Is it possible to define argument validity as a formula?

Question

Let A, B and C be propositions. Define ARG(A, B, C) as the following argument:

1. A.
2. B.
3. Therefore, C.

My goal is to create a formula whose truth value is equivalent to "ARG(A, B, C) is valid". In other words, I am looking for a formula that yields true if and only if the argument is valid.

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Attempt #1

ARG(A, B, C) is valid if and only if (A ∧ B) → C

Unfortunately, this attempt does not work. Proof: the following argument is invalid (source: third example on wikipedia here), but (A ∧ B) → C yields true (since A is false):

1. All men are immortal.
2. Socrates is a man.
3. Therefore, Socrates is mortal.

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Attempt #2

ARG(A, B, C) is valid if and only if A ∧ B ∧ ((A ∧ B) → C)

Unfortuntately, this attempt also does not work, because although it works for the example above, this attempt yields false for any argument that has a false premise (and it is known that there are valid arguments with false premises).

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I tried to create other attempts but I'm stuck.

Is this impossible? If yes, why (can you give a proof)? If it's possible, what would be the formula?

Answer 1

Obviously, you have to work in the meta-language (see : tautology) :

Valid[ARG(A,B,C)] iff Taut[(A ∧ B) → C].

We may define the function Taut assuming a certain amount of set-theory: we have to define the object truth-valuation i.e. a function

v : Prop → { 0,1 },

where Prop is the set of propositional atoms (or propositional variables) of the language : P1, P2, P3,...

Then we extend the valuations to all formulas of the language using the usual truth-tables for the propositional connectives.

Example : if formula α is P1 ∧ P2 and we have v(P1)=1 and v(P2)=0, then v(α)=0, and so on.

Having defined the set Val of all valuations, we have :

Taut[α] iff ∀v ∈ Val : v(α)=1.

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All this "machinery" needs a meta-language; thus, it is difficult to manufacture a formula of the language itself that can express the "semantical" properties of the language.

See : Self-Reference, Liar Paradox, Tarski's Truth Definitions and Arithmetization of the Formal Language.

Answer 2

The general form of the answer would be:

[p, ξ, N(ξ)]

Ok, now that I’ve got my dumb Wittgenstein joke out of the way, a more serious answer.

The definition of logical validity is that an argument is valid if and only if the conclusion cannot be false when all the premises are true. We can symbolise that fairly straightforwardly:

ARG(A,...,Z, ξ) is valid if A ∧ ... ∧ Z ≡ ξ

Where A,...,Z is any number of premises, A ∧ ... ∧ Z is the conjunction of all the premises and ξ is the conclusion or, in other words, if the conjunction of all the premises is logically equivalent to the conclusion.

Informally, this should work since the above formula entails that the conclusion cannot be false when all the premises are true and, equivalently, that the conclusion will be false if any of the premises are false. I’m not clever enough to come up with a formal proof for this, but I suspect such a proof is impossible.