How does a bill become a law (from the U.S. Constitution, Article 1, section 7)? My answer to this question (putting our propositional logic tools to use) is:

(H • S • P) • {[A ∨ (~R • ~C)] ∨ (V • O)}

Given: H = The bill passed the House; S = The bill passed the Senate; P = The bill is presented to the President; A = The President approves and signs the bill; R = The bill is returned by the President within ten days; C = The Congress by adjournment prevent the return of the bill; V = The President vetoes the bill (i.e. he returns it with objections to congress); O = The veto is overridden (i.e. the bill is reconsidered and approved by two-thirds majority of both houses).

Truth tree decomposition is a rarely used tool, but this is one time we want to pull it out of the toolbox. Decomposing that compound proposition shows remarkably clearly the three paths that a bill can take in becoming a law:

(H • S • P) • {[A ∨ (~R • ~C)] ∨ (V • O)}  ✔

H • S • P  ✔

[A ∨ (~R • ~C)] ∨ (V • O)  ✔

H

S

P

/        \

A ∨ (~R • ~C)  ✔              V • O  ✔     

|                               V

|                               O

      /        \

A            ~R • ~C  ✔

~R

~C

Now, wasn’t that fun?