You state: (V→T), (¬V→H), (¬T→¬H), Therefore: V

Based on the previous paragraph, I think you mean: (V→T), (¬V→H), (¬T→¬H), Therefore: T

Can you work the statements to get to: (V→T) & (¬V→T)

Yes, that's possible with these premises... and I guess now that I'm looking at it in that light, I see that it is impossible to have (¬V & ¬T), since the ¬T would trigger ¬H, which would be contradictory to the consequent of (¬V→H)... And since V has a consequent of T, there is only one possible truth value for T, if the premises are true.

The (¬T→¬H) acts as a contradiction for anything that could produce a False value for the conclusion...whew, that was cooking my brain there for a bit. Thanks for the hint!

I wonder if this valid form could have a sound instance?

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
  Proof:
• 4. H → T ..... 3, transposition
• 5. ¬V → T ..... 2,4, hypothetical syllogism
• 6. (V→T)^(¬V→T) ..... 1,5, conjunction

If V implies T and ¬V also implies T (as you concluded above), then T is implied by anything.
For T to be implied by anything, it must be true, because anything implying true is true.
Does this prove The argument's validity?
This is crazy, because in the truth table, the argument seems to be valid, but i found a lot of inconsistencies, like this:

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
• Proof:
• 4. ¬T→¬V ..... 1, transp.
• 5. ¬T→ H ..... 4,2, S.H.

5 and 3 are inconsistent

======================

• 1. V → T
• 2. ¬V → H
• 3. ¬T → ¬H // T
• Proof:
• 4. T v H ..... 1,2, resolution
• 5. ¬T → H ..... 4, disjunctive syllogism

3 and 5 are inconsistent

=======================

How can this be solved?

Is this valid by some odd technicality? Is it just straight up valid and I cant see the inference? — garberdude

Why does it strike you as odd? The first premise asserts that virtue is a sufficient condition for Tim's achieving heaven while the other two premises jointly entail that lack of virtue is a sufficient condition for Tim's achieving heaven. Since Tim must either be virtuous or lacking in virtue, a sufficient condition for Tim's achieving heaven is realized in all cases. The truth table reflects this.

Sounds like a contradiction to me.

"We know that Tim does not achieve Heaven therefore the argument fails."

Not a valid argument.

Validity obtains when it's impossible for all of the premises to be true and the conclusion false. That's the definition of validity. (Outside of relevance logics, by the way, the "and" there is actually more of an "or," which is why, outside of relevance logics, arguments with contradictory premises are considered valid regardless of what the conclusion is.)

One scenario in which all of those premises are true is when we assign "F" to all of V, T and H. (And that's still the case with biconditionals, too, by the way.)

If we assign "F" to all of V, T and H, then T is false (since we just stipulated that we're assigning "F" to T), and therefore it is NOT impossible for all of the premises to be true and the conclusion false. That's possible instead. Which means that it is NOT a valid argument.

In terms of informal logic, by the way--that is, the English version of the argument, where we're paying attention to English semantics, etc., the argument is also not at all valid. For one, we're stating a number of conditionals, we're only stating conditionals, and then we're claiming that a non-conditional fact follows from that. A non-conditional fact never follows only from conditionals. It can always be the case that the antecedents of the conditionals do not obtain. ("If such and such" . . . well, such and such didn't obtain . . . maybe Tim is or is not virtuous, Maybe Tim is or is not happy. We have no idea. The argument doesn't allow us to conclude anything at all about Tim's actual qualities.The argument only suggests what would follow IF something was one of Tim's actual qualities. This helps underscore that informal logic is not the same thing as formal logic, and it doesn't work to plug a natural language into formal logic, with natural language semantics. That's misleading instead.)

One scenario in which all of those premises are true is when we assign "F" to all of V, T and H. — Terrapin Station

If you assign "F" to all three propositional variables, then, in that case, the second premise, (¬V→H), evaluates as false.

You're right. I made a mistake and read it as (~V->~H)

One important thing for trying to translate formal logic into English is to remember that for ~P -> ~H, for example One scenario in which that's true is where it amounts to "If Tim achieves heaven, then Tim is not happy."