logically valid but not tautologically valid, or (c) invalid. In
PHIL12A
Section answers, 20 April 2011
Julian Jonker
1
From last time.
Compare especially (a), (d) and (e) below.
1. (Ex 11.19)
(c) No cube with nothing to its left is between two cubes.
x(Cube(x) yLeftOf(y,x) zwBetween(x,z,w)
(d) The only large cubes are
PHIL12A
Section questions, 27 April 2011
Julian Jonker
Revision problems
1. (Ex 11.39 and 11.40) Translate the following sentences into FOL.
(a) Everything is either a cube or a tetrahedron.
x(Cube(x) Tet(x)
(b) There are at least three tetrahedra.
xyz(Te
Phil12A, Jonker, 25 January 2011
1
How much do you know?
Consider a rst-order language that has the names Jesse, Kendall, Lou,
Maria and Nate, and the function symbols leftof(x) referring to the person
to the left of x (from your perspective), and rightof
Tautological Equivalence, Logical
Equivalence, and Tarskis World
Equivalence (Oh My!)
Brian van den Broek
20150226
A number of students have expressed to me that they are finding the
differences between tautological equivalence, logical equivalence, and T