1. If A, then B

2. ~ A

3. So, possible that B

Valid or not?

• My take: Not valid.
• Reason:
• Valid means if all the premises are true, the conclusion must be true
• That means adding new information should not alter the truth-value of the conclusion
• But in the above argument, if all these premises true, the conclusion could still be false
• (there may be other unknown conditions that make B not possible)

Concrete counter-example:

If I eat, then I'll be happy (assume true)

I don't eat

added info (If I smoke, I won't be happy)

So, I won't be happy (and it is not possible I'll be happy)

• Invalid, just take A=B with A impossible. – Conifold Jun 12 at 21:11

Quick way to refute, apply to B=A:

1. If A then A
2. not A
3. therefore A is possible

Obviously not true 2 and 3 contradict each other.

• In your version they do not, not A only applies to the actual world. – Conifold Jun 13 at 1:06

Indeed, `¬A`, `A→B` may be true and yet `◊B` may be false, so the semantic entailment is invalid.

To demonstrate: "If pigs could fly, then I would be the Queen. Pigs cannot fly. Anyway, I cannot possibly be the Queen."

The question is whether the following argument is valid:

1. If A, then B
2. ~ A
3. So, possible that B

Whether this could be a valid argument depends on what one means by "possible that B". If it means the modal sentence ◊B then the argument would not be valid. The counter model would be A is false and B is false. One could not even derive ◊~B since then the counter model would be A is false and B is true.

However, if one means ◊(B v ~B) then one could show by a derivation that the argument is valid. Furthermore, if one is starting with natural language, this may be what someone is likely intending when they say "possible" in such a circumstance: they would likely mean perhaps B perhaps not.

In the sentence "if A, then B", A is a sufficient condition for B. It is not a necessary condition. Suppose A symbolizes "It is raining on the grass" and B symbolizes "The grass is wet". Then if it is raining on the grass, the grass is wet. But if it is not raining on the grass, what could one say? The grass could be wet for other sufficient reasons besides rain - or it could be dry. Perhaps the lawn sprinklers are on? Then the grass would be wet. Suppose it is morning and there is dew on the grass. Then the grass would also be wet.

What a person who is not thinking about modal logic would likely mean by the phrase "possibly the grass is wet even though it is not raining" is simply it is possible that the grass is wet or that the grass is not wet if it is not raining. They would have to touch the grass and check. We might paraphrase that as "possibly B v ~B". Since B v ~B is a tautology, it follows from any premises and if the modal system has the T rule, then one can infer ◊(B v ~B) from that tautology. If that is what was intended, then the argument would be valid.

For a description of a natural deduction approach to modal logic and the System T see section 39.3 in forallx. The T rule allows one to derive A from □A. The following is a derivation starting with the tautology as a premise:

```1 B v ¬B
2  | □¬(B v ¬B)
3  | ¬(B v ¬B)    RT 2
4  | ⊥            ¬E 1,3
5 ¬□¬(B v ¬B)     ¬I 2-4
6 ◊(B v ¬B)       Def◊ 5
```

Assume to derive a contradiction line 2. Use the T rule to remove the necessity symbol on line 3. Derive the contradiction on line 4. Introduce the negation on line 5. Use the definition of ◊ as ¬□¬ to derive line 6.

P. D. Magnus, Tim Button with additions by J. Robert Loftis remixed and revised by Aaron Thomas-Bolduc, Richard Zach, forallx Calgary Remix: An Introduction to Formal Logic, Fall 2019. http://forallx.openlogicproject.org/forallxyyc.pdf