Posted: November 9th, 2015

Logic

Logic

The standard symbol for negation (‘not’) is ~ . So that one is easy. You have it on your keyboard.??The standard symbol for dis junction (‘or’) is wedge-shaped. So use lower-case ‘v’ on your keyboard.??The standard symbol for conjunction (‘and’) is a dot, larger than a period mark, and half-way between the top and bottom of a letter.?You can use a period ‘.’ or you can use the ampersand ‘&’??The standard symbol for implication (‘if … then …’) is a horseshoe shaped symbol. You can use —-> for it. What you really need is a U-shaped symbol that is rotated 90 degrees counterclockwise.??The standard symbol for equivalence (‘if and only if’) is a triple-bar. Imagine = with another bar on top of it. That gives you a triple-bar. You can use = if you want.

Exercise 1

Run both sentences through all of the sentence forms and note whether it is or is not a substitution instance of that sentence form.  You don’t have to write out the sentence form–just the letter which precedes it.

1.    ~ [(A v B)  –> C]
2.    ~ [(~A v B) –> C]

Sentence forms
a)    p
b)    ~p
c)    p v q
d)    p –> q
e)    ~ p v q
f)    ~ p –> q
g)    ~ p –> ~ q
h)    ~ (p v q)
i)    ~ (p –> q)
j)    ~ (~p –> q_
k)    (p v q) –> r
l)    p v (q –> r)
m)    (~p v q) –> r
n)    ~ (p v q) –> r
o)    (p v q) –> ~ r
p)    ~ [p v (q –> r)]
q)    ~ [(p v q) –> r]

Exercise 2

Construct truth tables from the following sentences:

1)    ~A = [(B –> A) . [(A –> B) –> ~A)]
2)    A= – A
3)    ( A . ~ A) –> B

Exercise 3

Use MP,  MT, DS and HS to prove that the following arguments are valid.  You must include the premises in your proof.

(1)    1. ~ R
2. S –> R / :. ~ S

(2)    1. ~ (H . K)
2. R v (H . K) / :. R

(3)     1. R –> S
2. T –> R
3. ~S / :. ~ T

(4)    1. ~G –> ( A v B)
2. ~B
3. A –> D
4. ~ G /:. D

(5)    1. A –> (B –> C)
2. ~C
3. ~D –>A
4. C v ~ D /:. ~ B

Exercise 4

Using the eighteen valid argument forms, prove that the following arguments are valid (These proofs are very basic. None requires more than six additional lines to complete).  You must include the premises in your proof.

(1)    1. (A . B) –> C
2. A/:. B  –> C

(2)    1. ~ M v N
2. ~ R –> ~ N /:. M –> R

(3)    1. ~ (H v ~ K)
2. L –> H /:. L –> M

(4)    1. (A . B) v (C . D)
2. ~A/:. C

(5)    1. (A . B) –> C
2. A . ~ C /:. ~ B

Exercise 5

Complete the following proofs using the rules for adding and removing quantifiers where appropriate.  You must include the premises in your proof.
(1)    1. (x) Fx v (x) ~ Gx        p
2. ~(x) Fx            p
3. (x) (Dx –> GX)        p
/:. (?x) ( ~Dx v G x)

(2)    1. (x) [~Ax v (Bx . Cx)]    p
2. (x) [(Ax –> Cx) –> Dx    p
3. (x) (Dx –> ~ Cx)        p /:. (?x) ~ Ax

(3)     1. (z) [Az –> (~ Bz –> Cz)]    p
2. ~Ba                p/:. Aa –> Ca

Exercise 6

Prove valid.  You must include the premises in your proof.
(1)    1. Ka                    p
2. (x) [Kx –> (y) Hy]        p /:. (x)Hx

(2)    1. (?x) (Ax . Bx)            p
2. (y) (Ay –> Cy)            p /:. (?x)(Bx . Cx)

(3)     1. (x) [(Fx v Rx) –> ~ Gx]        p
2. (?x) ~ (~fx . ~ Rx)    p /:. (?y) ~ Gy

(4)     1. (x) (Fx –. Gx)            p
2. (y) (Ey –> Fy)            p
3. (z) ~ (Dz . ~ Ez)            p/:. (?x) Bx

Exercise 7

Prove valid (note that these problems are not necessarily in order of difficulty) You must include the premises in your proof.

(1)     1. ~ (x) Ax            p
/:. (?x)(Ax –> BX)

(2)     1. (?x) Fx –> (x) ~ Gx    p
2. (?x) Ex –>~(x) ~ Fx    p
/:.(?x)Ex –>~(?x)Gx

(3)    1. (x)[(Fx v Hx) –> (Gx . Ax)]        p
2. ~(x)(Ax . Gx)                p
/:.(?x) ~Hx

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