Using Syntax to Distinguish Good Inferences from Bad

Rules of Inference

  • Abstraction

  • Patterns of reasoning

  •  

  •  

  •  

In these slides

The word "abstraction" gets a bad rap. Abstraction is just looking for patterns.

Abstraction

  • The first two are about particular numbers.

  • The third abstracts away from these specific values. This allows it to instead represent the pattern that the first two have in common.

  • Abstraction means we're ignoring details (i.e., generalizing).

  • Consider these three equations:

 2 + 4 = 4 + 2
   3.9 - 17.4 = -17.4 + 3.9
 x + y = y + x

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.

Made with Slides.com