Truth Tables
Abbreviations
We have already used a number of connectives for joining propositions
into more complex propositions: Negation ('not'), Conjunction ('and'),
Disjunction ('or') and Hypotheticals, also known as Conditionals ('if'
. . . 'then'). We will add one more: Biconditionals ('if and only if').
These connectives are known as truth functional because the truth
of the resulting complex proposition is a function of the truth or falsity
of the component propositions. Indeed we include negation in this group
because it is truth functional; it does not really connect component
propositions even if it does create a more complex proposition. To save
space time and effort each of these connectives may be abbreviated using
symbols.
Type

Connective

Statement Form

Reads

Negation (not) 
~

~p

It is not the case that p.

Conjunction (and) 
&

p & q

p and q

Disjunction (or) 
v

p v q

p or q

Conditional (if . . . then) 
⊃

p ⊃q

If p then q

Biconditional (if and only if) 
≡

p ≡ q

p if and only if q

We can use tables to indicate how the truth or falsity of the resulting
complex proposition will be determined by (as this is a function of) all
the possible combinations of truth and falsity of the components.
Negation
In the case of negation only one component exists, so the total number
of possibilities for truth and falsity are two: the component could be
true or it could be false. So the table will need only two lines.
This table shows that whenever the component (p) is true, stating that
it is not the case that p will be false. And whenever the component is
false, stating that it is not the case that p will be true. In other words,
negating any proposition, say by adding a 'not' (a '~' in symbols), will
result in a proposition with the opposite truth value. The truth
value of a proposition is the assignment of truth or falsity to it.
For example, if a proposition is said to be false, then its truth value
is false. Each line in the above table represents a consistent assignment
of truth values on its own. The truth values of both lines could not be
consistently applied to a single proposition at the same instant. The two
lines in the table do account for all the possible consistent assignments
of truth values for any proposition and its negation. In other words, no
other consistent assignments of truth values for a proposition and its
negation exist. For the purpose of truth tables the above table defines
negation.
Conjunction
Conjunctions involve two components, each of which could be true or could
be false, so there are four possible combinations of truth values among
the components. The table will need four lines.
p

q

p & q

T

T

T

F

T

F

T

F

F

F

F

F

The gray cells in the table represent all the possible combinations
of truth values for the two components. Each white cell under "p &
q" represents the only truth value for the conjunction consistent with
the truth values assigned to the components on the same line. In
other words, the table indicates that if both conjuncts are true then the
conjunction as a whole will be true, but if either conjunct is false, or
if both conjuncts are false, then the conjunction as a whole will be false.
For the purpose of truth tables the above table defines conjunction.
Disjunction
Disjunctions also involve two components, each of which could be true or
could be false, so there are four possible combinations of truth values
among the components. The table will need four lines.
p

q

p v q

T

T

T

F

T

T

T

F

T

F

F

F

The gray cells in the table represent all the possible combinations
of truth values for the two components. Each white cell under "p v q" represents
the only truth value of the disjunction consistent with the truth values
assigned to the components on the same line. In other words, the
table indicates that if both disjuncts are false then the disjunction as
a whole will be false, but if either disjunct is true, or if both disjuncts
are true, then the disjunction as a whole will be true. This table defines
'or' in the inclusive sense for truth tables; the table for the exclusive
'or' would be different (in particular the line where both disjuncts are
true would be different).
Conditional (Hypothetical)
Conditionals also involve two components, each of which could be true or
could be false, so there are four possible combinations of truth values
among the components. The table will need four lines.
p

q

p ⊃ q

T

T

T

F

T

T

T

F

F

F

F

T

The gray cells in the table represent all the possible combinations
of truth values for the two components. Each white cell under "p ⊃
q" represents the only truth value of the conditional consistent with the
truth values assigned to the components on the same line. In other
words, the table indicates that if the antecedent (p) is true while the
consequent (q) is false, the conditional as a whole will be false, but
in all other possible combinations of truth value the conditional as a
whole will be true. For the purposes of truth tables this table defines
the hypothetical relation.
Note: The actual truth values given in the white column here
may disagree somewhat with your intuitions, unlike the earlier tables which
probably seemed obviously right. Consider an example:
p: Snow is white
q: Grass is green
Both p and q are true. Looking at the first line of the table we see that
the hypothetical "If snow is white then grass is green" must also be true.
And one may object that the greenness of grass is in no way connected to
the whiteness of snow. It would be possible for God to create a world where
it was true that snow was white but grass was not green. Such a possibility
is, however, totally irrelevant to our question. Our question is: Given
a situation (world) where they are both true, is the claim true that if
the one (p) is true the other (q) is true? Certainly this claim is true,
because the other (q) is true, and the claim said that it
would be.
Similar reasoning would support claiming the conditional is true on
the second line. The consequent is true, and so the relationship the conditional
claims exists between the antecedent and the consequent, namely that if
the first is true the second must also be true, can not be false, as we
are referring only to those cases where the consequent is
in fact true.
This reasoning will not support claiming the conditional is true on
the last line, since the consequent is false there. But the claimed relation
is only that if the antecedent is true the consequent is.
The antecedent in the last line is false, so given that this line in the
table refers only to those situations where the antecedent condition
is false, the relation itself can not be false. The conditional relation
would never lead one to falsely think the consequent true in those cases
where the antecedent is false. (The relation does not specify any truth
value for the consequent in the cases where the antecedent is false.)
Biconditional
Biconditionals also involve two components, each of which could be true
or could be false, so there are four possible combinations of truth values
among the components. The table will need four lines.
p

q

p ≡ q

T

T

T

F

T

F

T

F

F

F

F

T

The gray cells in the table represent all the possible combinations
of truth values for the two components. Each white cell under "p ≡
q" represents the only truth value of the biconditional consistent with
the truth values assigned to the components on the same line. In
other words, the table indicates that when the two components have the
same truth value, the biconditional as a whole will be true, but in all
other possible cases the biconditional will be false. For the purposes
of truth tables this table defines the biconditional relation.
We can test this table's assignment of truth values by seeing whether
it agrees with the table that represents what "if and only if" literally
means. In the expression "p if and only if q" the first "if" would makes
the claim "p if q" which becomes "if q then p" in standard form. The "only
if" in the expression makes the claim "p only if q" which becomes "if p
then q" in standard form. The "and" makes the claim that both "if q then
p" and "if p then q" are true.
p

q

q ⊃ p

p ⊃ q

(q ⊃ p) & (p ⊃
q)

p ≡ q

T

T

T

T

T

T

F

T

F

T

F

F

T

F

T

F

F

F

F

F

T

T

T

T

Notice that under the "q ⊃ p" column
the second line is given the truth value of false because the antecedent
is true while the consequent is false. (Remember that the second column
represents the antecedent here and the first column the consequent.) The
column for the conjunction is determined by looking at the columns for
the conjuncts. The conjuncts here are "q ⊃
p" and "p ⊃ q". On any
line where there is a T in both of these columns, we place a T otherwise
we place an F. The result is a column that has the exact same arrangement
of T's and F's as the column for the biconditional. The propositions over
the last two columns therefore are equivalent in the sense that any circumstance
which would make the one true, would equally make the other true, and any
circumstance which would make the one false, would equally make the other
false.
Note that in the complex expression involving more than one connective
we used parentheses to punctuate the expression so as to make it unambiguous.
The arrangement of the parentheses make clear that the statement as a whole
is a conjunction, each conditional being limited to the elements in its
set of parentheses and so not ranging over the whole proposition. Except
for negation, each connective should apply to two propositions, where one
or both of these component propositions may itself be a compound proposition
marked off by parentheses. Parentheses are used to make clear to which
two propositions any such connective applies. The main connective of the
proposition as a whole will stand outside all parentheses. Negation differs
only in that it applies to one proposition. A negation sign in front of
a component denies that component alone, while a negation sign in front
of a parenthetical expression (i.e. outside the parentheses) denies the
expression as a whole. Note that the reason for the two columns representing
the expressions within the parentheses is to facilitate the calculating
of the truth values for the conjunction which is too complex to safely
compute directly from the gray cells (pun intended: both meanings hold
true). As a general rule when computing the truth values for complex propositions
one should create columns for the most embedded elements (those formed
by connectives with components not requiring parentheses) first and use
these to calculate columns for less embedded elements and repeat until
until one reaches the complete proposition. The calculations are based
on the definitional tables for each basic connective given above.
Truth Tables and Validity
Truth tables provide a useful method of testing the validity of any
argument. To use a truth table to test an argument:

make a column for each of the most basic components used in the argument.
There must be a line for each possible combination of truth values for
these components. Each additional component will double the number of lines
needed. A single component will need two lines. Thus, if there are n
components there must be 2^{n}
lines. By using a set pattern we can be sure to have included all the possible
combinations and that no combination occurs more than once. The set pattern
is to alternate true with false in the first column, in the second column
(if needed) alternate 2 trues with 2 falses, in the third column (if needed)
alternate 4 trues with four falses, and so on (doubling the number kept
together for alternation with each column one adds) until the final component's
column consists of the top half true and the bottom half false.

Add a column for each premise and for the conclusion. This may require
additional columns to enable the calculation of complex premises or conclusions.
Calculate the truth values for each line in the premises' and conclusion's
columns.

Identify the lines where the conclusion is false and check those lines
to see whether there is at least one false premise on that line.

If there is at least one false premise on every line where the conclusion
is false, the argument is valid. Otherwise, you have demonstrated the possibility
of all the premises being true at the same time as the conclusion is false,
which is the mark of an invalid argument.
Examples:
1.
p

q

p v q

~p

q

T

T

T

F

T

F

T

T

T

T

T

F

T

F

F

F

F

F

T

F

The last two lines in this table assign false as the truth value of
the conclusion (last column). On the second last line the premise ~p is
also false and on the last line the premise p v q is false. Thus we see
there is no consistent assignment of truth values possible by which this
argument could have all true premises with a false conclusion. This means
that the argument is valid.
2.
p

q

p v q

p

~q

T

T

T

T

F

F

T

T

F

F

T

F

T

T

T

F

F

F

F

T

The first two lines of this table assign false as the truth value of
the conclusion (last column). On the first line, neither premise is false.
There is not at least one false premise on the first line. This shows that
even when the premises are all true, a false conclusion is still possible,
so the argument is invalid.
3.
p

q

r

~q

~r

~q v ~r

p ⊃(~q v ~r)

p & r

~q

T

T

T

F

F

F

F

T

F

F

T

T

F

F

F

T

F

F

T

F

T

T

F

T

T

T

T

F

F

T

T

F

T

T

F

T

T

T

F

F

T

T

T

F

F

F

T

F

F

T

T

T

F

F

T

F

F

T

T

T

T

F

T

F

F

F

T

T

T

T

F

T

At the risk of repeating myself: the gray columns are filled in using
the pattern spoken of above, and the white columns are filled in by calculating
consistently from the gray columns according to the connectives definitional
tables above. Thus the ~q column is determined by taking the opposite of
whatever is in the q column on the same line. The ~q and the ~r columns
then form the basis for the ~q v ~r column. In this column we put an F
only when both disjuncts were false, namely the first two lines. (These
are the only two lines where ~q and ~r are both false.) The p column and
the ~q v ~r column then became the basis for calculating the p ⊃(~q v ~r) column. Since a conditional is false only when the antecedent
is true but the consequent false, only the first line is false. To test
for validity we must look at the four lines where the conclusion is false:
the first, second, fifth and sixth lines. In each of these lines at least
one premise has a false truth value (see on each of these lines the italic
F). Thus we see the argument is valid.
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