Solved Exercises on Logical Reasoning in Grade 8

Answer: This is a common point of confusion! The key is understanding what "If P then Q" means logically:

The statement "If P then Q" is really saying: "It's never the case that P is true while Q is false."

So when P is false:

• We don't have a situation where P is true and Q is false
• The condition "P true and Q false" is not violated
• Therefore, the statement holds true

Think of it like a promise: "If you clean your room, then you can go out to play." If you don't clean your room, the promise isn't broken regardless of whether you go out to play or not. The promise only fails if you clean your room but still can't go out to play.

Answer: In formal logic, "OR" always means inclusive OR unless explicitly stated otherwise:

Inclusive OR (standard logic): "P OR Q" is true when P is true, Q is true, or both are true

Exclusive OR (special notation): "P XOR Q" or "P ⊕ Q" is true only when exactly one of P or Q is true

In word problems:

• Look for phrases like "either...or" which often implies exclusive
• Look for "at least one" which implies inclusive
• When in doubt, assume inclusive OR in formal logic contexts

For example: "You can have soup OR salad" typically means you can have both in logic, but might mean exclusive in everyday conversation.

Answer: Here's a systematic approach:

1. Count variables: If you have n variables, you'll need 2^n rows
2. Create columns: One for each variable, then for each subexpression
3. Fill variable columns: Use binary counting pattern (00, 01, 10, 11 for 2 variables)
4. Evaluate step by step: Calculate simpler expressions before complex ones
5. Complete the table: Work from left to right, top to bottom

Example for (P ∧ Q) ∨ R:

P | Q | R | P∧Q | (P∧Q)∨R
--|---|---|-----|--------
0 | 0 | 0 |  0  |   0
0 | 0 | 1 |  0  |   1
0 | 1 | 0 |  0  |   0
0 | 1 | 1 |  0  |   1
1 | 0 | 0 |  0  |   0
1 | 0 | 1 |  0  |   1
1 | 1 | 0 |  1  |   1
1 | 1 | 1 |  1  |   1
        

Always evaluate parentheses first, then follow operator precedence!

Answer: These are distinct concepts:

Valid argument: The conclusion logically follows from the premises. If the premises are true, the conclusion must be true. Validity is about the logical structure.

Sound argument: An argument that is both valid AND has all true premises. Soundness is about both structure and content.

Example of valid but not sound:

• Premise 1: All birds can fly
• Premise 2: Penguins are birds
• Conclusion: Penguins can fly

This argument is valid (the conclusion follows logically), but not sound (the first premise is false).

For an argument to be sound, it must be valid and all premises must actually be true.

Answer: Use this systematic approach:

1. List all entities: Identify all people, places, or things involved
2. Identify constraints: Write down all the rules and restrictions
3. Start with definite positions: Look for clues that place someone definitely
4. Use process of elimination: Cross off impossible arrangements
5. Test assumptions: If stuck, try a possibility and see if it leads to contradiction
6. Track implications: When you place someone, update what else must be true

For example, if you know "Alex is not in the middle" and there are only 3 positions, Alex must be in position 1 or 3. This immediately eliminates one possibility for Alex and might help place others.

Always double-check that your final answer satisfies all the original constraints!