I am reading a book called "The Haskell Road to Logic, Maths and Programming"

A question in the book is: "How many truth tables are there for 2-letter formula's"

The answer in the answer sheet is: "A two-letter formula has a truth table with four rows. The value at every row can be either t of f, so there are 2^4 = 16 truth tables altogether."

Now I don't understand, how a 2 letter formula, can have 16 truth tables? Let's say P ⇒ Q It will look like:

P    Q    (P⇒Q)
T    T    T
T    F    F
F    T    T
F    F    T

Now how can this have 16 tables? It looks like one table to me.

  • $\begingroup$ Their explanation rightfully (but unjustifiably) assumes that every truth table with four rows is attainable from a two letter formula. $\endgroup$ – Git Gud May 5 '16 at 19:16

For different formulae, the third column above can have different 4-tuples of T or F. So there are $2^4$ different formulae. For example, $P\wedge Q$,

$$\begin{array}{cc|c} P&Q&P\wedge Q\\\hline T&T&T\\ T&F&F\\ F&T&F\\ F&F&F \end{array}$$

is another truth table with a different 4-tuple $(T,F,F,F)$.

In fact, for any integer $0\le n\le 2^4-1$, let $b_i\in\{0,1\}$ be the $i$th bit of $n$ ($i = 0,1,2,3$), then $f_n$ is a new function defined by

$$\begin{array}{cc|c} P&Q&f_n(P,Q)\\\hline 0&0&b_0\\ 0&1&b_1\\ 1&0&b_2\\ 1&1&b_3 \end{array}$$

  • $\begingroup$ Thank you for answering, I get it now :) $\endgroup$ – Mazzy May 5 '16 at 19:12

What they are asking is how many possible relationships like this are there. You have 4 rows, first 2 columns are fixed, last column has 4 entries with T/F each, so $2^4=16$ total choices.


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