Truth Tables How to build them How to use them to determine logical properties of statements How to use them to determine logical relations between statements How to use them to determine validity or invalidity
A truth table exhibits all the possible truth values of a set of simple statements The number of lines the table requires, to show all possible truth values, depends on the number of simple statements in the example. The number of lines needed is 2 to the nth power, where “n” is the number of distinct simple statements (p . q) v ~(p . q) p > q (p v q) . r 4 lines 8 lines 4 lines
Fill in the truth values for the simple statements Start on the left, and halve the number of rows Make the first half True, the second half False At the next simple statement, halve the number you halved. Make the first half of those True, the next half false Continue until you get a column reading T F T F
~ ( p v q) T F
~ ( p v q) T t F f
~ ( p v q) f T t F
p v ( q . r) t f
p v ( q . r) t f
p v ( q . r) t f
(p > q) . ( r > s) t f
(p > q) . ( r > s) t f
(p > q) . ( r > s) t f
(p > q) . ( r > s) t T F f
All three examples just seen have a feature in common: Each statement is TRUE on some lines of the table, but FALSE on others Look under the MAIN OPERATOR to see the final truth value for the statement
A statement that is T sometimes, and F other times is called “CONTINGENT” This means that its truth value depends upon (is contingent upon) the truth value of the simple statements.
Tautology Self-Contradiction There are two other possibilities The value under the main operator is TRUE in all cases Tautology Or: the value under the main operator is FALSE in all cases Self-Contradiction
(A . B) v ~ ( A . B) T F
(A . B) v ~ ( A . B) T F
(A . B) v ~ ( A . B) T F
(A . B) v ~ ( A . B) T F T
(p > q) . (~q . p) T F
(p > q) . (~q . p) T F
When a statement is always TRUE or always FALSE, regardless of the truth value of its component statements, this is due to its structure or form – not to its content. This is similar to the point made about DEDUCTION in general: validity is a function of structure or form.
Contingent Tautologous Self-contradictory Logical properties of truth-functional compound statements Contingent Tautologous Self-contradictory
Truth tables can also show relationships between statements or sets of statements ~ (p v q) (~ p v ~ q) Take a minute, and write up the truth tables for these
~ (p v q) (~p v ~q) T F
~ (p v q) (~p v ~q) F T
~ (p v q) (~p v ~q) F T Sometimes these are the same as one another, but not always: they are not equivalent.
Between any two (or three or four, etc) statements, there are four possibilities: They may be EQUIVALENT: having the same value under the main operator on every line They may be CONTRADICTORY: having opposite values under the main operator on every line They may be CONSISTENT: all TRUE on at least one line (in at least one case; under at least one set of circumstances) They may be INCONSISTENT: never all TRUE under any set of circumstances
VALIDITY Truth tables define the logical operators They exhibit logical properties of statements: contingent; tautology; self-contradiction They exhibit logical relations between statements: equivalent contradiction consistent inconsistent And they let us test arguments for VALIDITY
what is validity? The property of : it being impossible for a false conclusion to follow from true premises
what is invalidity? The property of: IT BEING POSSIBLE FOR A FALSE CONCLUSION TO FOLLOW FROM TRUE PREMISES
How to lay out a table for an argument (p > q) / ~p // ~q T F Single slash “/” to separate premises Double slash “//” to set off the conclusion (far right)
The possibility --or the impossibility-- of a False What does this table show? (p > q) / ~p // ~q T F The possibility --or the impossibility-- of a False conclusion coming from True premises?
p > q / p // q
p > q / p // q T F
p > q / p // q T F
p v q / ~p // q T F
p v q / ~p // q T F
p v q / ~p // q T F Disjunctive Syllogism What is this called?
T F p > q / q > r // p > r What is this called? Hypothetical Syllogism
p > q / q > r // p > r T F
Russell was either a realist or an empiricist. If the former, then he was not an idealist, so he was not an empiricist. R v E / R > ~I // ~ E
Russell was either a realist or an empiricist. If the former, then he was not an idealist, so he was not an empiricist. R v E / R > ~I // ~ E T F t F T f
Russell was either a realist or an empiricist. If the former, then he was not an idealist, so he was not an empiricist. R v E / R > ~I // ~ E T F F t T f
T F If Sartre's an existentialist, then Wittgenstein wrote the Tractatus, therefore if Wittgenstein wrote the Tractatus, Sartre's an existentialist. S > W // W > S S > W // W > S T F
T F If Sartre's an existentialist, then Wittgenstein wrote the Tractatus, therefore if Wittgenstein wrote the Tractatus, Sartre's an existentialist. S > W // W > S S > W // W > S T F