Ethical Realism

January 14, 2013

Logic Part 3: Truth Tables

Filed under: philosophy — JW Gray @ 7:15 am
Tags: , ,

Truth tables are visual aids to help us determine all the truth value possibilities of various statements. Learning about truth tables can help us better understand logic. Truth tables are used to define logical connectives, and to help us identify various distinctions (such as tautologies, self-contradictions, consistent statements, equivalent statements, and valid arguments).

Logical connectives

The five connectives used in propositional logic are the following: “and” (∧), “not” (¬), “or” (∨), “implies” (→), and “if and only if” (↔). Each of the logical connectives has a precise definition, which is provided by a truth table:

Conjunction

p q p ∧ q
T T T
T F F
F T F
F F F

Lower case letters represent “predicate constants.” These lower case letters stand for any possible statement, such as “rocks exist” or “if rocks exist, then bananas are pink.”

The first row contains various statements (“p,” “q,” and “p ∧ q”). “p ∧ q” roughly translates to mean “both p and q.” For example, “p” can mean “rocks exist” and “q” can mean “bananas exist.” In that case “p ∧ q” means “rocks and bananas exist.”

The following image tells us how to read the truth table:

truth table of conjunction

There is a column (vertical area) under each statement, which contains every possible truth value. The column under “p” has “T, T, F, F” (true, true, false, false). The column under “q” is “T, F, T, F” (true, false, true, false). The column under “p ∧ q” contains “T, F, F, F” (true, false, false, false).

Every row (horizontal area) beneath the statements contains every combination of truth values. The first row of truth values states that “p,” “q,” and “p ∧ q” are all true. The second row states that “p” is true, “q” is false, and “p ∧ q” is false. The third states that “p” is false, “q” is true, and “p ∧ q” is false. The fourth states that “p,” “q” and “p ∧ q” are all false.

We can replace “p” and “q” with statements of the English language to clarify how the truth table works. “p” could stand for “life used to exist on Mars” and “q” could stand for “life will exist on Mars in the future.” We don’t currently know if either of those statements are true, but we can talk about all the possibilities.

Life used to exist on Mars. Life will exist on Mars in the future. Life used to exist on Mars and life will exist on Mars in the future.
T T T
T F F
F T F
F F F

Each row states the following possibilities:

  • Row 1: It’s true that “life used to exist on Mars.” It’s true that “life will exist on Mars in the future.” In that case it’s also true that “Life used to exist on Mars, and that life will exist on Mars in the future.”

  • Row 2: It’s true that “life used to exist on Mars.” It’s false that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”
  • Row 3: It’s false that “life used to exist on Mars.” It’s true that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”
  • Row 4: It’s false that “life used to exist on Mars.” It’s false that “life will exist on Mars in the future.” In that case it’s also false that “Life used to exist on Mars, and that life will exist on Mars in the future.”

The truth table makes it clear that “p ∧ q” is only true when both “p” is true and “q” is true. For example, “humans are mammals and they are animals” is true because “humans are mammals” is true and “humans are animals” is true.

Consider what happens if one of these statements is false. “p” can mean “humans are reptiles” and “q” can mean “humans are animals.” In that case we will have the statement “humans are reptiles and they’re animals.” That statement is false because one part of the statement is false.

Negation

p

¬p

T F
F T

Roughly speaking, “p” is any possible statement and “¬p” means “it’s not the case that p.”

Each box on the top row contains a logical statement. (In this case “p” or “¬p.”) Each box below a statement tells us the possible truth values of that statement. “p” can be true or false, and “¬p” can be false or true.

Each row of boxes below the logical statements contains the possible combinations of truth values of the statements above. The first row down says “p” is true and “¬p” is false. Whenever “p” is true, “¬p” will be false. For example, “p” can stand for “rocks exist.” In that case the statement is true, and “¬p” is false because it stands for “it’s not the case that rocks exist.”

The final row says “p” is false and “¬p” is true. Whenever “p” is false, “¬p” will be true. For example, “p” could stand for “1+1=3,” which is false. In that case “¬p” is true because it means “it’s not the case that 1+1=3.”

Truth tables provide every possible combination of truth values that logical statements can have. The only two truth values needed here are true and false, so there are only two rows beneath the logical statements.

Disjunction

p q p ∨ q
T T T
T F T
F T T
F F F

“p ∨ q” roughly translates as “either p or q.” For example, “p” can be “dogs are mammals” and “q” can be “dogs are reptiles.” In that case “p or q” will be “dogs are mammals or dogs are reptiles.

The truth table indicates that every “p ∨ q” statement is true unless both “p” and “q” are false, which is shown on the final row down. For example “p” can be “dogs are reptiles” and “q” can be “dogs are lizards.” In that case “p ∨ q” stands for “either dogs are reptiles or they’re lizards.” That statement is false.

Conditional

p q p → q
T T T
T F F
F T T
F F T

“p → q” roughly translates as “if p, then q.” For example, “if humans are mammals, then humans are animals.”

The truth table indicates that “p → q” is true unless “p” is true and “q” is false. “p” can be “the President of the USA is a human” and “p” can be “the President of the USA is a reptile.” In that case “p → q” will mean “if the President of the USA is a human, then the President of the USA is a reptile.” That statement is false.

We can also consider a true statement where “p” is false and “q” is false. For example, “p” can be “the President of the USA is a lizard” and “q” can be “the President of the USA is a reptile.” In that case “p → q” will be “if the President of the USA is a lizard, then the President of the USA is a reptile.” That statement is true.

Finally, let’s consider a conditional statement where “p” is false and “q” is true. “p” can stand for “the President of the USA is a lizard” and “q” can stand for “the President of the USA is an animal.” In that case the statement is “if the President of the USA is a lizard, then the President of the USA is an animal.” That statement is true.

Equivalence

p q p ↔ q
T T T
T F F
F T F
F F T

“p ↔ q” roughly translates as “p if and only if q.” For example, “p” can stand for “1+2=3” and “q” can stand for “2+1=3.” In that case “p ↔ q” stands for “1+2=3 if and only if 2+1=3.”

The table above makes it clear that “p ↔ q” is only true when “p” and “q” have the same truth values. They must both be true or false. If not, the statement is false.

Consider when “p” stands for “dogs are animals” and “q” stands for “dogs are reptiles.” In that case “p ↔ q” stands for “dogs are animals if and only if dogs are reptiles.” That statement is false.

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:

  1. Socrates is a human and Socrates isn’t a vampire.

  2. If Socrates is a human and Socrates isn’t a vampire, then Socrates is mortal.

  3. 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:

  1. Socrates is an animal.
  2. If Socrates is a mammal, then Socrates is an animal.
  3. 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.

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