From textbook
Ex. 7.5
1, 4
Symbolize the following sentences in PL plus identity:
Note that not every symbolization will require you to use an identity symbol.
UD: The set that includes Canada, all its provinces, cities, and people
Cx: x is a city
Bxy: x
1
Assuming a domain of people, symbolize and prove 1 - 5.
1. Assuming that all friends like each other, show that from the fact that John is a friend
of William, and that William likes John only if John is a friend of Mary, it follows that
Mary likes John
1
Assuming a domain of people, symbolize and prove 1 - 5.
1. Assuming that all friends like each other, show that from the fact that John is a friend
of William, and that William likes John only if John is a friend of Mary, it follows that
Mary likes John
1
Prove the following:
1. Gs, x(Gx Mx) xMx
2. x(Fx Gx), x(Gx Hx) x (Fx Hx)
3. x(Fx Gx), x(Gx Hx) x(Fx Hx)
4. Gs, xGx Fm xFx
5. x(Fx yGy), Ga Fa
6. x(Fx Fx)
In the following 5 theorems assume that P is a sentence which does not contain the name
letter a.
7
1
Prove the following:
1. Gs, x(Gx Mx) xMx
1
2
2
1,2
1,2
1) Gs
2) x(Gx Mx)
3) Gs Ms
4) Ms
5) xMx
P
P
2, UI
1,2, MP (or: SL)
4, EG
2. x(Fx Gx), x(Gx Hx) x (Fx Hx)
1
2
1
2
1,2
1,2
1) x(Fx Gx)
2) x(Gx Hx)
3) Fa Ga
4) Ga Ha
5) Fa Ha
6) x(Fx Hx)
P
P
1, UI
2, U
1
Predicate logic rules (Text p. 293)
In these rules A(x/), indicates that where the sentence A has a variable such as x, the
sentence A(x/) has any name letter, indicated by , in place of the variable x.
UI (Universal Instantiation)
, xA
, A(x/)
Example:
Rules of thumb for Sentential Logic Derivations
Begin by writing down, using rule P, all the premises you are given. Then apply the
following rules of thumb to
within a derivation,
each
derivation,
and
to each move
in the order given
.
Do not ignore rule
1
Some suggested possible symbolizations. Often others are possible. If you have a
different symbolization, check (using truth tables) to see if your symbolization is
equivalent to the one offered here. If it is not, check to see if one or more of the Eng
1
SL symbolization practice exercises. Give a clear key!
If you are having trouble with any given term to be symbolized such as 'unless' or 'only if'
check out the sentences involving those terms and make sure you have tried symbolizing
all of them.
Canad
FACULTY OF ARTS
DEPARTMENT OF PHILOSOPHY
PHIL 279 Lec 01 Logic I
Fall Term 2013
Course Outline
Lectures: MW 8:00 - 9:15, ST 141
Instructor: J. J. MacIntosh
Teaching Assistant: Matt
Notes on consequence, satisfiability, etc.:
A B B is a consequence of A, or: A implies B.
A |= B B is not a consequence of A; A does not imply B.
/
A iff there is no model in which A is false and no sentence in is false. For present
purposes we may consi
1
Disjunctive and Conjunctive Normal Form
(For more on DNF and CNF see pp 129-32 in our text)
For DNF we form, for each TRUE line of the truth table, a conjunction consisting solely
of atomic sentences and their negations. And we choose in each such line
Philosophy 279
Soundness and Completeness
1
Soundness and Completeness
Soundness: If A then A. Text 108-9, 169-71.
Note that each line of a deduction preserves consequence, thus proving soundness.
Completeness: If A then A. Text p. 201 ff.
Step 1: To see
Syntax of PL:
1) Every atomic sentence is a sentence of PL.
An atomic sentence of PL consists of an n-place predicate letter followed by n occurrences of names.
Note: Since sentence letters are zero place predicate letters followed by zero occurrences of
Proofs for Truth Tree Claims
The truth tree method is committed to the following two key claims:
TT1: If has a completed open branch then is consistent.
TT2: If has a closed tree then is inconsistent.
Assuming these claims, we can prove other claims that
Semantics of PL
A sentence of PL has a truth-value relative to some interpretation. An interpretation of PL sentences
consists of two things: 1) a universe of discourse (UD), which is some non-empty set of objects, and 2)
an assignment of values/extension
Test 2 Exercise Questions
PHIL 279
Winter 2015
1. Proofs
You will be responsible for reproducing the proofs from Proofs for test 2, which is available on
D2L.
2. Derivations
Ex. 5.3
1 10
Note You should be familiar with the alternative ways of asking thes
CHAPTER NINE
Section 9.1E
a. 1.
2.
3.
4.
(x)Fx
(x) Fx
Fa
Fb
o
SM
SM
1 D
2 D
The tree has a completed open branch.
c. 1.
2.
3.
4.
5.
6.
7.
(x)(Fx & Gx)
(x)(Fx Gx)
Fa & Ga
Fa
Ga
Fa Ga
SM
SM
1 D
3 &D
3 &D
2 D
Fa
6 D
Ga
The tree is closed.
e. 1.
2.
3.
4.
Derivation Rules (text pp. 140-141:
P
_
cfw_A, A
PC
_, B_
- cfw_A, A B
RAA
_, B; , B_
( ) - cfw_A, A
MP
, A B; ,
,
CONJ
, A; , _
,
SIMP
, A B
, A
, A B
, B
ADD
, A_
, A B
, B_
, A B
DIL
, A B; , A C; , B C
, C
BI
, A B; , B A
, A B
BE
, A B
, A B
1
Metatheory practice questions
Sample answers or sketches of answers
1. Prove that cfw_Q P iff P Q
Answer: L R: if cfw_Q P there is no assignment of truth values which makes
all the members of cfw_Q true and P false. That is, there is no assignment of tr
1
Metatheory practice questions
1. Prove that cfw_Q P iff P Q
2. Given that is a finite set of sentences show that P iff cfw_P is consistent.
3. Given that is a maximally consistent set show that P iff P .
4. If 1 is a subset of some maximally consistent
1
Name_
ID#_
Philosophy 279.01 Sample Test 3
November 2015
Time which will be allowed: 75 minutes.
Marks
4
1. Prove that a relation, R, is asymmetric only if it is irreflexive.
1
1
1
1
6
1) xy(Rxy Ryx)
2) Raa Raa
3) Raa
4) xRxx
P
1, UI x 2
2, SL
3, UG
2)