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Logical Structures in Natural Language: Exercises

Propositional Logic

Universita di Trento

Exercise 1: PL language

For each of the following expressions, say whether it's a well formed formula or not.

1.:(:p_q)

2.p_(q)

3. ( p!p)!(p!q) 4. ( p_q^r)

5.:p^ ::q

6.: ^p

Exercise 2: Construction trees

(a) Build the construction tree of the following formulas (p$r)_ :qandp$(r_ :q). For each formula, list

its sub-formulae.

(b) Classify each of the following formulas as atomic, negation, conjunction, disjunction, implication or equiv-

alence. 1.p!q 2.:p 3.p 4. ( p_q)^(p_q)

5.:(p!q)

6.:(p^q)^ :r

7.p!(q^ :r)

(c) List of all the formulas that can be obtained by means of parentheses from the following string of symbols:

p^ :q!r. 1

Exercise 3: Truth tables

Build truth tables of the following formulas:

1.::A

2.A^(B^ :A)

3. ( A!B)! :B

Exercise 4: Tautology

Which of the following formulas are tautologies?

1.A!A

2.A!(B!A)

3. ( B!A)!A

4.::A!A

Exercise 5: Equivalence

Show that the following formulas are equivalent

1.A!B :A_B

2.A^(A_B)A

3.A^(:A_B)A^B

Exercise 6: Translation from English into PLL

Translate the following English sentences into PL. Try to use the same structure of the sentence and give the

translation keys.

Eg. If you don't sleep then you will be tired.

Translation keys: p = you sleep, q= you will be tired. Formula::p!q. 1. If it rains while the sun shines, a r ainbowwill app ear 2. Charles comes if Elsa do esand the other w ayaround 3.

Johan comes just when P etersta ysat home

4.

W eare going, unless it is raining

5. Charles and Elsa are brother an dsister or nephew and niece 6. If I ha velost if I cannot mak ea mo ve,then I ha velost. 2

Solutions

0.1 Exercise : Language

1. Y es. 2.

No, but p_qis a w.

3. Y es. 4.

No, but ( p_q)^randp_(q^r) are w.

5. Y es. 6. No.

0.2 Exercise : Construction trees

(a) (p$r)_ :q.

Construction Tree

_$ p r :q

Subformula:p;r;q;:q;p$r;(p$r)_ :q.

p$(r_ :q).

Construction Tree

$p_r:q

Subformula:p;r;q;:q;r_ :q;p$(r_ :q).

(b) 1. implication 2. negation 3. atomic form ula 4. conjunction 5. negation 6. conjunction 7. implication (c) 1. (p^ :q)!r, 2.p^(:q!r), 3.p^ :(q!r). 3

0.3 Exercise : Truth Tables

1.::A

A: :ATT F

FF T (1)

2.A^(B^ :A)

ABA^(B^ :A)TTF F F

TFF F F

FTF T T

FFF F T

(1) 3. ( A!B)! :B

AB(A!B)! :BTTT F F

TFF T T

FTT F F

FFT T T

(1)

0.4 Exercise : Tautologies

1.A!A. Tautology.

AA!ATT

FT (1)

2.A!(B!A). Tautology.

ABA!(B!A)TTT T

TFT T FTT F FFT T (1) 3. ( B!A)!A

AB(B!A)!ATTT T

TFT T FTF T FFT F (1)

4.::A!A. Tautology.

A: :A!ATT F T

FF T T

(1) 4

0.5 Exercise : Equivalence

1.A!Band:A_B. Yes

AB(A!B):A_BTTTF T

TFFF F

FTTT T

FFTT T

(1)(2)

2.A^(A_B) andA. Si.

ABA^(A_B)TTT T

TFT T FTF T FFF F (1)(2)

3.A^(:A_B) andA^B. Yes

ABA^(:A_B)A^BTTT F TT

TFF F FF

FTF T TF

FFF T TF

(1)(2)

0.6 Exercise: Translation

1. p = it rains, r = the sun shines, q = a rain bowwill app ear (p^r)!q 2. p = Charles comes, q = Elsa come s (q!p)^(p!q) or (p$q) 3. p = Johan comes, q = P etersta ysat home p$q 4. p = W eare going, q = it rains :p$q 5. p = Charles and Elsa are brother and sister, q = Charles and Elsa are nephew and niece. p_q 6. p = I can mak ea mo ve,q = I ha velost. (:p!q)!q 5quotesdbs_dbs9.pdfusesText_15