Predicate Logic

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Predicate Logic
What is a predicate?
A property that can be attributed to --or said
of a thing.
Being green Being contingent Being four feet tall
Being smelly Being the inventor of the
truth-table
But not Being (or Existence)
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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 dont exist.
This would mean there are things that are not.
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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.)
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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.)
p Paris
B is beautiful
Bp Paris is beautiful
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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
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In Categorical logic, wed have said All things
identical to Paris are French cities.
Universal affirmative predication
In Predicate logic, well treat this as a
singular predication.
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Expressing Categorical predications using
techniques of Propositional logic
ASP (All S is P) S ? P (If its S, then
its P)
ESP (No S is P) S ? P (If its S, then its
not P)
All cats are mammals If its a cat, its a
mammal
No demons are friendly If its a demon, its
not friendly
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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
No demons are friendly
(x) (Dx ? Fx)
(x) (Ox ? Cx)
Oranges are citrus
(x) (Ox v Lx) ? Cx
Oranges and lemons are citrus
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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)
and
No D wins
No R wins
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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 dont
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.
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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
I statement
(?x) (Sx . Rx)
Some philosophers are not logicians
(?x) (Px . Lx)
O statement
There is at least one thing that is a Philosopher
but not a Logician
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A (x) (Sx gt Px) E (x) (Sx
gt Px) universal universal affirmative
negative I (?x) (Sx . Px)
O (?x) (Sx . Px) particular particular a
ffirmative negative
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All men are mortal, and Socrates is a man,
so hes mortal
M S / O
Clearly invalid
In predicate representation
(x) (Mx gt Ox) Ms / Os
looks like MP might work on it
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Bp

• Paris is beautiful.
• 2. Tokyo is overcrowded.
• 3. If Paris is beautiful then it's popular.
• 4. If Gonzales isnt tortured, hell
• never talk.
• 5. All lawyers are members of the
• Bar Association.

Ot
Bp gt Pp
Tg gt Ag
(x)(Lx gt Mx)
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(?x) (Fx . Px)
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)(Cx . Lx) gt Bx)
(x)(Sx gt Cx) . Sm
(x)(Sx gt Cx) or (?x)(Sx . Cx)
Ro . Rc
Co . Cc or (Co v Cc)
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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 gt Mx)
(?x) (Sx . Lx)
(x)Tx gt (Sx v Lx)
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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 gt (Rx gt Dx)
(x) (Px . Rx) gt Dx
(x)Dx gt (Px gt Rx)
(x) Dx gt (Fx v Hx) gt Bx
(x) (Bx v Ex) gt Tx
Lp . Lc
(x) (Vx gt Ax)