Logical Connectives - Intermediate Level: negation INTERMEDIATE

Step 1: Understand 'neither...nor' statements
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q

Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T

Step 3: Apply given values
p = T, q = T
¬p = F, ¬q = F
¬p ∧ ¬q = F

Answer: False

Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.

Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)

Step 2: Apply the given values
p = True, q = True

Step 3: Evaluate p ↔ q
Since p and q have the same truth value (True), p ↔ q = True

Step 1: Understand Exclusive OR (XOR)
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.

Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)

Step 2: Apply the given values
p = False, q = True

Step 3: Evaluate the exclusive OR
Since exactly one of p or q is True, the exclusive OR is True

Step 1: Understand the disjunction (OR) operator
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.

Step 2: Apply the given values
p = True, q = True

Step 3: Evaluate p ∨ q
Since at least one of p or q is True, p ∨ q = True

Step 1: Break down the expression
Expression: (p ∧ q) → r
Given: p=T, q=T, r=F

Step 2: Evaluate inner expressions first
p ∧ q = T ∧ T = T

Step 3: Evaluate outer expression
(T) → F = False
Remember: Implication is False only when antecedent is True and consequent is False

Rule: Disjunctive Syllogism

If P ∨ Q is true and P is false, then Q must be true.

Answer: q

Step 1: Break down the expression
Expression: (p ∧ q) → r
Given: p=T, q=F, r=T

Step 2: Evaluate inner expressions first
p ∧ q = T ∧ F = F

Step 3: Evaluate outer expression
(F) → T = True
Remember: Implication is False only when antecedent is True and consequent is False

Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.

Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)

Step 2: Apply the given values
p = True, q = False

Step 3: Evaluate p ↔ q
Since p and q have different truth values, p ↔ q = False

Step 1: Break down the compound expression
Expression: p → (q ∧ r)

Step 2: Evaluate inner expression first
q ∧ r = True ∧ False = False

Step 3: Evaluate outer expression
p → (False) = True → False = False
Implication is False only when antecedent is True and consequent is False

Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'

Step 2: Identify components
Original: The plant will die unless you water it
p: The plant will die, q: You water it

Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p

Step 4: Write in English
Equivalent statement: If you do not water it, then the plant will die

Argument form: Modus Ponens

Modus Ponens: P → Q and P together force Q to be true.

Conclusion: This argument is VALID.

Step 1: Understand the implication (→) operator
The implication p → q is False ONLY when p is True but q is False.
In all other cases, it is True.

Truth table for p → q:
p=T, q=T → Result=T
p=T, q=F → Result=F (the only False case)
p=F, q=T → Result=T
p=F, q=F → Result=T

Step 2: Apply the given values
p = False, q = True

Step 3: Evaluate p → q
Since this is not the case where p is True and q is False, p → q = True

Step 1: Understand necessary and sufficient conditions
• P is NECESSARY for Q: Q cannot be true without P (Q → P)
• P is SUFFICIENT for Q: P being true guarantees Q (P → Q)
• P is BOTH: P if and only if Q (P ↔ Q)

Step 2: Analyze the relationship
P: Being a square
Q: Being a rectangle

Step 3: Determine the condition type
All squares are rectangles (sufficient), but not all rectangles are squares (not necessary)

Answer: Sufficient but not necessary

Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.

Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)

Step 2: Apply the given values
p = True, q = False

Step 3: Evaluate p ↔ q
Since p and q have different truth values, p ↔ q = False

Argument form: Affirming the Consequent

This is a fallacy! The ground could be wet for other reasons (sprinklers, flood, etc.). P → Q and Q does NOT guarantee P.

Conclusion: This argument is INVALID.

Step 1: Understand the classifications
• Tautology: Always True for all possible truth values
• Contradiction: Always False for all possible truth values
• Contingent: True for some values, False for others

Step 2: Analyze the expression
Expression: (p ∧ q) ∧ ¬(p ∧ q)

Step 3: Test all possible combinations
A statement cannot be both true and false

Valid: No A are B, all C are A → No C are B.

Step 1: Analyze A's statement
A says: 'We are both knaves'

Step 2: Test if A is a knight
If A is a knight, then A tells the truth.
But then 'We are both knaves' would be true.
This means A is a knave, which contradicts our assumption.
Therefore, A cannot be a knight.

Step 3: Test if A is a knave
If A is a knave, then A lies.
A's statement 'We are both knaves' must be false.
For 'both knaves' to be false, at least one must be a knight.
Since A is a knave, B must be a knight.

Step 4: Verify
A (knave) lies: 'We are both knaves' is indeed false ✓
B is a knight ✓

Answer: A is a knave, B is a knight

Argument form: Modus Tollens

Modus Tollens: If P → Q and ¬Q, then ¬P follows necessarily.

Conclusion: This argument is VALID.

Step 1: Understand the original statement
Original: p → q means "If it is raining, then the ground is wet"

Step 2: Understand Contrapositive
Contrapositive switches AND negates both parts: ¬q → ¬p
If the original is p → q, the contrapositive is ¬q → ¬p
Important: A conditional and its contrapositive are LOGICALLY EQUIVALENT

Step 3: Apply to our statement
Contrapositive: If the ground is wet is false, then it is raining is false