To solve this problem, I'll guide you through filling in the truth table step by step. I'll also explain the logic for each column.

---

Key Logical Operators and Their Meanings:

1. Negation (-): Reverses the truth value.
   • -P is true if P is false, and vice versa.
2. Conjunction ( ∧ ): True only if both operands are true.
   • P ∧ Q is true if both P and Q are true.
3. Disjunction ( ∨ ): True if at least one operand is true.
   • P ∨ Q is true if either P or Q (or both) are true.
4. Implication ( → ): False only when the first operand is true, and the second is false.
   • P → Q is true except when P is true and Q is false.
5. Biconditional ( ↔ ): True if both operands are the same.
   • P ↔ Q is true if P and Q are both true or both false.

---

Truth Table Framework:

The provided variables are:

• P, Q, R
• Derived columns: -P, -R, P∧Q, Q∧R, P∨Q, P∨R, P→Q, Q→R, (P→Q)∧(P→R), (Q→R)∨(P→R), and P↔R.

Step 1: Set up the truth values of P, Q, and R.

The truth table includes all possible combinations of truth values for three variables (2³ = 8 rows):

P	Q	R
T	T	T
T	T	F
T	F	T
T	F	F
F	T	T
F	T	F
F	F	T
F	F	F

Step 2: Compute -P and -R.

Negate P and R for each row:

P	Q	R	-P	-R
T	T	T	F	F
T	T	F	F	T
T	F	T	F	F
T	F	F	F	T
F	T	T	T	F
F	T	F	T	T
F	F	T	T	F
F	F	F	T	T

Step 3: Compute P∧Q, Q∧R, P∨Q, and P∨R.

• P∧Q: True if both P and Q are true.
• Q∧R: True if both Q and R are true.
• P∨Q: True if at least one of P or Q is true.
• P∨R: True if at least one of P or R is true.

Fill these values:

P	Q	R	P∧Q	Q∧R	P∨Q	P∨R
T	T	T	T	T	T	T
T	T	F	T	F	T	T
T	F	T	F	F	T	T
T	F	F	F	F	T	T
F	T	T	F	T	T	T
F	T	F	F	F	T	F
F	F	T	F	F	F	T
F	F	F	F	F	F	F

Step 4: Compute Implications (P→Q, Q→R).

• P→Q: False only when P = T and Q = F.
• Q→R: False only when Q = T and R = F.

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

Step 5: Combine (P→Q)∧(P→R) and (Q→R)∨(P→R).

1. (P→Q)∧(P→R):
   • True only if both P→Q and P→R are true.
2. (Q→R)∨(P→R):
   • True if at least one of Q→R or P→R is true.

Step 6: Compute P↔R.

• P↔R: True if P and R are either both true or both false.

Let me know if you'd like me to complete the table or focus on a specific part!