Philosophy - Proofs

Truth Functions and Truth Table Basics

NOTE: The Truth Tables below may appear slightly different from when they are on the board or in the book. This is because I am using MSWord Tables to make writing out truth values easier. I do not mind if you write them like they are below, with each variable, statement, and premise given a unique column. Alternatively, you could construct them like they are in the book, which is a bit quicker. See pages 303 - 329

Truth tables are models, similar to a graph or Venn diagram, which let us visually represent the truth values of statements.

They express all the logically possible truth values of the logical operators for all the logically possible truth values of the (relevant) statements.

Truth table for a simple statement, ‘x’.

x

T

F

Truth table for two simple statements, x, y (no operator)

x	y
T	T
T	F
F	T
F	F

Truth table for 3 distinct statements, x, y, z

x	y	z
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

NOTE: A truth table for 4 distinct statements would have 16 possible combinations of truth values. A table with 5 statements would have 32. 6 statements would have 64. Truth tables get unwieldy when using more than 3 statements.

Below are the truth tables for our five logical operators.

NOTE: Truth values for logical operators are written under the operators, not the variables.

NEGATION: True when x is false, false when x is true

x	∼x
T	F
F	T

CONJUNCTION: True only when x and y are both true, false otherwise

x	y	x ⦁ y
T	T	T
T	F	F
F	T	F
F	F	F

DISJUNCTION: Only false when x and y are both false, true otherwise

x	y	x ⌵ y
T	T	T
T	F	T
F	T	T
F	F	F

CONDITIONAL: Only false when x is true and y is false, true otherwise

x	y	x → y
T	T	T
T	F	F
F	T	T
F	F	T

BICONDITIONAL: True when both x and y have same truth value, false when truth values are different.

x	y	x ↔ y
T	T	T
T	F	F
F	T	F
F	F	T

Complex Statements and Identifying the Main Operator

Consider the truth table for the following statement: A → B ⦁ ∼A

Since this statement isn’t well formed, we can’t tell which operator is the main operator. When this happens, there will be more than one way to make the truth table. So, we need to know the order of operations when filling in the truth tables. Let’s say the above means (A → B) ⦁ ∼A. The main operator is now the conjunction, “⦁”. Here is a truth table below.

A	B	(A → B) ⦁ ∼A
T	T	T F F
T	F	F F F
F	T	T T T
F	F	T T T

I’ve bolded the truth values for the main operator, because they tell you the truth values for the entire statement. The negation, “∼” and conditional, “→” were minor operators. I determined their truth values first. In the above case, it wouldn’t matter which of the minor operators you wrote the truth values for first. But in general, follow these rules:

1. You will always determine the truth value of the main operator last .

2. First, solve for operators within parenthesis or brackets.

3. If there are two operators within a pair of parenthesis, for example (P ⌵ ∼Q), solve for the minor operator first and the main operator last. In the example, the negation is minor so we will solve for it first. After that, we determine the truth value of the main operator, the disjunction.

4. If there are both parenthesis and brackets, for example [(P ⌵ ∼Q) ⦁ Q], start with the innermost parenthesis. So, we would solve for (P ⌵ ∼Q), and then solve for the conjunction ⦁ Q.

5. When two negations are side by side, the main operator is to the far left. So you would solve for the inside negation first, and the outside negation second.

a. For example: ∼∼P

6. If there are two or more operators outside parenthesis (or brackets), and one is a negation, the other operator is the main operator. So, determine truth values for the negation first, and the main operator second.

a. For example: ∼(P ⌵ ∼Q) ⦁ Q {Conjunction is main}

b. For example: P ⌵ ∼Q {Disjunction is main}

c. For example: ∼∼(P ⌵ ∼Q) → ∼∼(Q ⌵ ∼P) {Conditional is main}

7. There are many possible shortcuts one can take when doing truth tables. There is nothing wrong with this once you are familiar with them. On a test, I’ll expect you to be able to show that you know how to construct them fully.

Arguments and Truth Tables

Writing the truth table for entire arguments isn’t difficult when you can do it for statements. See the book, Chapter 7.3 for more information. Below, I’ve written truth tables for arguments by merely adding additional columns, one for each premise and one for the conclusion. You can also separate statements by using a comma, which is how they are done in the book.

MODUS PONENS

1. P → Q

2. P

3. ∴ Q

P	Q	1. P → Q	2. P	3. ∴ Q
T	T	T	T	T
T	F	F	T	F
F	T	T	F	T
F	F	T	F	F

This is the truth table for Modus Ponens. I’ve included the premises numbers above for clarity, although you do not need to do this. The bolded P Q columns at the left represent all the logically possible truth values for those two variables. The line titled “1. P → Q” represents the first premise of modus ponens. Below it we see the regular truth values for the conditional. Since its form is the same as the x → y truth table above, it has the same truth values. The next column to the right expresses the second premise of Modus Ponens, “2. P”. It has the same truth values as the P column to the far left, since it merely restates a variable without adding any logical operator. The final column is the conclusion, “3. ∴ Q”. It also has the same truth value as the “Q” column to the left. Below are truth tables for other famous forms.

NOTE: A truth table can be made for any argument, valid or invalid.

NOTE: If a statement in an argument isn’t well formed, then you may not be able to determine the truth value for that statement.

MODUS TOLLENS

1. P → Q

2. ∼Q

3. ∴∼P

P	Q	P → Q	∼Q	∴∼P
T	T	T	F	F
T	F	F	T	F
F	T	T	F	T
F	F	T	T	T

DISJUNCTIVE SYLLOGISM

1. P ⌵ Q

2. ∼Q

3. ∴ P

P	Q	P ⌵ Q	∼Q	∴P
T	T	T	F	T
T	F	T	T	T
F	T	T	F	F
F	F	F	T	F

HYPOTHETICAL SYLLOGISM

1. P → Q

2. Q → R

3. ∴ P → R

P	Q	R	P → Q	Q → R	∴ P → R
T	T	T	T	T	T
T	T	F	T	F	F
T	F	T	F	T	T
T	F	F	F	T	F
F	T	T	T	T	T
F	T	F	T	F	T
F	F	T	T	T	T
F	F	F	T	T	T

CONSTRUCTIVE DILEMMA

1. P ⌵ Q

2. P → R

3. Q→ S

4. ∴ R ⌵ S

P	Q	R	S	P ⌵ Q	P → R	Q→ S	∴ R ⌵ S
T	T	T	T	T	T	T	T
T	T	T	F	T	T	F	T
T	T	F	T	T	F	T	T
T	T	F	F	T	F	F	F
T	F	T	T	T	T	T	T
T	F	T	F	T	T	T	T
T	F	F	T	T	F	T	T
T	F	F	F	T	F	T	F
F	T	T	T	T	T	T	T
F	T	T	F	T	T	F	T
F	T	F	T	T	T	T	T
F	T	F	F	T	T	F	F
F	F	T	T	F	T	T	T
F	F	T	F	F	T	T	T
F	F	F	T	F	T	T	T
F	F	F	F	F	T	T	F

Determining Validity:

The five famous forms are all valid. And we can see that on the truth table, because there is no line in which the conclusion is false and the premises are all true. (Remember what is needed for a counterexample?)

Validity: An argument is valid means that if the premises are true, then the conclusion must be true.

For truth tables, we can see every single logically possible way the premises can be true. In the above arguments, every time all the premises are true, then the conclusion is also true. This is because they are valid.

Invalid arguments will have at least one line on the truth table in which the premises are all true, and the conclusion is false. This tells us that the argument form is invalid.

For example:

1. A → B

2. B

3. ∴ A

A	B	A → B	B	∴ A
T	T	T	T	T
T	F	F	F	T
F	T	T	T	F
F	F	T	F	F

I’ve bolded the line in which the premises are both true, but the conclusion is false. This occurs when the antecedent “A” is false, and the consequent “B” is true. Think:

1. If it’s raining, then the sidewalks are wet.

2. The sidewalks are wet.

3. So, it is raining.

The sidewalks could be wet for some other reason (and the antecedent false), and hence the conclusion could be false even when the premises are true. It is invalid.

Abbreviated Truth Tables

Relatively quick way to show that an argument is valid or invalid using truth tables.

Useful for arguments containing 4 or more variables.

1. Write out the truth table but do not fill in the possible truth values for the variables.

2. Assume the conclusion is false.

3. Assume the premises (main operator of each) are true.

4. Try to consistently fill in truth values in such a way that the assumptions in 2 and 3 are confirmed. Start with minor operators and then work backwards to the variables.

5. If there is such a way, the argument is invalid. Fill in the truth values for the variables to the left.

6. If there is no consistent way to assign truth values such that the premises are all true, then the argument is valid. Do not fill in values for the variables to the left.

Exercises page 329

Tautology

· If an argument has a tautology for a conclusion, then it is valid even if the conclusion is unrelated to the premises.

· EX: P ⌵ Q, ∴ R ⌵ ∼R

Contradiction

· If an argument has a contradiction in the premises, then it is valid (but unsound).

Contingency

Logical Equivalency: Two statements are logically equivalent if and only if, for every row of their truth table, the truth value of those two statements is identical.

EX: P ↔ Q, (P → Q) ⦁ (Q → P)

Logical Contradictory: Two statements are logically contradictory if and only if, for every row of their truth table, the truth value of those two statements is different.

EX: P, ∼P

EX: P → Q, ∼(∼P ⌵ Q)

P	Q	P → Q	∼(∼P ⌵ Q)
T	T	T	F F T
T	F	F	T F F
F	T	T	F T T
F	F	T	F T T

· If an argument has contradictory premises, then it is valid (but unsound).

Logical Consistency: Two statements are logically consistent if and only if there exists at least one row of their truth table such that both statements are true.

EX: P, Q

EX: P → Q, P ⌵ Q

Logical Inconsistency: Two statements are logically inconsistent if and only if there does not exist a row of their truth table such that both statements are true.

EX: (P → Q) ⦁ P, ∼Q

· If an argument has inconsistent premises, then it is valid (but unsound).