Intermediate Logic

We will be moving away from categorical logic and moving into propositional logic (also called symbolic logic).

A proposition is a statement.

God loves the world.

The world is loved by God.

Deus mundum amat.

Truth-functional propositions - propositions in which the truth value of the proposition depends on the truth value of the component parts.

Truth-functional compound proposition:

"There are stars in the sky but there is no moon."

• If both components are true, the entire proposition is true.
• If either component is false, the entire proposition is false.
• If both components are false, the entire proposition is false.

Simple vs. Compound: How many component parts?

• "There are stars in the sky."
• "There are stars in the sky but there is no moon."
• "It is false that there are stars in the sky."

Words like "and," "but," and "or" are called logical operators.

What is a self-report?

Propositional constant - an uppercase letter that represents a single, given proposition. (known propositions)

Propositional variable - a lowercase letter that represents any proposition. (unknown propositions)

Negation

Negation is the logical operator that denies or contradicts a proposition.

What symbol have we used for this before?

We have talked about contradiction before.

• Draw the square of opposition. Label the 4 types of statements.
• Draw the lines to represent contradiction.
• What types of statements contradict?

Write the negation in words and symbols:

• R = "Everyone can read."
• B = "No one likes brussel sprouts."

If a proposition is true, then the negation will be false. And vice versa.

Truth tables

Conjunction

When two propositions come together using "and," "but," "still".

It is represented with a dot.

Conjunction is a logical operator that joins two propositions and is true if and only if both the propositions are true. Thus, if either component is false, the whole thing is false.

Represent the following statements into symbolic form:

• It is raining and I am cold.
• I am cold, but the sun is shining.
• I am cold and wet.
• You and I are both lost.

Truth Table for Conjunction

Disjunction:

When two propositions come together with "or."

We use a V to represent this.

Represent the following statements in symbolic form:

• I got us lost or you got us lost.
• Forest rangers look for the lost or injured.

In Logic we always use the disjunction in the inclusive sense (this or that or both).

Disjunction is the logical operator that joins two propositions and is true if and only if one or both of the propositions is true.

Truth Table for Disjunction

Symbolic Examples:

1. If I make a face, then baby will laugh.

2. Everyone will be disappointed if I change my mind.

3. The fact that he goes to Harvard implies that he is smart

4. When you finish shoveling the driveway I will give you fifty cents.

5. Decapitating my paper dolls is a sufficient condition for your death.

Another word for conditional is hypothetical.

Translate each of the sentences into if/then form:

1. If I make a face, then baby will laugh.

2. Everyone will be disappointed if I change my mind.

3. The fact that he goes to Harvard implies that he is smart

4. When you finish shoveling the driveway I will give you fifty cents.

5. Decapitating my paper dolls is a sufficient condition for your death.

The proposition that follows the "if" is called the antecedent.

The proposition that follows the "then" is called the consequent.

The symbol for a conditional statement is a horseshoe.

Rewrite the first statement in symbolic form.

In a conditional proposition the antecedent implies the consequent and the consequent necessarily follows from the antecedent.

If the antecedent is true, the consequent must also be true.

The only case when a conditional proposition is false is when the antecedent is true and the consequent is false.

1. If I make a face, then the baby will laugh.

2. If I change my mind, then everyone will be disappointed.

3. If he goes to Harvard, then he is smart.

4. If you shovel the driveway, then I will give you fifty cents.

5. If you decapitate my paper dolls, then I will have every right to kill you.

Truth table

"If a cat is a rat, then a cat is an animal."

"If a cat is a rat, then a cat is a rodent."

Symbolize the following statements using

W = My parents will worry and L = I'm out late.

• "My parents will worry if I'm out late."
• "My parents will worry only if I'm out late."

"My parents will worry if and only if I'm out late."

Biconditional statements are symbolized using 3 horizontal lines.

Truth Table for Biconditional

A biconditional is true if and only if the truth values of both parts are the same.

Describe the following words:

• Tautology
• Self-contradiction
• Biconditional

Biconditional are helpful to test whether two statements are logically equivalent.

Two propositions are logically equivalent if and only if they have identical truth values.

Write the following statement in symbolic form.

"The wind is blowing if and only if it is false that the wind is not blowing."

Create a truth table for this statement.

We see that B and ~~B are logically equivalent, because when one is true, the other is true. When one is false, the other is false.

The biconditional of B and ~~B is always true as seen in the truth table.

This is similar to what type of propositions?

Can you think of other ways to symbolically show tautology?

Can you think of a way to symbolically show self-contradiction?

The way to test if two statements are logically equivalent, contradictory or neither is by putting biconditional signs between them and solving the truth table for the biconditional.

Examples:

The premises and conclusion in an actual proof are much more complex than the premises and conclusions simplified in a rule.

Variables in the rules of inference can represent very complicated compound propositions.

Variables in the rules of inference can represent propositions which are similar or identical to those represented by other variables.

Temporary variables you use to simplify complex propositions change value from one step to the next.

Lesson 6

Lesson 2: Negation, Conjunction, and Disjunction

Intermediate Logic

Lesson 5: Biconditional

Lesson 14

Unit One

Review Activity:

• Create the truth tables for negation, conjunction, and disjunction
• Create a (difficult) compound symbolic statement

Lesson 1: Propositional Logic

Lesson 4: The Conditional