Predicate Logic What is a predicate? A property that can be attributed to --or said of –a thing. Being greenBeing contingent Being four feet tall Being smelly Being the inventor of the truth-table But not: Being (or Existence)

Being (existence) is not a predicate -- Immanuel KANT Why not? If we conceptualize “being” as something things can have or lack (like Redness), then we cannot avoid self-contradictory assertions like “Some things don’t exist.” This would mean: “there are things that are not.”

“There is…” “Il y a…” --there it has… “Es gibt…” --it gives… “existence” is treated as a “primitive” rather than as a predicate. Not a property a thing has, but as a condition. E.g., “There are no unicorns” is preferable to “Unicorns have non-being.” (Unicorns are things that have the property of not existing.)

Capital letters will now stand for predicates. Lower-case letters will now stand for things of which the predicates can be said. (a-w will stand for individual entities (constants) x, y, z will stand for variables.) B “is beautiful” p “Paris” Bp “Paris is beautiful”

C is a city F is French (Bp. Cp). Fp“Paris is a beautiful French city” “p” is a constant: each time it occurs it is naming the same thing: Paris 1. (Bp. Cp). Fp 2. Bp. Cp sm 1 3. Cp sm 2

In Categorical logic, we’d have said: All things identical to Paris are French cities. Universal affirmative predication In Predicate logic, we’ll treat this as a “singular predication.”

Expressing Categorical predications using techniques of Propositional logic ASP (All S is P) S  P (If it’s S, then it’s P) ESP (No S is P) S  ~P (If it’s S, then it’s not P) All cats are mammals: If it’s a cat, it’s a mammal No demons are friendly: If it’s a demon, it’s not friendly

Universal statements do not assert existence: they say “IF” something has a property, it has another one as well. (x) (Cx  Mx) All cats are mammals (x) (Dx  ~Fx) No demons are friendly Oranges are citrus (x) (Ox  Cx) Oranges and lemons are citrus (x) [(Ox v Lx)  Cx]

If neither a Republican nor a Democrat wins, Ron Paul will be President (x) [(Rx v Dx)  ~Wx]  Pr IF No R or D wins Ron will be Prez (x) (Rx  ~Wx). (x)(Dx  ~Wx) No R wins andNo D wins

Existence is not a predicate It is indicated by a unique symbol (  x) (  x) Gx Ghosts exist/ There are ghosts ~(  x)Gx It is false that ghosts exist. Ghosts don’t exist. There are no ghosts. It is not the case that there is an x such that x has the property of being a ghost.

In categorical logic, “some” indicated “there is at least one.” Some Senators are Republicans There is at least one thing, which is both a Republican and a Senator (  x) (Sx. Rx) Some philosophers are not logicians (  x) (Px. ~Lx) “I” statement “O” statement There is at least one thing that is a Philosopher but not a Logician

A: (x) (Sx > Px) E: (x) (Sx > ~Px) universal affirmative negative I: (  x) (Sx. Px) O: (  x) (Sx. ~Px)particular affirmativenegative

All men are mortal, and Socrates is a man, so he’s mortal M S / O Clearly invalid (x) (Mx > Ox) Ms / Os In predicate representation looks like MP might work on it

1.Paris is beautiful. 2. Tokyo is overcrowded. 3. If Paris is beautiful then it's popular. 4. If Gonzales isn’t tortured, he’ll never talk. 5. All lawyers are members of the Bar Association. Bp Ot Bp > Pp ~Tg > ~Ag (x)(Lx > Mx)

6. Some flowers are not pretty. 7. All laptop computers have batteries. 8. No students carry cellphones, but Mary is not a student. 9. Not all Senators are communists. 10. Obama is running for President and so is Clinton. 11. Obama's not a communist and neither is Clinton. (  x) (Fx. ~Px) (x)[(Cx. Lx) > Bx)] (x)(Sx > ~Cx). ~Sm ~(x)(Sx > Cx) or: (  x)(Sx. ~Cx) Ro. Rc ~Co. ~Cc or: ~(Co v Cc)

12. Either Clinton will be the candidate or there will be no woman candidate. 13. Horses exist, but not unicorns. 14. Sea lions are mammals. 15. Squirrels live on this campus. 16. Only snakes and lizards thrive in the desert Cc v ~(  x) (Cx. Wx) (  x) Hx. ~(  x) Ux (  x) (Sx. Lx) (x) (Sx > Mx) (x)[Tx > (Sx v Lx)]

17. Peaches are delicious unless they are rotten. 18. Dogs bite if they are frightened or harassed. 19. Bears and eagles are talked about alot on The Colbert Report. 20. Sean Penn and Steven Colbert love metaphors. 21. Only arguments can be valid. (x) [Px > (~Rx > Dx)] (x) [(Px. ~Rx) > Dx] (x)[Dx > (Px > ~Rx)] (x) {Dx > [(Fx v Hx) > Bx]} (x) [(Bx v Ex) > Tx] Lp. Lc (x) (Vx > Ax)