Lecture 4: Logic and Hypothetical Judgments

• How do we define the "meaning" of a proposition?
• Denotational - 27x37 and 999 denote the same integer
• Problem: 27x39 = 999, true, and FORALL n>z FORALL x,y,z != 0 x^n + y^n != z^n (i.e. Fermat's last theorem) are denotationally the same, but some are pretty clearly more interesting than others.
• What constitutes a proof of P?
• Atomic propositions and connectives and, or, not, implies, true, false.
• What constitutes a proof of A and B?
• A proof of A paired with a proof of B
• What constitutes a proof of A or B?
• A proof of A and a proof of B, along with a bit telling us which one we proved.
• What constitutes a proof of true?
• Well, it's already true!
• What constitutes a proof of false?
• You can't prove false.
• What constitutes a proof of A implies B?
• Assuming A, prove B - in other words, we can prove B using a hypothetical proof of A.
• What constitutes a proof of not A?
• Assuming A, prove false.

The introduction rules

A proved
B proved
----------------------------- and-i
A AND B proved

A proved
----------------------------- or-i1
A OR B proved

B proved
----------------------------- or-i2
A OR B proved

----------------------------- true-i
TRUE proved

A proved |- B proved
----------------------------- imp-i
A => B proved

The hypothetical judgment

It is a big point that we always may have some assumptions "floating around" on the left in all of our rules. Pierce would always explicitly mention these assumptions, but we will not, integrating the fact that whenever we write judgments then we might

This class takes the view that Pierce's choice is actually the less natural choice. However, it is so ingrained into the way we teach language theory that the SASyLF tool goes through great lengths to support the Pierce view - the tool would have been easier to write if we had taken this course's view of hypothetical judgments! You can see a presentation more in line with Pierce's by going to the SASyLF code here.

The elimination rules

A AND B proved
----------------------------- and-e1
A proved

A AND B proved
----------------------------- and-e2
B proved

A => B proved
A proved
----------------------------- imp-e (or "modus ponens")
B proved

A OR B proved
A proved |- C proved
B proved |- C proved
----------------------------- or-i1
C proved

FALSE proved
----------------------------- or-i1
A proved

Proof

Part 1

----------------------------------------------------------- hyp
(A OR B) AND (NOT A) proved |- (A OR B) and (NOT A) proved
----------------------------------------------------------- and-e1
(A OR B) AND (NOT A) proved |- A OR B proved

Part 2

----------------------------------- hyp    
... |- (A OR B) AND (NOT A) proved         
-----------------------------------         ------------------------- hyp
... |- A => FALSE proved                    ..., A proved |- A proved
--------------------------------------------------------------------- imp-e
(A OR B) AND (NOT A) proved, A proved |- FALSE proved
----------------------------------------------------- false-e
(A OR B) AND (NOT A) proved, A proved |- B proved

Part 3

------------------------------------------------- hyp
(A OR B) AND (NOT A) proved, B proved |- B proved

Wrap-up

    (part 1)                (part 2)                (part 3)

... |- A OR B proved    ... |- B proved        .... |- B proved
----------------------------------------------------------------- or-e
(A OR B) AND (NOT A) proved |- B proved
------------------------------------------- imp-i
((A OR B) AND (NOT A)) => B proved