Chapter 7: Propositional Logic Exercises (Symbolic Logic)

The basic units of reasoning in PROPOSITIONAL logic are STATEMENTS, whereas the basic units of reasonig in CATEGORICAL logic are CLASSES.

What is the main difference between propositional logic and categorical logic?

1. Letters

   1. lowercase italics represent variables

      1. variables can represent any statement

   2. UPPERCASE REPRESENT DEFINITE STATEMENTS

      1. ex: It’s raining outside. 

2. Operators

   1. ∼ (tilde

   2. ● (dot)

   3. ∨ (wedge)

   4. ⊃ (horseshoe)

   5. ≡ (triple bar)

What symbols are used in propositional logic?

simple statement

a statement that AFFIRMS ONE thing

It is TRUE when, and only when, what it asserts is actually the case. 

simple statement’s truth condition

compound statement

any statement that is NOT a simple statement

negation

statement that denies something

∼ (tilde)

Symbol Used for Negations

It is true when, and only when, what it DENIES is FALSE.

negation’s truth condition

disjunctive statement

“either… or” statements

∨ (wedge)

Symbol Used for Disjunctions

True when, and only when, AT LEAST ONE disjunct is true

disjunctive statement’s truth condition

conjunctive statement

“and” statement

It is true when, and only when, BOTH conjuncts are true

conjunctive statement’s truth condition

conditional statement

“if… then” statement

FALSE, when and only when, the ANTECEDENT is TRUE and the CONSEQUENT is FALSE

• TRUE IN EVERY OTHER CASE

conditional statement’s truth condition

antecedent

the part that comes before the horseshoe

consequent

part that comes after horseshoe

biconditional statement

“if, and only if” statement

It’s true when, and only when, each side has the SAME TRUTH VALUE

biconditional statement’s truth condition

• Type: Simple

• Truth Value: True

• Symbolized: B

For the statement below, indicate its type and truth value:

NIU has a business college. 

• Type: Negation

• Truth Value: False

  • The truth value of “H” is true, so the negation of “H” must be false.

• Symbolized: ∼H

For the statement below, indicate its type and truth value:

The Huskie is not NIU’s mascot. 

• Type: Disjunctive Statement

• Truth Value: true

  • because at least one disjunct is true (both are true in this case)

• Symbolized: S ∨ T

For the statement below, indicate its type and truth value:

Either Sacramento is CA’s capital, or Tallahassee is FL’s capital. 

• Type: Disjunctive Statement

• Truth Value: false

  • because both disjuncts are false

• Symbolized: S ∨ M

For the statement below, indicate its type and truth value:

Either Saturn is the planet closest to our sun, or Mars is. 

• Type: Conjunction

• Truth Value: False

  • because one of the conjuncts is false (“Earth has rings”)

• Symbolized: S ● E

For the statement below, indicate its type and truth value:

Saturn has rings and Earth has rings. 

• Type: Conditional

• Truth Value: false

  • because the antecedent is true and the consequent is false

• Symbolized: S ⊃ V

For the statement below, indicate its type and truth value:

If Saturn has rings, then Venus is the planet closest to our sun.

• Type: conditional

• Truth Value: true

  • because antecedent is false

• Symbolized: E ⊃ V

For the statement below, indicate its type and truth value:

If Earth has rings, then Venus is the planet closest to our sun. 

• Type: Biconditional

• Truth Value: True

  • because both sides have the same truth value

• Symbolized: E ≡ U

For the statement below, indicate its type and truth value:

Earth has seven continents if, and only if, the US has fifty states. 

• Type: Biconditional

• Truth Value: false

  • because the sides have different truth values (“Earth has seven oceans” is false, but “the US has fifty states” is true)

• Symbolized: E ≡ U

For the statement below, indicate its type and truth value:

Earth has seven oceans if, and only if, the US has fifty states.

well-formed formula

a grammatical statement in propositional logic

well-formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

A

not well formed

• need something on either side of triple bar

For the symbolized sentence below, indicate whether it’s a well formed formula:

A ≡

not well formed

• can’t put two operators next to each other

• also need a closing parenthesis

For the symbolized sentence below, indicate whether it’s a well formed formula:

A ∨ ● C)

well-formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

∼A ⊃ ∼B

not well-formed

• can’t put a tilde right next to closing parenthesis

For the symbolized sentence below, indicate whether it’s a well formed formula:

(A ● B) ∼C

not well-formed

• need parentheses to avoid ambiguity

For the symbolized sentence below, indicate whether it’s a well formed formula:

A ∨ B ● C

well formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

(A ● B) ⊃ ∼(C ⊃ D)

not well-formed

• can’t put tilde to the right of what it negates

  • tilde must be to left of A

For the symbolized sentence below, indicate whether it’s a well formed formula:

A∼ ≡ ∼B

well-formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

A ∨ [B ● (C ⊃ ∼D)]

well-formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

∼∼C

not well-formed

• can’t have tilde on right side

For the symbolized sentence below, indicate whether it’s a well formed formula:

∼C∼

well-formed

For the symbolized sentence below, indicate whether it’s a well formed formula:

A ∨ (B ∨ C)

Main Operator: ∼

Type: Negation

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall:

∼K

Main Operator: ⊃

Type: Conditional

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall:

∼R ⊃ (S ● P)

Main Operator: ∨

Type: Disjunctive Statement

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall:

(L ≡ M) ∨ (O ⊃ P)

Main Operator: Leftmost ∼ (Tilde)

∼{[D ⊃ (F ≡ ∼J] ● N}

Type: Negation

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall. 

∼{[D ⊃ (F ≡ ∼J] ● N}

Main Operator: ≡

Type: Biconditional

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall. 

∼(X ∨ Y) ≡ ∼(Q ⊃ Z)

Main Operator: ●

Type: Conjunction

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall. 

A ● ∼B

Main Operator: ∨

Type: Disjunctive Statement

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall. 

A ∨ ∼(B ⊃ C)

Main Operator: the rightmost ⊃

(B ⊃ C) ⊃ A

Type: Conditional

Identity the main operator in the statement below, and (based on the main operator) what type of statement it is overall. 

(B ⊃ C) ⊃ A

not not-A

aka A

Type: Negation

What is the statement below equivalent to?

∼∼A

not-A

Type: Negation

What is the statement below equivalent to?

∼∼∼A

● (dot)

Symbol Used for Conjunctions

⊃ (horseshoe)

Symbol Used for Condition Statements

≡ (triple bar)

Symbol Used for Biconditional Statements

• P ⊃ Q does NOT mean that P validly entails Q

• ⊃ represents material implication

  • means that in any world very much like ours, whenever P is true, Q will be true.

    • relation weaker than full necessitation

Material Implication versus Necessitation

• Because if we don’t, then some arguments we KNOW are invalid CANNOT be PROVEN invalid by any EFFECTIVE method (a method guaranteed to work in a finite number of mechanical steps)

• Basically, we use the system we use because it works

Why do we treat pardoxes of material implication as true?

1. Any simple statement by itself is a well-formed formula

2. If there’s an opening parenthesis, then there must be a closing one, and vice versa.

3. Compound statements with at least three different letters typically need parentheses, brackets, or braces to avoid ambiguity.

4. A tilde is always placed to the immediate left of what it negates.

5. Do not place two operators adjacent to one another unless at least one is a tilde.

6. Never place a tilde to the immediate right of any closing parenthesis, bracket, or brace.

7. Non-tilde operators always go between statements.

Rules for a Well-Formed Formula (7):

main operator

the operator that tells us what kind of statement it is overall

no

An even number of tildes is equivalent to ___ tildes.

one

An odd number of tildes is equivalent to ___ tilde.

1) If there is only one operator in a statement, then it’s the main
operator. A • B
2) If a tilde stands outside a set of parentheses, brackets, or braces,
and if everything else is inside, then that tilde is the main operator of
the statement. ~(A ∨ B)
3) If there are 2+ operators outside parentheses, brackets, or braces,
the main operator is the one that’s not a tilde. ~(A ≡ B) ∨ (C ⊃~D)
4) If there are no parentheses and two or more operators, the main
operator is the one that’s not a tilde. A • ~B
5) If there are multiple tildes next to a letter, as in ~~A, the main
operator is the left-most tilde.

Rules for Identifying the Main Operator (5)

∼L

Translate the statement below into a well-formed formula using the indicated letters:

Li is not a truck driver. (L)

T ⊃ C

Translate the statement below into a well-formed formula using the indicated letters:

If Li is a truck driver, then he has a commercial license. (T, C)

T ● S

Translate the statement below into a well-formed formula using the indicated letters:

Both Tammy and Sally are athletes. (T, S)

T ∨ S

Translate the statement below into a well-formed formula using the indicated letters:

Either Tammy is an athlete, or Sally is a band member. (T, S)

∼T ● ∼S or ∼(T ∨ S)

Translate the statement below into a well-formed formula using the indicated letters:

Neither Tammy nor Sally is an athlete. (T, S)

∼Y ⊃ ∼S

Translate the statement below into a well-formed formula using the indicated letters:

If this doesn’t burn yellow, then it’s not sodium. (Y, S)

∼D ● (R ⊃ V)

• dot is main operator

Translate the statement below into a well-formed formula using the indicated letters:

My stone is not a diamond; and if your stone is a ruby, then it’s valuable. (D, R, V)

C ● ∼S

Translate the statement below into a well-formed formula using the indicated letters:

This is made of corundum, but it’s not a sapphire. (C, S)

(R ⊃ A) ∨ (A ⊃ R)

Translate the statement below into a well-formed formula using the indicated letters:

Either being rich implies that you’re ambitious, or being ambitious implies that you’re rich. (R, A)

∼(A ⊃ R)

Translate the statement below into a well-formed formula using the indicated letters:

Being ambitious does not imply that you’re rich. (R, A)

(R ● V) ⊃ (S ∨ T)

Translate the statement below into a well-formed formula using the indicated letters:

If roses are red and violets are blue, then either sunflowers or tulips are yellow. (R, V, S, T)

∼[(R ● H) ≡ C]

Translate the statement below into a well-formed formula using the indicated letters:

That you’re rich and happy if and only if you’re a celebrity is not true. (R, H, C)

[P ⊃ (H ● S)] ● ∼P

Translate the statement below into a well-formed formula using the indicated letters:

If pigs can fly, then both horses and sparrows can talk; but pigs can’t fly. (P, H, S)

∼∼N OR N

Translate the statement below into a well-formed formula using the indicated letters:

It’s not the case that nine is not a square number. (N)

1. M ⊃ K

2. ∼K

3. So, ∼M

Symbolize this argument using the indicated letters:

If you’re a mechanic (M), then you know engines (K). So, it’s clear that you’re not a mechanic. For you don’t know engines.

1. S ∨ C

2. ∼S

3. So, C

Symbolize this argument using the indicated letters:

Either this was a simple accident (S) or a case of criminal negligence (C). But there’s no way this could have been a simple accident. So, this was criminal negligence.

1. (T ● E) ⊃ I

2. E

3. So, ∼I

Symbolize this argument using the indicated letters:

If this has three sides (T) and exactly two sides are equal in length (E), then it’s an isosceles triangle (I). This has exactly two sides that are equal in length. So, this is not an isosceles triangle.

1. ∼(L ∨ C)

2. L ⊃ M

3. So, M

Symbolize the argument (choose your own letters):

It’s not the case that either lithium or cobalt is the lightest element. If the lightest element is lithium, then the lightest element has more than one proton in the nucleus. So, the lightest element has more than one proton in the nucleus. 

1. Reduce the argument to its form.

2. Substitute the statements for the letters such that the premises are true and the conclusion is false.

Only difference is instead of letters = classes, now letters = statements.

Remember to look at the truth conditions to determine the statements’ types.

How is the counterexample method used to prove the invalidity of an argument in propositional logic?

True

True or False: The argument below is a counterexample to this argument: If this has three sides (T) and exactly two sides are equal in length (E), then it’s an isosceles triangle (I). This has exactly two sides that are equal in length. So, this is not an isosceles triangle.

1. If IL is a state and its state bird is the northern cardinal, then its state flower is the violet.

2. IL’s state bird is the northern cardinal.

3. So, IL’s state flower is not the violet.

1. If Professor Beaudoin is the President of NIU, then he works at NIU.

2. Professor Beaudoin is not the president of NIU.

3. So, Professor Beaudoin doesn’t work at NIU.

Provide a counterexample to the invalid argument below:

1. P ⊃ Q

2. ∼P

3. So, ∼Q

1. Either roses are red or violets are blue.

2. Violets are blue.

3. So, roses are not red.

Provide a counterexample to the invalid argument below:

1. P ∨ Q

2. Q

3. So, ∼P

1. Neil armstrong was an astronaut, or a lifelong farmer.

2. If Neil Armstrong was a lifelong farmer, then he didn’t step on the moon.

3. So, Neil Armstrong didn’t step on the moon.

Provide a counterexample to the invalid argument below:

1. M ∨ N

2. N ⊃ ∼O

3. So, ∼O