Hey there, let's work through this logic proof problem together. First off, I notice your third premise is cut off: "若控制灯亮起，则发动机……（内容未完整）" —— that missing piece is make-or-break for determining validity, so I’ll walk through how to handle this once you fill that gap, plus formalize what we already have to set you up.

Step 1: Formalize the Existing Premises #

First, let's assign simple propositional variables to make this concrete:

• B: Battery is not dead (so ¬B means the battery is dead)
• E: Engine works normally
• L: Control light is on

Translating your given premises into logical notation:

1. Premise 1: "若电池未没电，发动机正常工作则控制灯亮起" → B → (E → L) (logically equivalent to (B ∧ E) → L if you want to rewrite it)
2. Premise 2: "若电池没电，则发动机无法工作" → ¬B → ¬E

Step 2: Handling the Missing Third Premise #

Once you fill in the third premise (e.g., "若控制灯亮起，则发动机正常工作" or "若控制灯亮起，则发动机无法工作"), you can test validity in two ways:

If the Argument is Valid: Use Natural Deduction

Let's assume the third premise is "若控制灯亮起，则发动机正常工作" (translated to L → E) and prove a common conclusion like L → B (if the control light is on, the battery isn't dead):

1. B → (E → L) (Given Premise 1)
2. ¬B → ¬E (Given Premise 2)
3. L → E (Completed Premise 3)
4. Assume L (Conditional Proof assumption: we're testing what follows if the light is on)
   5. E (Modus Ponens: from 3 and 4)
   6. ¬¬E (Double Negation: from 5)
   7. ¬¬B (Modus Tollens: from 2 and 6)
   8. B (Double Negation: from 7)
5. L → B (Conditional Proof: from 4 to 8, we've shown light on implies battery isn't dead)

Since we can derive the conclusion using valid logical rules, the argument is valid in this case.

If the Argument is Invalid: Use a Truth Table

Suppose the third premise was "若控制灯亮起，则发动机无法工作" (translated to L → ¬E). We can build a truth table to find a counterexample (a scenario where all premises are true but the conclusion is false):

Here, all three premises are true, but the conclusion "battery is not dead" is false. This counterexample proves the argument is invalid.

Key Takeaway #

The missing third premise is critical — without it, we can't definitively judge validity. Once you complete it:

• For validity: Use natural deduction rules (modus ponens, modus tollens, conditional proof, etc.) to derive your conclusion from the premises.
• For invalidity: Build a truth table to find at least one row where all premises are true but the conclusion is false — that's your proof of invalidity.

内容的提问来源于stack exchange，提问作者lilmert