A complex truth table
We can use the above truth tables to create a more complex truth table. For example, “Socrates is a human and he’s not a vampire.” We can translate this into logical form as “A ∧ ¬B.” (We use capital letters because they stand for something specific.) In this case “A” stands for “Socrates is a human” and “B” stands for “Socrates is not a vampire.”
The truth table for this is the following:
| A | B |
¬B
|
A ∧ ¬B
|
| T | T | F | F |
| T | F | T | T |
| F | T | F | F |
| F | F | T | F |
We have “¬B” on the truth table because simpler statements must be resolved before we can find the truth values of more complicated statements. “¬B” is contained in “A ∧ ¬B.”
The truth value for “¬B” is the opposite of the truth vale for “B,” so we just write in the opposite values there.
The truth value for “A ∧ ¬B” will be true whenever both “A” and “¬B” are true. There’s only one place on the truth table where “A ∧ ¬B” is true.
Tautologies
A tautology is a statement that’s always true because of it’s logical form. For example, “there are life forms on other planets or there are no life forms on other planets.” That statement has the form “P ∨ ¬P.”
We can identify a tautology by looking at the truth table because all the possible truth values of a tautology are true. When looking at all the above truth tables, you will notice that none of the statements are tautologies because there’s always one possibility of each of the above statements to be false.
A truth table of a tautology:
| P |
¬P
|
P ∨ ¬P
|
| T | F | T |
| F | T | T |
“P ∨ ¬P” is true whenever at least on of those statements is true. That’s why it’s always true—Whenever P is false, ¬P is true and vise versa.
Self-contradictions
A self-contradiction is a statement that’s always false. For example, “there are life forms on other planets and there are no life forms on other planets.” That statement has the form “A ∧ ¬A.”
We can identify a self-contradiction on a truth table by seeing when a statement is always false. None of the above truth tables contain self-contradictions because none of those statements are always false.
A truth table of a self-contradiction:
| P |
¬P
|
P ∧ ¬P
|
| T | F | F |
| F | T | F |
“P ∧ ¬P” is false whenever either “P” or “¬P” is false. One of those simple statements is always false.
Consistent statements
Statements are logically consistent as long as they can all be true at the same time, and contradictory (or inconsistent) whenever they can’t be. “P” and “Q” are consistent because it’s possible they are both true, but “P” and “¬P” are inconsistent because it’s not possible that they’re both true.
Consider the following two statements:
¬P → Q
¬(P ∨ Q)
We can make a truth table for them:
| P | Q |
¬P
|
¬P → Q
|
P ∨ Q
|
¬(P ∨ Q)
|
| T | T | F | T | T | F |
| T | F | F | T | T | F |
| F | T | T | T | T | F |
| F | F | T | F | F | T |
“¬P” must be given truth values before we can find the truth values of “¬P → Q” because it is part of that more complex statement. We need to find the truth values for “P ∨ Q” before “¬(P ∨ Q)” because it’s also part of that more complex statement.
“¬P → Q” will only be false when “¬P” is true and “Q” is false. That only happens on the bottom row.
“¬(P ∨ Q)” will only be true when “P ∨ Q” is false. That only happens on the bottom row as well.
The truth table above shows that “¬P → Q” and “¬(P ∨ Q)” are contradictory statements because they’re never both true at the same time.
Equivalent statements
Equivalent statements always have the same truth values. For example, “all humans are mammals and all humans are animals” is logically equivalent to “all humans are animals and all humans are mammals.” “P ∧ Q” is logically equivalent to “Q ∧ P.”
Consider the following two statements:
¬(P ∧ Q)
¬P ∨ ¬Q
We can make a truth table for them:
| P | Q |
P ∧ Q
|
¬(P ∧ Q)
|
¬P
|
¬Q
|
¬P ∨ ¬Q
|
| T | T | T | F | F | F | F |
| T | F | F | T | F | T | T |
| F | T | F | T | T | F | T |
| F | F | F | T | T | T | T |
This truth table shows why they are equivalent—they always have the same truth values. Whenever “¬(P ∧ Q)” is false “¬P ∨ ¬Q” is also false, and they are both always true at the same time.
Valid arguments
An argument is logically valid whenever it’s impossible for the premises to be true and the conclusion false at the same time. They are logically invalid whenever it is possible for the premises to be true and the conclusion to be false at the same time.
Argument 1
Consider the following argument:
- Socrates is a human and Socrates isn’t a vampire.
- If Socrates is a human and Socrates isn’t a vampire, then Socrates is mortal.
- Therefore, Socrates is mortal.
We can translate this argument into the following logical statements:
A ∧ ¬B
(A ∧ ¬B) → C
C
Each letter stands for a specific statement:
A: Socrates is a human.
B: Socrates is a vampire.
C: Socrates is mortal.
The truth table for this argument is the following:
| A | B | C |
¬B
|
A ∧ ¬B
|
(A ∧ ¬B) → C
|
C
|
| T | T | T | F | F | T | T |
| T | T | F | F | F | T | F |
| T | F | T | T | T | T | T |
| T | F | F | T | T | F | F |
| F | T | T | F | F | T | T |
| F | T | F | F | F | T | F |
| F | F | T | T | F | T | T |
| F | F | F | T | F | T | F |
There’s only one spot on the table where both premises are true, and the conclusion is also true. Therefore, this argument is logically valid.
Argument 2
Consider the following argument:
- Socrates is an animal.
- If Socrates is a mammal, then Socrates is an animal.
- Therefore, Socrates is a mammal.
We can translate this argument into logical form:
A
B → A
B
The truth table for this argument is the following:
| A | B | B → A | B |
| T | T | T | T |
| T | F | T | F |
| F | T | F | T |
| F | F | T | F |
“B → A” is only false when “B” is true and “A” is false. That’s only on the second row from the bottom.
This truth table proves that the argument is invalid because there’s a row of true premises and a false conclusion.
Now, what about our first argument?, is it valid or invalid:
"they will kill me if I work
and they will kill me if I don´t work
either if I work or I don´t work
They will kill me."
1- firstly we will formalize it:
p= I work
q= they will kill me.
p → q
¬p → q
(p ∨ ¬p) → q
2- then we built the truth table:
| p | q | ¬p |
p → q
|
¬p → q
|
p ∨ ¬p
|
(p ∨ ¬p) → q
|
| T | T | |||||
| T | F | |||||
| F | T | |||||
| F | F |
3- Finally I figure out whether there are any row where the premises are true and the the conclusion is also true
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