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I have been racking my head for a few days trying to fix it, but when I check it with the truth table it's still wrong. I thought that since it's not (W and not X), and it's not (X and not W), it is both or neither, so I thought it would be:

~(W & ~X) / ~(X & ~W) //

(W & X) v (~X & ~X)

But the truth table says its wrong:

~(W & ~X) / ~(X & ~W)

F(T T FT) / F(T T FT)

T(T F TF) / T(F F FT)

T(F F FT) / T(T F TF)

T(F F TF) / T(F F TF)

// (W & X) v (~W & ~X)

// (T T T) T (FT F FT)

// (T F F) F (FT F TF)

// (F F T) F (TF F FT)

// (F F F) T (TF T TF)

It is invalid on line 2 and 3.

What did I do wrong?

Thank you

Dear Ilona,

0) I am a specialist in Eastern philosophies (Have you read my introduction note?), not general philosophy and logic. Yet, this is so simple that I would like to help you.

1) I am not sure what your notation means, in my country (Poland) we do not use signs like / and // in logic. In the English Wikipedia https://en.wikipedia.org/wiki/Truth_table they're not used, either.

But your argument is fomulated like this:

"Since it's not (W and not X), and it's not (X and not W), it is both or neither"

so I understand that / means "and", // means either "if - then" (i.e. it is an implication) or "is equivalent to" (i.e. is a logical equality)

I have added extra brackets to make it more clear, for simplification I have made the sentence A, B, C and D.

[~(W & ~X) / ~(X & ~W)] // [(W & X) v (~X & ~X)]

if [ A and B ] then [ C or D ]

[ A & B ] => [ C v D ]

or

[ A and B ] is equivalent to [ C or D ]

[ A & B ] = [ C v D ]

I also understand that your "or" (v) is the "alternative" (aka. "logical disjunction") and not the "exclusive disjunction" ("either ... or ...).

2) You say you've calculated the truth tables:

a)

~(W & ~X) / ~(X & ~W)

F(T T FT) / F(T T FT)

T(T F TF) / T(F F FT)

T(F F FT) / T(T F TF)

T(F F TF) / T(F F TF)

These truth tables (part a) were calculated wrong by you. They should read:

~(W & ~X) / ~(X & ~W)

T(T F FT) / T(T F FT)

F(T T TF) / T(F F FT)

T(F F FT) / F(T T TF)

T(F F TF) / T(F F TF)

b)

// (W & X) v (~W & ~X)

// (T T T) T (FT F FT)

// (T F F) F (FT F TF)

// (F F T) F (TF F FT)

// (F F F) T (TF T TF)

part b) was calculated correctly.

3) I still don't understand why you haven't calculated the whole phrase, you have not submitted any T-F vale for your functors / and //

/ = & - without that it won't work.

a)

~(W & ~X) & ~(X & ~W)

T(T F FT) T T(T F FT)

F(T T TF) F T(F F FT)

T(F F FT) F F(T T TF)

T(F F TF) T T(F F TF)

You can already see it will work.

4) If you don't see it yet, all you need now is to join part a) with part b)

if // is => - it works

~(W & ~X) / ~(X & ~W) // (W & X) v (~W & ~X)

~(W & ~X) & ~(X & ~W) => (W & X) v (~W & ~X)

T(T F FT) T T(T F FT) T (T T T) T (FT F FT)

F(T T TF) F T(F F FT) T (T F F) F (FT F TF)

T(F F FT) F F(T T TF) T (F F T) F (TF F FT)

T(F F TF) T T(F F TF) T (F F F) T (TF T TF)

if // is = - it also works

~(W & ~X) / ~(X & ~W) // (W & X) v (~W & ~X)

~(W & ~X) & ~(X & ~W) = (W & X) v (~W & ~X)

T(T F FT) T T(T F FT) T (T T T) T (FT F FT)

F(T T TF) F T(F F FT) T (T F F) F (FT F TF)

T(F F FT) F F(T T TF) T (F F T) F (TF F FT)

T(F F TF) T T(F F TF) T (F F F) T (TF T TF)

q.e.d.

All the best

MAciej

---

Ilona wrote:

Thank you, I'll see if it works with your corrected truth tables. The / and // just separate the first premise from the second premise, and the second premise from the conclusion. If I'm not mistaken, I either saw that from Hurley's text or Mark Thorsby's video.

Dear Ilona,

In that case I was absolutely right to substitute the / with &, and the // with =.

All the best

MAciej

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Comment | Thank you, I'll see if it works with your corrected truth tables. The / and // just separate the first premise from the second premise, and the second premise from the conclusion. If I'm not mistaken, I either saw that from Hurley's text or Mark Thorsby's video. Thank tou again |

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