You can type a tilde for "not" or "negation". Look above the TAB key
to the left of the 1 key. You have to hold the shift key down. ~
You can make a right arrow for conditional" or "if then" with a dash
- and a greater than > to get ->. Use a little "v" for "logical or".
For "logical and", you can use "&".
Determine if the argument is valid or invalid using a truth table.
Premise: p->~q
Premise: pvq
Conclusion: p
Put & between the premises and -> before the conclusion
[(p->~q)&(pvq)]->p
You have to use parentheses and brackets so that there are only two
things inside a parentheses or bracket with a v, &, or -> between
them.
Start with a column for each single letter
for a statement on the left, and build it
up piece by piece.
Each heading must have either
p
q
~p
~q
( ) v ( )
( ) & ( )
( ) -> ( )
Single letters or letters with a tilde before them can
replace the ( ), and the ( ) can have ~ before them.
You can use brackets [ ] to enclose ( ). But remember
each parentheses and each brackets can at most only
contain two things with v, &, or -> between them.
Start with this:
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| | | | | | || |
| | | | | | || |
| | | | | | || |
| | | | | | || |
Put TTFF in the p column and TFTF in the q column:
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | | | | || |
| T | F | | | | || |
| F | T | | | | || |
| F | T | | | | || |
Next we have a ~q column. We put the opposite of what was in the q
column:
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | F | | | || |
| T | F | T | | | || |
| F | T | F | | | || |
| F | T | F | | | || |
Next we have a p->~q column. The rule for -> is
"put T for everything except T->F, where there's a T on the
left of -> and an F on the right of ->. That's the only time you
put F." You have to look at the previous columns to see what's on
the left of -> and what's on the right of it.
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | F | F | | || |
| T | F | T | T | | || |
| F | T | F | T | | || |
| F | F | T | T | | || |
Next we have a pvq column. The rule for v is
"put T for everything except FvF, where there's an F on both
sides of the v. That's the only time you put F." You have to
look at the previous columns to see what's on the left of v
and what's on the right of it.
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | F | F | T | || |
| T | F | T | T | T | || |
| F | T | F | T | T | || |
| F | F | T | T | F | || |
Next we have a (p->~q)&(pvq) column. The rule for & is
"put F for everything except T&T, where there's an T on both
sides of the &. That's the only time you put T." You have to
look at the previous columns to see what's on the left of &
and what's on the right of it.
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | F | F | T | F || |
| T | F | T | T | T | T || |
| F | T | F | T | T | T || |
| F | F | T | T | F | F || |
Finally we have a [(p->~q)&(pvq)]->p column. We already had a ->
column. The rule, remember, for -> is "put T for everything except
T->F, where there's a T on the left of -> and an F on the right of
->. That's the only time you put F." You have to look at the
previous columns to see what's on the left of -> and what's on the
right of it.
| p | q | ~q | p->~q | pvq | (p->~q)&(pvq) || [(p->~q)&(pvq)]->p |
| T | T | F | F | T | F || T |
| T | F | T | T | T | T || T |
| F | T | F | T | T | T || F |
| F | F | T | T | F | F || T |
Oh, oh! there's a F in that last column, so the argument is invalid.
There must be only T's in the last column for an argument to be
valid.
Be sure to learn those rules for ~,v,->, and &. There is only one
other, and that's equivalence <->. The rule for that is to put T if
what are on both sides are the same (both T's or both F's), and F if
they aren't the same.
Edwin
(