SL: Basic syntax - formulae
•
Atomic formulae – have no connectives
•
If P is a formula, so is ~P
•
If P and Q are formulae, so are
(P&Q), (PQ), (PQ), (PQ)
•
nothing else is a formula
PL: Basic syntax – formulae of PL
•
Atomic formulae of PL are formulae
•
If P is a formula, so is ~P
•
If P and Q are formulae, so are
(P&Q), (PQ), (PQ), (PQ)
•
If P is a formula with at least one occurrence
of x and no x-quantifier, then
xPx and xPx are formulae
•
nothing else is a formula
PL: Basic syntax – definitions of important notions
•
Atomic formulae
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
•
(Main) Logical operator
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
•
(Main) Logical operator
•
Scope of a quantifier
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
•
(Main) Logical operator
•
Scope of a quantifier
•
Bound/Free variable
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
•
(Main) Logical operator
•
Scope of a quantifier
•
Bound/Free variable
•
Sentence
PL: Basic syntax – definitions of important notions
•
Atomic formulae
•
Formulae
•
(Main) Logical operator
•
Scope of a quantifier
•
Bound/Free variable
•
Sentence
•
Substitution instance
Aristotle
Aristotle’s Syllogistic
A
Universal affirmative:
x(Ax  Bx)
All As are B
E
Universal negative:
No As are B
x(Ax  ~Bx) or ~x(Ax & Bx)
I
Particular affirmative:
x(Ax & Bx)
O
Particular negative:
Some As are not B
x(Ax & ~ Bx) or ~x(Ax  Bx)
Some As are B
Logical Square
x(Sx  Px)
Logical Square
x(Sx & ~ Px)
or ~x(Sx  Px)
x(Sx  Px)
Logical Square
x(Sx  ~Px) or
~x(Sx & Bx)
x(Sx & Px)
x(Sx & ~ Px)
or ~x(Sx  Px)
All Syllogisms
First Figure: L, M, S.
1. 1. If every M is L and every S is M, then every S is L (Barbara).
2. 2. If no M is L and every S is M, then no S is L (Celarent).
1. 3. If every M is L and some S is M, then some S is L (Darii).
1. 4. If no M is L and some S is M, then some S is not L (Ferio).
Second Figure: M, L, S.
2. 1. If no L is M and every S is 11.I, then no S is L (Cesare).
2. 2. If every L is M and no S is M, then no S is L (Camestres).
2. 3. If no L is M and some S is M, then some S is not L (Festino).
2. 4. If every L is M and some S is not M, then some S is not L (Baroco).
Third Figure: L, S, M.
3. 1. If every M is L and every M is S, then some S is L (Darapti).
3. 2. If no M is L and every M is S, then some S is not L (Felapton).
3. 3. If some M is L and every M is S, then some S is L (Disamis).
3. 4. If every M is L and some M is S, then some S is L (Datisi).
3. 5. If some M is not L and every M is S, then some S is not L (Bocardo).
3. 6. If no M is L and some M is S, then some S is not L (Ferison).
Barbara
1.
1. If every M is L and every S is M, then every S is L (Barbara).
All Ms are L
All Ss are M
So all Ss are L
Barbara
1.
1. If every M is L and every S is M, then every S is L (Barbara).
All Ms are L
A
All Ss are M
A
So all Ss are L
A
bArbArA
Chrysippus
Greece
7.6E2
a. (∀y)(By ⊃ Ly)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]
n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]
n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx)
o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]
n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx)
o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw)
p. [(∃w)Sw & (∃w)Ow] & ∼ (∃w)(Sw & Ow)
7.6E2
a. (∀y)(By ⊃ Ly)
b. (∀x)(Rx ⊃ Sx)
c. (∀z)(Rz ⊃ ∼ Lz)
d. (∃w)(Rw & Cw)
e. (∃x)Bx & (∃x)Rx
f. (∃y)Oy & (∃y) ∼ Oy
g. [(∃z)Bz & (∃z)Rz] & ∼ (∃z)(Bz & Rz)
h. (∀w)(Rw ⊃ Sw) & (∀x)(Gx ⊃ Ox)
i. (∃y)By & [(∃y)Sy & (∃y)Ly]
j. ∼ (∃z)(Rz & Lz)
k. (∀w)(Cw ⊃ Rw) & ∼ (∀w)(Rw ⊃ Cw)
l. (∀z)Rx & [(∃y)Cy & (∃y) ∼ Cy]
m. (∀y)Ry ∨ [(∀y)By ∨ (∀y)Gy]
n. ∼ (∀x)Lx & (∀x)(Lx ⊃ Bx)
o. (∃w)(Rw & Sw) & (∃w)(Rw & ∼ Sw)
p. [(∃w)Sw & (∃w)Ow] & ∼ (∃w)(Sw & Ow)
q. (∃x)Ox & (∀y)(Ly ⊃ ∼ Oy)
1
Everyone that Michael likes likes either Henry or Sue
2
Michael likes everyone that both Sue and Rita like
3
Michael likes everyone that either Sue or Rita like
4
Rita doesn’t like Michael but she likes everyone that Michael likes
5
Grizzly bears are dangerous but black bears are not
6
Grizzly bears and polar bears are dangerous but black bears are not
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
3
Michael likes everyone that either Sue or Rita like
4
Rita doesn’t like Michael but she likes everyone that Michael likes
5 Grizzly bears are dangerous but black bears are not
6
Grizzly bears and polar bears are dangerous but black bears are not
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
4 Rita doesn’t like Michael but she likes everyone that Michael likes
5
Grizzly bears are dangerous but black bears are not
6
Grizzly bears and polar bears are dangerous but black bears are not
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
x(LsxLrx  Lmx)
4
Rita doesn’t like Michael but she likes everyone that Michael likes
5
Grizzly bears are dangerous but black bears are not
6
Grizzly bears and polar bears are dangerous but black bears are not
7 Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
x(LsxLrx  Lmx)
4
Rita doesn’t like Michael but she likes everyone that Michael likes
~Lrm & x(Lmx  Lrx)
5
Grizzly bears are dangerous but black bears are not
6 Grizzly bears and polar bears are dangerous but black bears are not
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
x(LsxLrx  Lmx)
4
Rita doesn’t like Michael but she likes everyone that Michael likes
~Lrm & x(Lmx  Lrx)
5
Grizzly bears are dangerous but black bears are not
x(Gx  Dx) & x(Bx  ~Dx)
6
Grizzly bears and polar bears are dangerous but black bears are not
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
x(LsxLrx  Lmx)
4
Rita doesn’t like Michael but she likes everyone that Michael likes
~Lrm & x(Lmx  Lrx)
5
Grizzly bears are dangerous but black bears are not
x(Gx  Dx) & x(Bx  ~Dx)
6
Grizzly bears and polar bears are dangerous but black bears are not
x(Gx  Px  Dx) & x(Bx  ~Dx)
7
Every self-respecting polar bear is a good swimmer
1
Everyone that Michael likes likes either Henry or Sue
x(Lmx  LxsLxh)
2
Michael likes everyone that both Sue and Rita like
x(Lsx&Lrx  Lmx)
3
Michael likes everyone that either Sue or Rita like
x(LsxLrx  Lmx)
4
Rita doesn’t like Michael but she likes everyone that Michael likes
~Lrm & x(Lmx  Lrx)
5
Grizzly bears are dangerous but black bears are not
x(Gx  Dx) & x(Bx  ~Dx)
6
Grizzly bears and polar bears are dangerous but black bears are not
x(Gx  Px  Dx) & x(Bx  ~Dx)
7
Every self-respecting polar bear is a good swimmer
x(Px & Rxx  Sx)
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10
Everyone who likes Sue likes Rita
11
If anyone likes Sue, Michael does
12
If everyone likes Sue, Michael does
13
If anyone likes Sue, he or she likes Rita
14
Michael doesn’t like everyone
15
Michael doesn’t like anyone
16
If someone likes Sue, then s/he likes Rita
17
If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
12 If everyone likes Sue, Michael does
13 If anyone likes Sue, he or she likes Rita
14 Michael doesn’t like everyone
15 Michael doesn’t like anyone
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
13 If anyone likes Sue, he or she likes Rita
14 Michael doesn’t like everyone
15 Michael doesn’t like anyone
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
14 Michael doesn’t like everyone
15 Michael doesn’t like anyone
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
x(Lxs  Lxr)
14 Michael doesn’t like everyone
15 Michael doesn’t like anyone
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
x(Lxs  Lxr)
14 Michael doesn’t like everyone
15 Michael doesn’t like anyone
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
~xLmx
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
x(Lxs  Lxr)
14 Michael doesn’t like everyone
~xLmx
15 Michael doesn’t like anyone
~xLmx
16 If someone likes Sue, then s/he likes Rita
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
x(Lxs  Lxr)
14 Michael doesn’t like everyone
~xLmx
15 Michael doesn’t like anyone
~xLmx
16 If someone likes Sue, then s/he likes Rita
x(Lxs  Lxr)
17 If someone likes Sue, then someone likes Rita
UD = people
9
Anyone who likes Sue likes Rita
x(Lxs  Lxr)
10 Everyone who likes Sue likes Rita
the same as 9
11 If anyone likes Sue, Michael does
xLxs  Lms
12 If everyone likes Sue, Michael does
xLxs  Lms
13 If anyone likes Sue, he or she likes Rita
x(Lxs  Lxr)
14 Michael doesn’t like everyone
~xLmx
15 Michael doesn’t like anyone
~xLmx
16 If someone likes Sue, then s/he likes Rita
x(Lxs  Lxr)
17 If someone likes Sue, then someone likes Rita
xLxs  xLxr