Editing Truth function (section)

== Definition ==

Using the functions defined above, we can give a formal definition of a proposition's truth function.<ref>{{Cite web |title=An Introduction to Mathematical Logic |url=https://store.doverpublications.com/products/9780486497853?srsltid=AfmBOoo9mFyD06QpIypwYtJnNn2CYOf-Ps2CCwMYl_IgAfLRwgeh7v1s |access-date=2025-02-20 |website=Dover Publications |language=en}}</ref>

Let ''PROP'' be the set of all propositional variables, 

: <math> PROP = \{p_1,p_2,\dots\}</math>

We define a '''truth assignment''' to be any function <math>\phi:PROP\to \{T,F\}</math>.  A truth assignment is therefore an association of each propositional variable with a particular truth value.  This is effectively the same as a particular row of a proposition's truth table.

For a truth assignment, <math>\phi</math>, we define its '''extended truth assignment''', <math>\overline\phi</math>, as follows.  This extends <math>\phi</math> to a new function <math>\overline \phi</math> which has domain equal to the set of all propositional formulas.  The range of <math>\overline\phi</math> is still <math>\{T,F\}</math>.

# If <math>A \in PROP</math> then <math>\overline\phi(A) = \phi(A)</math>.
# If ''A'' and ''B'' are any propositional formulas, then 
## <math>\overline\phi(\neg A) = f_{\text{not}} (\overline\phi(A))</math>.
## <math>\overline\phi(A\land B) = f_{\text{and}} (\overline\phi(A),\overline\phi(B))</math>.
## <math>\overline\phi(A\lor B) = f_{\text{or}} (\overline\phi(A),\overline\phi(B))</math>.
## <math>\overline\phi(A\to B) = \overline\phi(\neg A\lor B)</math>.
## <math>\overline\phi(A\leftrightarrow B) = \overline\phi((A\to B)\land (B\to A))</math>.

Finally, now that we have defined the extended truth assignment, we can use this to define the truth-function of a proposition.  For a proposition, ''A'', its '''truth function''', <math>f_A</math>, has domain equal to the set of all truth assignments, and range equal to <math>\{T,F\}</math>.  

It is defined, for each truth assignment <math>\phi</math>, by <math>f_A(\phi) = \overline\phi(A)</math>.  The value given by <math>\overline\phi(A)</math> is the same as the one displayed in the final column of the truth table of ''A'', on the row identified with <math>\phi</math>.