Logic question (conjunction of implications)

[tribas]

hi all,

I'm no logician but am interested in sorting out this problem.

Say you've shown that

1. x implies y
and
2. z implies w

what steps/assumptions are required, in classical logic, to get from 1&2 to:

3. x&z implies y&w

Do the steps require some sort of separability assumption, or something of the sort?

Thanks!

 

[Stephen Tashi]

"x implies y" is logically equivalent to "(not x) or y"
"z impllies w" is logically equivalent to "(not z) or w".

The statement you want to prove is logically equivalent to
"not (x and z) or (y and w)"

Are you skilled enough at manipulating "and","or" and "not" to do the proof?

If not, you could use a truth table.

 

[tribas]

thanks very much stephen!

I am not skilled at all with manipulating operators, so cannot do the proof myself.

Can you suggest a website or textbook I could look at that would help?

 

[Stephen Tashi]

I don't know of a good website, off hand. Perhaps another forum member will.

An outline of what you need to know is

Distributive laws:

"A and (B or C)" is equivalent to "(A and B) or (A and C)"
"A or (B and C)" is equivalent to "(A or B) and (A or C)".


DeMorgan's Laws:

"not (A and B)" is equivalent to "(not A) or (not B)"
"not (A or B)" is equivalent to "(not A) and (not B)"

Double Negation

"not (not A)" is equivalent to "A"


Material Implication
"if A then B" is equivalent to "(not A) or B"

Or you can search for "Truth Tables" and find a explanation of how to do them.

 

[tribas]

ok thanks very much