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Default Logic

  • Abstraction

  • Patterns of reasoning

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In these slides

Additional information can sometimes undermine initial conclusions.

Nonmonotonic reasoning

  • Monotonic means "only ever grows in one direction" (alternatively, "never grows smaller" or "never grows larger").

  • Most logics are monotonic:

\hbox{If}\quad\Gamma\vdash\phi\quad\hbox{then}\quad\Gamma,\Gamma'\vdash\phi\,.

if year mod 1000 == 0:

    return True

else if year mod 100 == 0:

    return False

else if year mod 4 == 0:

    return True

else:

    return False

Reasons that can conflict

Default rules of inference

  • Here's the simplest description of leap years:

    • Years divisible by 4 are leap years, except...

    • Years divisible by 100 are not leap years, except...

    • Years divisible by 1000 are leap years.

\hskip.3em\hbox{Rule 1.}\qquad\phantom{divisible by 1000}\llap{divisible by 4}\rightarrow\rlap{leap year}\phantom{not leap year}\hskip.6em\\ \hskip.3em\hbox{Rule 2.}\qquad\phantom{divisible by 1000}\llap{divisible by 100}\rightarrow\rlap{not leap year}\phantom{not leap year}\hskip.6em\\ \hskip.3em\hbox{Rule 3.}\qquad\phantom{divisible by 1000}\llap{divisible by 1000}\rightarrow\rlap{leap year}\phantom{not leap year}\hskip.6em\\ \hskip1em\\ \hskip.16em{}\rlap{Relative strength:}\phantom{Rule 0.\qquad divisible by 1000}\phantom{{}\rightarrow{}}\phantom{not le}\llap{3 > 2 > 1}\phantom{ap year}\hskip.6em

Rules of inference

  • The core idea of proof theory is to break complex reasoning down into simple, obviously-good patterns of reasoning known as rules of inference.

  • Examples:

Directly proving that an inference/argument is good.

(I)

 If X, then Y. 

 If Y, then Z. 

 If X, then Z. 

(II)

 If X, then Y. 

 If X, then Z. 

 If X, then Y and Z. 

  • If you can get from the premises of an argument to its conclusion using only those rules, you know it's a good argument.

Using the rules

  • First, we abstract away from specific statements.

  1.  If A, then B.
  2.  If C, then D.
  3.  If C, then A.
     If C, then B and D. 
  1. If I've completed level 1, I have a sword.
  2. If I've completed level 3, I have a shield.
  3. If I've completed level 3, I've completed level 1.
    If I've completed level 3, I have a sword and a shield. 

  • Next, we use the rules repeatedly until we reach the conclusion.

(I)

 If X, then Y. 

 If Y, then Z. 

 If X, then Z. 

(II)

 If X, then Y. 

 If X, then Z. 

 If X, then Y and Z. 

(I)

  1.  If A, then B.
  2.  If C, then A.
  3.  If C, then B. 

(II)

  1.  If C, then D.
  2.  If C, then B.
     If C, then B and D. 

Context matters

  • First, we abstract away from specific statements.

  1.  If A, then B.
  2.  If C, then D.
  3.  If C, then A.
     If C, then B and D. 
  1. If I've completed level 1, then I have a sword.
  2. If I've completed level 3, then I have a shield.
  3. If I've completed level 3, then I've completed level 1.
    If I've completed level 3, then I have a sword and a shield. 

  • Next, we use the rules repeatedly until we reach the conclusion.

(I)

 If X, then Y. 

 If Y, then Z. 

 If X, then Z. 

(II)

 If X, then Y. 

 If X, then Z. 

 If X, then Y and Z. 

(I)

  1.  If A, then B.
  2.  If C, then A.
  3.  If C, then B. 

(II)

  1.  If C, then D.
  2.  If C, then B.
     If C, then B and D. 

Patterns of reasoning can be good in one context and bad in another.

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