Questions about material implication and object language.

Question

Exercise 2.73
(d) ⊥ ㅑ ψ

Exercise 2.74
Let Γ be a set of formulas. Show that Γ ㅑ ⊥ if and only if Γ is not satisfiable.

Theorem 2.8 Proof by contradiction
Let ∆ be a set of formulas and φ a formula. If ∆, ¬φ ㅑ ⊥ then ∆ ㅑ φ

Proof
Suppose that ∆ ∪ {¬φ} ㅑ ⊥. Then by the result of Exercise 2.74, there are no
truth assignments making all of the formulas in ∆ ∪ {¬φ} true. This means
that if the truth assignment v makes all the formulas of ∆ true, v(¬φ) must
be false, so that v(φ) must be true; that is, every truth assignment satisfying
∆ also satisfies φ. Thus ∆ ㅑ φ.

The first question is regarding Exercise 2.73. According to the definition of logical consequence, if every truth assignment that makes ⊥ true also makes ψ true, then ⊥ entails ψ. However, such truth assignments do not exist. So, what conclusion should be drawn? According to material implication, we would say it is true, but this only applies to formulas and should not be applied to everyday language, right?

The second question pertains to Theorem 2.8. In the proof of this theorem, it used Exercise 2.74. However, when proving Exercise 2.74, I could only use proof by contradiction. This seemed somewhat contradictory. because I used proof by contradiction to prove "proof by contradiction". However, in Kleene's book, it is mentioned that we should not confuse the logic of the object language with the logic of the meta language. So, when I used proof by contradiction to prove Exercise 2.74, was it the logic of the meta language, while Theorem 2.8 pertains to the logic of the object language?

I am currently studying the book "Propositional Calculus and Predicate Calculus" by Derek Goldrei, and these questions are from exercises in that book.

Answer 1

Let us sort out the concepts in question distinctly without getting bogged down in the details and a variety of viewpoints.

We have a primary understanding of what logical consequence is. We see that logical consequence is a comprehensive concept when we attempt to define it. A list of definitions from various aspects is given by Stewart Shapiro in his article "Necessity, Meaning, and Rationality: The Notion of Logical Consequence" (p. 233) as follows (I recommend this article; it is possible to find a free pdf of it on the Web):

(M) $\Phi$ is a logical consequence of $\Gamma$ if it is not possible for the members of $\Gamma$ to be true and $\Phi$ false.

(PW) $\Phi$ is a logical consequence of $\Gamma$ if $\Phi$ is true in every possible world in which every member of $\Gamma$ is true.

(S) $\Phi$ is a logical consequence of $\Gamma$ if the truth of the members of $\Gamma$ guarantees the truth of $\Phi$ in virtue of the meanings of the terms in those sentences.

(FS) $\Phi$ is a logical consequence of $\Gamma$ if the truth of the members of $\Gamma$ guarantees the truth of $\Phi$ in virtue of the meanings of the logical terminology.

(Sub) $\Phi$ is a logical consequence of $\Phi$ if there is no uniform substitution of the non-logical terminology that renders every member of $\Gamma$ true and $\Phi$ false.

(R) $\Phi$ is a logical consequence of $\Gamma$ if it is irrational to maintain that every member of $\Gamma$ is true and that $\Phi$ is false. The premises $\Gamma$ alone justify the conclusion $\Phi$.

(Ded) $\Phi$ is a logical consequence of $\Gamma$ if there is a deduction of $\Phi$ from $\Gamma$ by a chain of legitimate, gap-free (self-evident) rules of inference.

The last item in the list states the deductive one; we usually denote it by $\vdash$. The others are the ones which we would denote by $\vDash$.

Goldrei tidies up the $\vDash$-class with the definition on p. 74 (except for wording style, this definition is standard):

Let $\Gamma$ be a set of formulas and $\phi$ a formula involving propositional variables in a set $P$. Then $\phi$ is a logical consequence of $\Gamma$, or equivalently $\Gamma$ logically implies $\phi$, when for all truth assignments $v$ on $P$, if $v(\gamma) = T$ for all $\gamma\in\Gamma$, then $v(\phi) = T$. We write this as $\Gamma\vDash\phi$.

It should be remarked that in logical discourse, the words and phrases signifying universal quantification ("all", "every", etc.) do not claim existential import. Therefore, $\bot\vDash\psi$ becomes vacuously true, since there is no valuation that would make $\bot$ true and $\psi$ false. Actually, this is not all too remote from everyday language behaviour. For example, suppose we deny the statement 'there is a red apple'. What we assert is that, if there is an apple, it is not red, regardless of there being an apple or not at all.

Essentially the same idea underlies the valuation of material implication which is usually denoted by $\rightarrow$. A frequent mistake is to take a statement $\phi\rightarrow\psi$ as a derivation $\phi\vdash\psi$. It might be convenient to deem $\phi\rightarrow\psi$ as stating that our world is fashioned such that there is no situation in which $\phi$ holds, but $\psi$ does not hold, whether there is a meaningful (for us) connection between them or not. The logical behaviour of implication is smoothly squared with mathematical discourse. Some of the non-mathematical cases that material implication seems paradoxical actually stem from the mistake I have mentioned.

Once we have proved a theorem, we can reuse it in the proof of another theorem, either by inserting the former proof into the latter one line by line, or by taking it as a schema. In both cases, we may need to make the appropriate uniform substitutions, a legitimate operation. Hence, there is no confusion of object/metalanguage there.