sl: basic syntax - formulae

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), (PQ), (PQ), (PQ)

• 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), (PQ), (PQ), (PQ)

• 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


• Atomic formulae

• Formulae


• Atomic formulae

• Formulae

• (Main) Logical operator


• Atomic formulae

• Formulae


• Scope of a quantifier


• Atomic formulae

• Formulae



• Bound/Free variable


• Atomic formulae

• Formulae




• Sentence


• Atomic formulae

• Formulae




• Sentence

• Substitution instance

Aristotle

Aristotle’s Syllogistic

A Universal affirmative: All As are Bx(Ax Bx)

E Universal negative: No As are Bx(Ax ~Bx) or ~x(Ax & Bx)

I Particular affirmative: Some As are Bx(Ax & Bx)

O Particular negative: Some As are not Bx(Ax & ~ Bx) or ~x(Ax Bx)

Logical Square

Logical Squarex(Sx Px)

x(Sx & ~ Px)

or ~x(Sx Px)

Logical Squarex(Sx Px)

x(Sx & ~ Px)

or ~x(Sx Px)

x(Sx & Px)

x(Sx ~Px) or ~x(Sx & Bx)

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).

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 bArbArA

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


x(Lmx LxsLxh)








x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)







x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)


x(LsxLrx Lmx)






x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)


x(LsxLrx Lmx)


~Lrm & x(Lmx Lrx)





x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)


x(LsxLrx Lmx)


~Lrm & x(Lmx Lrx)


x(Gx Dx) & x(Bx ~Dx)




x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)


x(LsxLrx Lmx)


~Lrm & x(Lmx Lrx)




x(Gx Px Dx) & x(Bx ~Dx)



x(Lmx LxsLxh)


x(Lsx&Lrx Lmx)


x(LsxLrx Lmx)


~Lrm & x(Lmx Lrx)




x(Gx Px Dx) & x(Bx ~Dx)


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


x(Lxs Lxr)


the same as 9








UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms







UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms






UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms


x(Lxs Lxr)





UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms


x(Lxs Lxr)

14 Michael doesn’t like everyone ~xLmx




UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms


x(Lxs Lxr)


15 Michael doesn’t like anyone ~xLmx



UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms


x(Lxs Lxr)



16 If someone likes Sue, then s/he likes Rita x(Lxs Lxr)


UD = people


x(Lxs Lxr)


the same as 9


xLxs Lms


xLxs Lms


x(Lxs Lxr)



16 If someone likes Sue, then s/he likes Rita x(Lxs Lxr)

17 If someone likes Sue, then someone likes Rita xLxs xLxr

sl: basic syntax - formulae

Documents

yby lyb

zrz lzd

xrx sxc

zrz lz7

yby ly7

wrw cwe

zbz zrz zbz rz7

xrx sx7