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Philosophy/Philosophy - Logic - Why is my argument invalid?


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 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   ]
   [  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:

~(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)

// (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.

~(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)


All the best


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



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Maciej St. Zięba


I can answer questions concerning Eastern (Oriental) philosophies and philosophers (Indian, Tibetan, Indonesian, Chinese, Korean, Japanese, Vietnamese: Hinduist, Buddhist, Confucianist, Taoist and other; alas not Islamic or Jewish) - both in terms of notions and facts (history of their development). I can write in English, French, Esperanto, Polish and Russian, German, Dutch and Norwegian. I can also understand questions in Spanish and Italian.


I have been teaching Indian and Chinese philosophies since 1987, during 1999-2009 I co-ordinated a project on Oriental philosophies within the scope of the Universal Encyclopedia of Philosophy (Powszechna Encyklopedia Filozofii) published in Polish by SITA-PL in Lublin (10 volumes, containing ca. 500 entries in Eastern philosophies, written by a team of a dozen of Polish scholars).

Polish Oriental Society; International Association of Buddhist Studies; Klingon Language Institute; Learned Society of the John Paul II Catholic University of Lublin; Polish Philosophical Association; Universala Esperanto-Asocio.

Books: "Origin of the World According to Rigveda" (Montreal 1996); "Our Bug. Creating Conditions for Development of the Border Areas of Poland, Ukraine and Belarus through Enhancement and Preservation of Natural and Cultural Heritage" (Lublin 2008); "Migration - a Challenge to the 21st century" (Lublin 2008); "Migracja zarobkowa do Woch" (Job migration to Italy) (Lublin 2008); more than 100 articles in "Powszechna Encyklopedia Filozofii" (Universal Encyclopedia od Philosophy) vol. 1-10 (Lublin 2000-2009); Contributions to the history of the Buddhist classifications of dharmas: Pancavastuka of Vasumitra (Bulletin, Polish Institute and Library, Montreal 1997); many more in Polish; some of them available online, see: here and here (a list up to 2012

philosophy (KUL, Lublin, 1976-81); M.A. in history of Indian philosophy (KUL, 1981); Ph.D. in history of Indian philosophy (KUL, 1989); other studies: Indian and Chinese philosophies (Institut Catholique, Paris, 1985-6); Tibetan language (INALCO, Paris, 1985-6); Chinese language (McGill University, Montreal, 1995-7).

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