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 AUniversal affirmative:All As are B  x(Ax  Bx) EUniversal negative:No As are B  x(Ax  ~Bx) or ~  x(Ax & Bx) IParticular affirmative:Some As are B  x(Ax & Bx) OParticular negative:Some As are not B  x(Ax & ~ Bx) or ~  x(Ax  Bx)

Logical Square

 x(Sx  Px)  x(Sx & ~ Px) or ~  x(Sx  Px)

Logical Square  x(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 If every M is L and every S is M, then every S is L (Barbara) If no M is L and every S is M, then no S is L (Celarent) If every M is L and some S is M, then some S is L (Darii) If no M is L and some S is M, then some S is not L (Ferio). Second Figure: M, L, S If no L is M and every S is 11.I, then no S is L (Cesare) If every L is M and no S is M, then no S is L (Camestres) If no L is M and some S is M, then some S is not L (Festino) If every L is M and some S is not M, then some S is not L (Baroco). Third Figure: L, S, M If every M is L and every M is S, then some S is L (Darapti) If no M is L and every M is S, then some S is not L (Felapton) If some M is L and every M is S, then some S is L (Disamis) If every M is L and some M is S, then some S is L (Datisi) If some M is not L and every M is S, then some S is not L (Bocardo) 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 LA All Ss are MAbArbArA So all Ss are LA 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)

1Everyone that Michael likes likes either Henry or Sue 2Michael likes everyone that both Sue and Rita like 3Michael likes everyone that either Sue or Rita like 4Rita doesn’t like Michael but she likes everyone that Michael likes 5Grizzly bears are dangerous but black bears are not 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like 3Michael likes everyone that either Sue or Rita like 4Rita doesn’t like Michael but she likes everyone that Michael likes 5Grizzly bears are dangerous but black bears are not 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like 4Rita doesn’t like Michael but she likes everyone that Michael likes 5Grizzly bears are dangerous but black bears are not 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like  x(Lsx  Lrx  Lmx) 4Rita doesn’t like Michael but she likes everyone that Michael likes 5Grizzly bears are dangerous but black bears are not 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like  x(Lsx  Lrx  Lmx) 4Rita doesn’t like Michael but she likes everyone that Michael likes ~Lrm &  x(Lmx  Lrx) 5Grizzly bears are dangerous but black bears are not 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like  x(Lsx  Lrx  Lmx) 4Rita doesn’t like Michael but she likes everyone that Michael likes ~Lrm &  x(Lmx  Lrx) 5Grizzly bears are dangerous but black bears are not  x(Gx  Dx) &  x(Bx  ~Dx) 6Grizzly bears and polar bears are dangerous but black bears are not 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like  x(Lsx  Lrx  Lmx) 4Rita doesn’t like Michael but she likes everyone that Michael likes ~Lrm &  x(Lmx  Lrx) 5Grizzly bears are dangerous but black bears are not  x(Gx  Dx) &  x(Bx  ~Dx) 6Grizzly bears and polar bears are dangerous but black bears are not  x(Gx  Px  Dx) &  x(Bx  ~Dx) 7Every self-respecting polar bear is a good swimmer

1Everyone that Michael likes likes either Henry or Sue  x(Lmx  Lxs  Lxh) 2Michael likes everyone that both Sue and Rita like  x(Lsx&Lrx  Lmx) 3Michael likes everyone that either Sue or Rita like  x(Lsx  Lrx  Lmx) 4Rita doesn’t like Michael but she likes everyone that Michael likes ~Lrm &  x(Lmx  Lrx) 5Grizzly bears are dangerous but black bears are not  x(Gx  Dx) &  x(Bx  ~Dx) 6Grizzly bears and polar bears are dangerous but black bears are not  x(Gx  Px  Dx) &  x(Bx  ~Dx) 7Every self-respecting polar bear is a good swimmer  x(Px & Rxx  Sx)

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita 11If anyone likes Sue, Michael does 12If everyone likes Sue, Michael does 13If anyone likes Sue, he or she likes Rita 14Michael doesn’t like everyone 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does 12If everyone likes Sue, Michael does 13If anyone likes Sue, he or she likes Rita 14Michael doesn’t like everyone 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does 13If anyone likes Sue, he or she likes Rita 14Michael doesn’t like everyone 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita 14Michael doesn’t like everyone 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita  x(Lxs  Lxr) 14Michael doesn’t like everyone 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita  x(Lxs  Lxr) 14Michael doesn’t like everyone~  xLmx 15Michael doesn’t like anyone 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita  x(Lxs  Lxr) 14Michael doesn’t like everyone~  xLmx 15Michael doesn’t like anyone~  xLmx 16If someone likes Sue, then s/he likes Rita 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita  x(Lxs  Lxr) 14Michael doesn’t like everyone~  xLmx 15Michael doesn’t like anyone~  xLmx 16If someone likes Sue, then s/he likes Rita  x(Lxs  Lxr) 17If someone likes Sue, then someone likes Rita

UD = people 9Anyone who likes Sue likes Rita  x(Lxs  Lxr) 10Everyone who likes Sue likes Rita the same as 9 11If anyone likes Sue, Michael does  xLxs  Lms 12If everyone likes Sue, Michael does  xLxs  Lms 13If anyone likes Sue, he or she likes Rita  x(Lxs  Lxr) 14Michael doesn’t like everyone~  xLmx 15Michael doesn’t like anyone~  xLmx 16If someone likes Sue, then s/he likes Rita  x(Lxs  Lxr) 17If someone likes Sue, then someone likes Rita  xLxs   xLxr