Hardware Freedom Day is April 20th!

Topic: Boolean Algebra

http://www.digilogwiki.com/index.php?title=Minterms_and_Maxterms

has IMO inconsistent "Given" conditions set up for the gas pipeline example.

Comments anyone?

I don't think the given condtions are inconsistent, but I think the first condition,

If El Paso is open, then Phoenix and San Diego must both be either open or closed.

could be simplified to just

If El Paso is open, then Phoenix and San Diego must both be closed.

So the "or open" is superfluous/redundant, because it can never happen, but that doesn't make it inconsistent.

However, I am troubled by the first row of the truth table.  I would have expected T=1 because

neither condition is violated.

I'm also confused by the Tuscon pipe.  I would have expected that the flow in the Tuscon pipe

would be the sum of the flows in its 3 feeder pipes, but clearly that isn't what column T represents.

There seems to be an implied condition that the Tuscon pipe flows if and only if the 2 conditions are

satisfied and at least one of the feeders is flowing.

coder27 wrote:

 

I don't think the given condtions are inconsistent, but I think the first condition,

If El Paso is open, then Phoenix and San Diego must both be either open or closed.

could be simplified to just

 

If El Paso is open, then Phoenix and San Diego must both be closed.

 

So the "or open" is superfluous/redundant, because it can never happen, but that doesn't make it inconsistent.

 

However, I am troubled by the first row of the truth table.  I would have expected T=1 because

neither condition is violated.

 

I'm also confused by the Tuscon pipe.  I would have expected that the flow in the Tuscon pipe

would be the sum of the flows in its 3 feeder pipes, but clearly that isn't what column T represents.

There seems to be an implied condition that the Tuscon pipe flows if and only if the 2 conditions are

satisfied and at least one of the feeders is flowing.

I agree that the Givens are inconsistent.  The first given is OK as written: it says that if El Paso is open, the other two are equal (exclusive NOR).  However, the second given conflicts with the first, because if El Paso is open then both Phoenix and San Diego could be open, but if Phoenix is open then El Paso must be closed.  It appears to be one of those examples that's so simple that its author didn't think he or she needed to check it.

John,

   What do you call a boolean logic condition that says

      A implies (False or B)

Clearly the False is redundant and the condition could be simplified to

     A implies B

But I wouldn't call that inconsistent, because you wouldn't have any trouble creating

a unique truth table for it.

In the original example we have

     E implies ((P and S) or (~P and ~S))

     P implies (~E and ~S)

The first condition can be simplified (in light of the second) to

     E implies (False or (~P and ~S))

which can be simplified to

     E implies (~P and ~S)

which leaves you with two consistent conditions that make for a unique truth table.

It wouldn't surprise me if the author unintentionally had a redundant (P and S) in

the first condition, but I don't see that it really matters at all.

John,

   What do you call a boolean logic condition that says

      A implies (False or B)

Clearly the False is redundant and the condition could be simplified to

     A implies B

But I wouldn't call that inconsistent, because you wouldn't have any trouble creating

a unique truth table for it.

In the original example we have

     E implies ((P and S) or (~P and ~S))

     P implies (~E and ~S)

The first condition can be simplified (in light of the second) to

     E implies (False or (~P and ~S))

which can be simplified to

     E implies (~P and ~S)

which leaves you with two consistent conditions that make for a unique truth table.

It wouldn't surprise me if the author unintentionally had a redundant (P and S) in

the first condition, but I don't see that it really matters at all.

Assuming that the site http://www.digilogwiki.com/index.php?title=Minterms_and_Maxterms is intended to be tutorial, it's very important to avoid verbiage that might be contradictory.  For that matter, one could interpret:

If El Paso is open, then Phoenix and San Diego must both be either open or closed.

to mean that Phoenix must be either open or closed, and San Diego must likewise be either open or closed.  Since there's no "partially open" in Boolean algebra, then Phoenix is always either open or closed, and San Diego is always either open or closed, so they both simplify to TRUE, so the whole given simplifies to "if El Paso is open, then TRUE".  Not amusing enough to be a sillygism, so it's just confusing.

They also neglect to say whether 0 means open and 1 means closed, or vice-versa.  You need to know this to construct the truth table.

It's hard enough to learn logic without trying to make sense of ambiguous examples.  In particular, the premises have got to be clear and unambiguous.

agreed, the example is anything but clear and unambiguous!

Working backwards from the truth table, I have concluded the the Givens need to be changed from

If El Paso is open, then Phoenix and San Diego must both be either open or closed. If Phoenix is open, then both El Paso and San Diego must be closed.

to

If El Paso is open, then Phoenix and San Diego must both be either open or closed. If Phoenix is open, then both El Paso and San Diego must both be either open or closed.

Happy Trails

Bill,

   I think you're on to something.  Your change would make the last row of the truth table make sense.

But the first row still wouldn't make much sense.