Asyllogistic Inference

1
Asyllogistic Inference

• Kareem Khalifa
• Department of Philosophy
• Middlebury College

2
Overview

• What are asyllogistic inferences?
• Why do such inferences matter?
• Translation tricks
• Proofs
• Exercises

3
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 Fs are Gs E No Fs are Gs I
  Some Fs are Gs O Some Fs are not Gs.
• 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.

4
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.

5
A teaser about proofs

• Theres nothing new here!
• You apply your same four rules (UI, UG, EI, EG)
  along with all the rules for propositional.
• But
• BEWARE OF TRANSLATION!!!

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First Translation Trap

• All Fs are either G or H. (All foxes are either
  gentle or theyre hungry).
• Temptation (x)(Fx ? Gx) v (x)(Fx ? Hx)
• Wrong!
• 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 doesnt allow this.

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Second Translation Trap

• Fs and Gs are H. (Fools and goons are horrible
  people)
• Temptation (x)((Fx Gx) ? Hx)
• Wrong!
• 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.

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Third Translation Trap

• All except Fs are Gs. (Except for the French
  people, everyone was genuine.)
• Temptation (x)(Fx ? Gx)
• Half Wrong!
• Correct (x)(Fx ? Gx)
• Translation 1 permits non-French people to be not
  genuine. However, the English statement denies
  this.

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Returning to proofs

• If you translate right, theres 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.

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Example

• Except for the French speakers, everyone was
  genuine. None of the Swiss students were genuine.
  So the Swiss students speak French.
• (x)(Fx ? Gx) A
• (x)(Sx ? Gx) A
• ?(x)(Sx ? Fx)
• Fa ? Ga 1 UI
• Sa ? Ga 2 UI
• Ga ? Fa 3 ?E
• Sa ? Fa 4, 5 HS
• (x)(Sx ? Fx) 6 UG

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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 -gt (Wx ? Lx))
• A6 A boxer who wins if and only if he is lucky
  is not skillful.
• (x)((Bx (Wx ? Lx)) ? Sx)

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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 its forbidden.
  In Russia, everything is forbidden even if its
  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))