# Asyllogistic Inference by c2e7V2

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```									Asyllogistic Inference

Kareem Khalifa
Department of Philosophy
Middlebury College
Overview
•   What are asyllogistic inferences?
•   Why do such inferences matter?
•   Translation tricks
•   Proofs
•   Exercises
What are Asyllogistic Inferences?
• A syllogistic inference is any inference consisting
only of singular propositions and/or quantified
statements that can be turned into A, E, I, or O.
– Recall A = All F’s are G’s; E = No F’s are G’s; I =
Some F’s are G’s; O = Some F’s are not G’s.
• An asyllogistic inference is any inference
containing at least one statement that is neither
a singular proposition nor a quantified statement
that can be turned A, E, I, or O.
Why do they matter?
• Quick answer: because we use asyllogistic
inferences all the time, e.g.,
– Northern New England states are rural and
have lax gun control laws.
– Some northern New England states are
Democratic strongholds.
– So some rural states with lax gun control laws
are Democratic strongholds.
A teaser about proofs…
• There’s nothing new here!
– You apply your same four rules (UI, UG, EI,
EG) along with all the rules for propositional.
• But…

BEWARE OF
TRANSLATION!!!
First Translation Trap
•  All F’s are either G or H. (All foxes are either
gentle or they’re hungry).
1. Temptation: (x)(Fx  Gx) v (x)(Fx  Hx)
–   Wrong!
2. Proper Translation: (x)(Fx  (Gx v Hx))
• Clearly, the English statement allows some
foxes to be gentle but not hungry, so long as
the rest are hungry.
• However, Translation 1 doesn’t allow this.
Second Translation Trap
•  F’s and G’s are H. (Fools and goons are
horrible people)
1. Temptation: (x)((Fx & Gx)  Hx)
• Wrong!
2. Correct: (x)((Fx v Gx)  Hx)
• The English statement clearly requires
someone who is a fool but not also a goon to
be horrible, but Translation 1 prohibits this.
• Ditto for goons who are not fools.
Third Translation Trap
•  All except F’s are G’s. (Except for the
French people, everyone was genuine.)
1. Temptation: (x)(Fx  ~Gx)
• Half Wrong!
2. Correct: (x)(Fx  ~Gx)
• Translation 1 permits non-French people
to be not genuine. However, the English
statement denies this.
Returning to proofs…
• If you translate right, there’s no
difference between asyllogistic and
syllogistic inferences…
• Use EI and UI to turn the predicate logic
proof into a propositional logic proof, prove
it accordingly, and then use EG and UG to
turn it back into predicate logic.
Example
•   Except for the French speakers, everyone was
genuine. None of the Swiss students were genuine. So
the Swiss students speak French.
1. (x)(Fx  ~Gx)                             A
2. (x)(Sx  ~Gx)                             A
(x)(Sx  Fx)
3. Fa  ~Ga                                  1 UI
4. Sa  ~Ga                                  2 UI
5. ~Ga  Fa                                  3 E
6. Sa  Fa                                   4, 5 HS
7. (x)(Sx  Fx)                              6 UG
Sample Exercises, p. 473
• A3: No car is safe unless it has good
brakes.
– (x)((Cx  (~Sx v Bx))
• A5: A gladiator wins if and only if he is
lucky.
– (x)(Gx -> (Wx  Lx))
• A6: A boxer who wins if and only if he is
lucky is not skillful.
– (x)((Bx & (Wx  Lx))  ~Sx)
More sample exercises
• A8: Not all tools that are cheap are either soft or
breakable.
– (x)((Tx & Cx) & (~Sx & ~Bx))
• A11: In America, everything is permitted that is not
forbidden. In Germany, everything is forbidden that is not
permitted. In France, everything is permitted even if it’s
forbidden. In Russia, everything is forbidden even if it’s
permitted.
– This says everything you would need:
•   (x) ((((Ax & ~Nx)  Px) & ((Gx & ~Px)  Nx)) & ((Fx  Px) & (Rx  Nx)))
– But if you want to get a more literal translation of the “even if”
statements, you could conjoin the following:
• (x) (((Fx & Nx)  Px) & ((Rx & Px)  Nx))

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