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In Kleene's IM page 149 the reader is asked to supply the details for the following two examples.

Example 5. Given $A(x) \vdash B$ and $A(x) \vdash \lnot B$ with x held constant, then $\vdash \lnot A(x)$ and $\vdash \forall x \lnot A(x)$.

Example 6. Given $A(x) \vdash^x B$ and $A(x) \vdash^x \lnot B$ with x not necessarily held constant, then $\vdash \lnot \forall x A(x)$.

I think that the proofs go roughly like below also see this question and answer:

For example 5:

  1. $A(x) \vdash B$ - given
  2. $A(x) \vdash \lnot B$ - also given
  3. $\vdash \lnot A(x)$ - by negation introduction (on 1 and 2) page 99
  4. $\lnot A(x) \vdash^x \forall x \lnot A(x)$ - by generality introduction page 99
  5. $\vdash \lnot A(x) \vdash^x \forall x \lnot A(x)$ - from 3 and 4 we have this chain
  6. $\vdash \forall x \lnot A(x)$ - from 5 (notice the lack/discharge of superscript, i.e. $\vdash$ instead of $\vdash^x$, that's one example of discharge)

For example 6:

  1. $A(x) \vdash^x B$ - given
  2. $\forall x A(x) \vdash A(x)$ - by generality elimination page 99
  3. $\forall x A(x) \vdash A(x) \vdash^x B$ - from 1 and 2 we have this chain
  4. $\forall x A(x) \vdash B$ - from 3 (notice the lack/discharge of superscript, that's the second example of discharge)
  5. $\forall x A(x) \vdash \lnot B$ similarly from the other given deduction
  6. $\vdash \lnot \forall x A(x)$ - by negation introduction (on 4 and 5)

The superscript was discharged twice:

  • in step 6 for the proof of example 5
  • in step 4 for the proof of example 6

What's the justification for this discharge of the superscript? (Or are there similarly simple proofs that don't involve this discharge?)

Mircea Baja
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  • Re.Ex.5: the "discharge" in step 6 is due to the fact that there are no more assumptions (nothing on the left of the $\vdash$ symbol). Thus, the superscript is no more needed. – Mauro ALLEGRANZA Sep 21 '21 at 11:43
  • Re.Ex.6: IMO, the discharge will occur only at step 6 (same reason as above). – Mauro ALLEGRANZA Sep 21 '21 at 11:50
  • I agree with your comment re.Ex.5. More specifically, the way I understand it is that the superscript meaning is that the variable was varied as per the definition on page 95 that says that y is varied in a deduction for some assumption if a) the variable occurs free in the assumption and b) the deduction contains an application of rule 9 or 12 with respect to the variable on a formula depending on the assumption. So no assumption on the left of turnstile means variable not varied. – Mircea Baja Sep 22 '21 at 00:32
  • But I disagree your comment re.Ex 6, I don't think the discharge occurs at step 6. I might be wrong and I appreciate your feedback, but I think it is discharged at step 4 because $x$ is not varied because it does not occur free in $\forall x A(x)$ (i.e. condition a) is not met for the superscript to be needed) – Mircea Baja Sep 22 '21 at 00:50
  • And again, I'm happy to corrected, but I think your answer to Ex.6 in https://math.stackexchange.com/questions/581663/examples-from-kleenes-introduction-to-metamathematics-1952-intro-and-elimi has mistakes: if truly your step 4) has superscript i.e. $\forall x A(x) \vdash^x B$ then you cannot apply $\to$-introduction to get step 5) because that requires that the free variables are held constant for the discharged assumption. It's just that $x$ is not free in $\forall x A(x)$ as whatever is free in $A(x)$ is now bound by the $\forall x$ therefore your step 4) does not have the superscript etc. – Mircea Baja Sep 22 '21 at 01:07
  • I don't have your book, but to clarify your thought, you can design concrete formulas for A(x) and B according to your example 5 & 6, respectively. Your example 5 is treating x as a common parameter in both antecedents of your sequents, while your example 6 is treating x as 2 independent variables... For example for the harder example 6, you can let $A(x): x=2, B: x^2 \ge 1$ for 1st sequent, $A(x): x=0$ for 2nd sequent then you'll see immediately that unlike example 5, here you may satisfy both sequents (no magic at all simply b/c here entailment can be dependent on the value x ranges over). – cinch Sep 22 '21 at 17:35
  • There are useful similar posts on this site... The issue is that Kleene's book has no semantics, but there is implicitly an "intuitive" semantics. $A(x) \vdash^x B$ is "equivalent" to $\forall x A(x) \vDash B$ while the lack of superscript $A(x) \vdash B$ means that the variable $x$ must be interpreted as a constant along the whole proof. This corresponds to the modern approach to free variables with variable assignment functions. – Mauro ALLEGRANZA Sep 23 '21 at 06:47
  • The various restrictions on quantifier inference rules are a "cumbersome" attempt to manage the interplay between "unrestricted" Gen rule: $A(x) \vdash \forall x A(x)$ and Deduction Th. If we want an unrestricted Gen rule, like Principia Mathematica and Mendelson's textbbok) we have to be careful with DT in order to avoid the fallacious $\vdash (x=0) \to \forall x (x=0)$. – Mauro ALLEGRANZA Sep 23 '21 at 06:50
  • @MauroALLEGRANZA are you sure $A(x) \vdash^x B$ is "equivalent" to $\forall x A(x) \models B$? In his example 6, Kleene writes "Given $A(x) \vdash^x B$ and $A(x) \vdash^x \lnot B$ with $x$ not necessarily held constant...", so seems to me as a non-trivial assumption in this example, the $x$-dependent inference here doesn't mean it must be the case for all variables in the domain, otherwise these 2 assumptions are immediately inconsistent by LNC (law of non-contradiction)... – cinch Sep 23 '21 at 19:32
  • @mohottnad to be honest at this point I can't clarify directly the equivalence question. I found that this answer https://math.stackexchange.com/a/580692/643181 was very good to clarify differences in approaching free variables. In particular the examples above 5 and 6 are from a subchapter called "Interpretation of formulas with free variables" where Kleene also list the choices between the generality and the conditional interpretation. – Mircea Baja Sep 23 '21 at 22:02
  • @mohottnad as far as particular examples for Ex 6. the closest I can is as follows $b \ne 0 \vdash^b 0\ne 0$ which he uses in Ex 3 and 4 the same subchapter page 149 where b is definitely varied (i.e. not held constant). And because $a = a$ can be proven from axioms, by introducing a dummy assumption $b \ne 0 \vdash 0 = 0$. So we have $A(x)$ is $x \ne 0$, $B$ is $0 = 0$, and we end up with $\vdash \lnot \forall b b \ne 0$ – Mircea Baja Sep 23 '21 at 22:16
  • Though in the end I'd like to know:
    • are my proofs correct as per Kleene's 1952?
    • in particular with the places where I discard the superscript, do I do it at the right places (e.g. I seem to do it at different places for Ex 6. than @MauroALLEGRANZA)
     – Mircea Baja Sep 23 '21 at 22:17
  • thx for your elaboration and further ref, and certainly we're near the end. In this book apparently free variable has the generality interpretation when there's superscript for turnstile while only acts as an arbitrary constant (term) when there's no superscript. So both your examples are direct explosions from trivial inconsistency assumptions, but the key is the difference of the order of negation connective and universal quantifier for these 2 situations emphasized by Kleene when you want to do universal introduction as conclusion. So your example 5 doesn't need to introduce superscript – cinch Sep 24 '21 at 01:20
  • ... it's essentially same as usual universal introduction. For your example 6, there's no need of such a concept called "superscript discharge". You should just view superscript as general universal quantified formula $\forall x A(x) \vdash...$ as modern version, then the key is the place of negation when you reach conclusion via RAA... – cinch Sep 24 '21 at 01:27

1 Answers1

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The superscript notation $\vdash^x$ is introduced by Kleene to keep track of the application of postulate Rule 9 and 12 with regards to the superscript variable $x$ for an assumption formula, when the deduced formula depends on the assumption. It is a bit ambiguous because it does not identify the assumption formula, but that's usually clear from context (e.g. when there is just one assumption formula). The reason for tracking is to ensure that we meet (or not meet) restrictions to rules, such as the deduction theorem, reductio ad aburdum, where they require that the discharged formula is not varied.

Now the definition for varied involves: an assumption, a variable free in the assumption, a dependent formula in a deduction and the application of Rule 9 or 12 for the variable, assumption to the dependent formula.

So the tracking via the superscript is not required, can be discharged, for some circumstances like:

  • there is no assumption (step 6 example 5 above)
  • the variable does not occur free (e.g. 4 in example 6 above, $x$ is not free in $\forall x A(x)$ because $\forall x$ bounds all free occurrences of $x$ in $A(x)$

In fact Kleene uses the discharge mechanism for the two circumstances above for Example 2 at step 2 and Example 3 at step 3 (IM same page 149)

Mircea Baja
  • 133