Ash
TNPer
Logic Lesson 3
Truth Tables
As you've seen truth tables are incredibly useful for finding out what happens to a statement when you consider all the possible truth values for the constants. Truth tables make it possible to identify tautologies and contradictions. They also allow one to figure out whether a set of statements are consistent and whether two statements are semantically equivalent.
Suppose you were given this:
P?(~Q?(P & ~Q))
And told to say something about the statement. What would you do?
Well, it may or may not come to you but if you assume P is true and Q false, then you conclude that P is true and Q is false. The statement is always true.
We'll use that statement and work through it. First lets set up the truth table.
Each constant on the left (P , Q) and the statement on the right [ P?(~Q?(P & ~Q)) ]
To find out how many rows you need follow this simple formula:
Number of rows = 2NUMBER_OF_CONSTANTS
We have 2 constants so we need four rows.
Next write down every possible set of truth values:
Now copy down the value of each constant into the proper statement columns
Write down the corresponding negation down in the column with ~
Starting with the innermost set of parentheses, fill in the column directly below the operator for that part of the statement.
We can now see the truth value for the entire statement under all the possible values for P and Q. It is clear, once the truth table is done, that no matter what values P and Q have, the statement is always true.
The statement P?(~Q?(P & ~Q)) is a tautology.
There are three categories:
Tautologies : A tautology is a statement that is always true regardless of the values of its constants.
Contradictions : A contradiction is a statement that is always false regardless of the values of its constants.
Contingent statement : A contingent statement is a statement that is true under at least one interpretation and false under at least one interpretation.
SEMANTIC EQUIVALENCE
When two statements are semantically equivalent, they have the same truth value under all interpretations.
For example H and ~~H are semantically equivalent. If H is true then ~~H is also true. If H is false then ~~H is also false.
EXERCISE 6
Q1)Are the following statements semantically equivalent?
A ? B
~A V B
Show your working (you do not have to attempt to render a truth table, simply going through each interpretation [for which you might want to use a truth table on paper] is sufficient).
Simply stating the answer will receive no credit.
CONSISTENT STATEMENTS
When a set of statements is consistent, at least one interpretation makes all of those statements true. When a set of statements is inconsistent, no interpretation makes all of them true.
EXERCISE 7
Q1)Are the following statements consistent?
A ? B
~A V B
A ? ~B
Show your working (you do not have to attempt to render a truth table, simply going through each interpretation [for which you might want to use a truth table on paper] is sufficient).
Simply stating the answer will receive no credit.
ARGUMENT VALIDITY
When an argument is valid, no interpretation exists for which all of the premises are true and the conclusion is false. When an argument is invalid, however, at least one interpretation exists for which all of its premises are true and the conclusion is false.
Truth tables can be used to test for argument validity. Here's an argument:
PREMISES
A & B
C ? ~ A
CONCLUSION
~B ? C
Let's construct a truth table.
Because for a valid argument, the premises all need to be true, we can strike away the interpretations in which one premises evaluates to false.
The only row that the true premises also has a true conclusion, therefore this argument is valid.
EXERCISE 8
Q1)What happens when you negate a tautology?
Q2) What happens when you negate a contradiction?
Q3) What happens when you link two semantically equivalent statements with a biconditional operator?
Q4)Write each of the following in symbolic form, and then decide whether it is a tautology or not:
If I am hungry and thirsty, then I am hungry.
For me to bring my umbrella it's necessary and sufficient that it rain; therefore if it does not rain I will not bring my umbrella.
To get good grades it is necessary to study, and if you get good grades you will get a good job; therefore, it is sufficient to study to get a good job.
To get good grades it is necessary to study, but John did not get good grades; therefore John did not study.
Truth Tables
As you've seen truth tables are incredibly useful for finding out what happens to a statement when you consider all the possible truth values for the constants. Truth tables make it possible to identify tautologies and contradictions. They also allow one to figure out whether a set of statements are consistent and whether two statements are semantically equivalent.
Suppose you were given this:
P?(~Q?(P & ~Q))
And told to say something about the statement. What would you do?
Well, it may or may not come to you but if you assume P is true and Q false, then you conclude that P is true and Q is false. The statement is always true.
We'll use that statement and work through it. First lets set up the truth table.
Each constant on the left (P , Q) and the statement on the right [ P?(~Q?(P & ~Q)) ]
To find out how many rows you need follow this simple formula:
Number of rows = 2NUMBER_OF_CONSTANTS
We have 2 constants so we need four rows.
Next write down every possible set of truth values:
Now copy down the value of each constant into the proper statement columns
Write down the corresponding negation down in the column with ~
Starting with the innermost set of parentheses, fill in the column directly below the operator for that part of the statement.
We can now see the truth value for the entire statement under all the possible values for P and Q. It is clear, once the truth table is done, that no matter what values P and Q have, the statement is always true.
The statement P?(~Q?(P & ~Q)) is a tautology.
There are three categories:
Tautologies : A tautology is a statement that is always true regardless of the values of its constants.
Contradictions : A contradiction is a statement that is always false regardless of the values of its constants.
Contingent statement : A contingent statement is a statement that is true under at least one interpretation and false under at least one interpretation.
SEMANTIC EQUIVALENCE
When two statements are semantically equivalent, they have the same truth value under all interpretations.
For example H and ~~H are semantically equivalent. If H is true then ~~H is also true. If H is false then ~~H is also false.
EXERCISE 6
Q1)Are the following statements semantically equivalent?
A ? B
~A V B
Show your working (you do not have to attempt to render a truth table, simply going through each interpretation [for which you might want to use a truth table on paper] is sufficient).
Simply stating the answer will receive no credit.
CONSISTENT STATEMENTS
When a set of statements is consistent, at least one interpretation makes all of those statements true. When a set of statements is inconsistent, no interpretation makes all of them true.
EXERCISE 7
Q1)Are the following statements consistent?
A ? B
~A V B
A ? ~B
Show your working (you do not have to attempt to render a truth table, simply going through each interpretation [for which you might want to use a truth table on paper] is sufficient).
Simply stating the answer will receive no credit.
ARGUMENT VALIDITY
When an argument is valid, no interpretation exists for which all of the premises are true and the conclusion is false. When an argument is invalid, however, at least one interpretation exists for which all of its premises are true and the conclusion is false.
Truth tables can be used to test for argument validity. Here's an argument:
PREMISES
A & B
C ? ~ A
CONCLUSION
~B ? C
Let's construct a truth table.
Because for a valid argument, the premises all need to be true, we can strike away the interpretations in which one premises evaluates to false.
The only row that the true premises also has a true conclusion, therefore this argument is valid.
EXERCISE 8
Q1)What happens when you negate a tautology?
Q2) What happens when you negate a contradiction?
Q3) What happens when you link two semantically equivalent statements with a biconditional operator?
Q4)Write each of the following in symbolic form, and then decide whether it is a tautology or not:
If I am hungry and thirsty, then I am hungry.
For me to bring my umbrella it's necessary and sufficient that it rain; therefore if it does not rain I will not bring my umbrella.
To get good grades it is necessary to study, and if you get good grades you will get a good job; therefore, it is sufficient to study to get a good job.
To get good grades it is necessary to study, but John did not get good grades; therefore John did not study.