Always let letters represent the positive idea, not the negative idea.
Let S = "I will wear a Speedo at the beach"
Let E = "I will be embarrassed"
I will not wear a Speedo at the beach or I will be embarrassed.
That's the 1st premise: ~S ∨ E
I am not embarrassed.
That's the 2nd premise: ~E
Therefore, I did not wear a Speedo at the beach.
That's the conclusion: ~S
That written in symbols is [(~S ∨ E) ● ~E] ⊃ ~S
Put ● between the premises and ⊃ between the set of premises
and the conclusion.
Put "not" (negation) indicated by ~ before the positive
premises if necessary to make them negative
Put TTFF under each S and TFTF under each E
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
T T T T
T F F T
F T T F
F F F F
Under each ~ put the opposite of what follows it. That is,
if a T follows the ~ put an F under the ~, and if an F
follows the ~, put a T under the.
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
FT T FT FT
FT F TF FT
TF T FT TF
TF F TF TF
Erase what followed the ~
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
F T F F
F F T F
T T F T
T F T T
Under the ∨ put F only if there are F's
immediately on BOTH sides of the ∨.
Otherwise put a T.
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
F T T F F
F F F T F
T T T F T
T T F T T
Erase what the ∨ was between
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
T F F
F T F
T F T
T T T
Under the ● (and) put T only if there are T's
immediately on BOTH sides of the ●. Otherwise
put a F.
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
T F F F
F F T F
T F F T
T T T T
Erase what the ● was between:
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
F F
F F
F T
T T
Under the ⊃ (conditional (if-then) put F only
if there is a T on the LEFT of the ⊃ and an F
on the RIGHT. Otherwise put a T.
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
F T F
F T F
F T T
T T T
Erase what the ⊃ was between
[(~S ∨ E) ● ~E] ⊃ ~S
-----------------------
T
T
T
T
This tells us the argument is totally valid.
If you are left with a line of all T's, the
argument is valid. If there had been any F's,
it would only be conditionally valid.
[Notice we started with T's and F's under the letters
only, and then got them all under the symbols and none
under the letters, then we ended up with only one column.
Edwin
