Bits and Bobs
What is Quantified Logic?
A mix of sentential and categorical logic; using "all" "no" "some"
Sir Lancelot was a member of the Round Table.
Rs
If anything is alive, then it is conscious of its environment.
(x) (Ax > Cx)
What is this called:
(x)
Universal Quantifier
1. (∃x)Hx
2. (x) ~Px
so, ~(x) (Hx > Px)
3. (x) (Hx > Px) AR
4. Ha 1EI
5. ~Pa 2UI
6. Ha > Pa 3UI
7. Pa 4,5 IF
8. ~(x) (Hx > Px) 3-7 RAA
What is the difference between:
~(∃x) & ~(x)
~(∃x) = no, none, nothing, etc.
~(x)= not all
Cabbage is an odoriferous food when cooked.
Oc
Anything that is either sweet or crunchy is tasty.
(x)[(Sx v Cx) >Tx]
What is this called:
(∃x)
Existential Quantifier
All humans are mortal. Socrates is human. Therefore, someone is mortal
1. (x)(Hx > Mx)
2. Hs
Therefore, (∃x) Mx
Valid
Which of these is a quantified statement?
1. (x) Px > Qx
2. (x) ~(Px) v Qx
3. (∃x) Px & (Qx & Zx)
4. ~(∃x)(Px & Qx)
4. ~(∃x)(Px & Qx)
Cell phones are not universally admired products.
~Uc
Whole numbers are even or odd.
(x)[Wx > (Ex v Ox)]
What are these called:
x, y, z
Variables
(x)(Bx > Ax)
(∃x)(Bx & Dx)
so, (∃x)(Ax & Dx)
3. (~∃x)(Ax & Dx) - AR
4. (X) ~(Ax & Dx) - 3QN
5. (Ba & Da) - 2 EI
6. (Ba > Aa) - 1 UI
7. ~(Aa & Da) - 4 UI
8. Ba - 5 AND
9. Da - 5 AND
10. Aa - 6, 8 IF
11. ~Aa - 7,9 IF
What do each of the following receipts stand for?
QN
EI
UI
Quantifier Negation
Existential Instantiation
Universal Instantiation
Only if Joe runs the mile under four minutes will he qualify for the rally.
Qj > Uj
The whale is neither a killer nor a fish.
(x)[Wx > ~(Kx v Fx)]
What are these called and what do they refer to:
a - u
Constants; individual subjects
1. (∃x) (Wx & Jx)
2. ~(∃x) (Ax & Jx) (∃x)(Wx & ~Ax)
3. ~(∃x)(Wx & ~Ax) - AR
4. (x) ~(Wx & ~Ax) - 3QN
5. (x) ~(Ax & Jx) - 2QN
6. (Wa & Ja) - 1EI
7. ~(Wa & ~Aa) - 4 UI
8. ~(Aa & Ja) - 5UI
9. Wa - 6 AND
10. Ja - 6 AND
11. Aa - 7, 9 NAND
12. ~Aa - 8, 10 NAND
13. (∃x)(Wx & ~Ax) - 3-13 RAA
What is the special restriction that only applies to Existential Instantiation?
You must use a different constant for each instance of EI in a proof.
If Bill is honest and loyal, then he will get the job
(Hb & Lb) > Jb
If all humans are mortal but Socrates is not mortal, then Socrates is not human.
(x)[(Hx>Mx) & ~Ms] > ~Hs
What are these called and what do they stand for:
A-Z
Predicates; represents the predicate of a statement.
(x)[(Fx & Gx) > Hx]
(∃x) Fx
(∃x) Gx
Therefore (∃x)Hx
Invalid
4. ~(∃x) Hx - AR
5. (x)~Hx - 5QN
6. Fa - 2UI
7. Gb - 3UI
8. [(Fa & Ga) > Ha] - 1UI
9. [(Fb & Gb) > Hb] - 1UI
10. ~Ha - 5UI
11. ~Hb - 5UI
12. (Fa & Ga) - 8, 10 IF
13. (Fb & Gb) - 9, 11 IF
14. Fa - 12 AND
15. Ga - 12 AND
16. Fb - 13 AND
17. Gb - 13 AND