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I'm looking at this note (page 3) and I don't really understand the substitution principle stated there. Specifically,

  • What are "assumptions that are not discharged in the subproof"? In the example of implication on the same page, it is stated that $u$ is not discharged in $\mathcal E$. What does it mean and how to see that?

  • (Probably related) Why does the hypothetical proof of $B \ true$ in the statement of the substitution principle have a horizontal line with a label $u$ on top, i.e., $$\rule{3cm}{0.4pt}\textit{u}\\ A \ true \\ \mathcal E \\ B \ true $$ as opposed to just having the form $$A \ true \\ \mathcal E \\ B \ true$$ ? The horizontal line with label $u$ is introduced in this note and my understanding was that the horizontal line with label $u$ introduces a "local subproof" (e.g. if we want to prove that $A\to B \ true$, then we need to "locally" assume $A$ and deduce $B$ from the local assumption $A$ and previous "more global" assumptions, if any). I think of it as an extra vertical line in Fitch-style deduction system. Why does the substitution principle is only stated for this kind of "local" assumption -- does it not work for global deduction trees (the ones without a horizontal line with label $u$) such as the one displayed above?

  • I'm not sure if I understand what "substituting $\mathcal D$ for all uses of the hypothesis labelled $u$ in $\mathcal E$" means. As far as I understand, $\mathcal D$ is itself a proof tree. What is "hypothesis labelled $u$? The thing labeled $u$ is this derivation tree $$\rule{3cm}{0.4pt}\textit{u}\\ A \ true \\ \mathcal E \\ B \ true $$ and this tree has the empty hypothesis, as far as I can tell.

  • For the first question, see https://math.stackexchange.com/questions/3527285/what-does-discharging-an-assumption-mean-in-natural-deduction – Naïm Camille Favier Aug 18 '24 at 08:11
  • You are misreading the tree: the thing labelled $u$ is the inference of $A$ from no hypotheses; in other words, it's the hypothesis of $A$. Then $\mathcal{E}$ derives $B$ from $A$. You can see the whole proof tree as the application of $\mathcal{E}$ to a variable $u$; substitution is then the process of replacing the variable $u$ with an actual derivation of $A$. – Naïm Camille Favier Aug 18 '24 at 08:14
  • So is it just saying that if we can prove "B true" from assumption "A true", and if we can prove "A true" from some unspecified assumptions (that are the leaves of the proof tree D), then we can in fact prove "B true" from the same unspecified assumptions by combining proof tree D with the proof tree which witnesses that from "A true" we can prove "B true"? I still don't understand why there is a need for a horizontal line (labeled $u$) and would go wrong (if anything) if we didn't have that line. –  Aug 18 '24 at 15:43
  • Mostly an inessential matter of syntax. Different authors might use different conventions. – Naïm Camille Favier Aug 18 '24 at 16:29
  • But in lecture 2 the author seems to be making a big deal of the difference between the presence of a line/bar and its absence (see e.g. his example on implication on page 4). I suppose there must be a reason why the line/bar notation matters in this substitution lemma (?) –  Aug 18 '24 at 17:44
  • Okay, so in lecture 2 they seem to establish that the horizontal bars denote discharged hypotheses, and in lecture 3 they present an undischarged hypothesis with a horizontal bar. That is confusing, and probably an oversight. Then again, it's hard to be precise with natural deduction, because it is by nature an "informal" formalism. Using sequents makes all these notational issues go away. You should probably not focus on this too much. – Naïm Camille Favier Aug 18 '24 at 18:52

2 Answers2

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I can how the presentation in the notes is somewhat confusing; here is my attempt at making sense of it.

First some general remarks:
In the tree notation, branches are local subproofs, and corrsepond to a new level of indented vertical line in Fitch notation. The leaves of the branches, i.e. the formulas without any other formulas above them, are the assumptions/hypotheses, and correspond to the formula written on top of the horizontal line in a subproof in Fitch.
Typically (for example as you've found it on Wikipedia), assumptions are just undecorated leaves on a tree, and are only specially marked once discharged, which is then commonly done by putting the formula inside square brackets, $[A]$, and coindexing the assumption and the rule which discharged them (more see below) with the same label. The correspondence in Fitch notation to discharging an assumption is leaving the subproof and continiuing on the next higher level of indentation.
Inutitively, discharging an assumption means that the conclusions obtained from it further down in the tree no longer depend on this assumption. For example, in the case of $\subset I$, once we weakened the conclusion $B$ to a conditional $A \subset B$, the truth of this implication no longer depends on whether $A$ is actually true, so we can remove $A$ from our open hypotheses.
Formally, an assumption is discharged once we marked it as such my putting a horizontal line on top of it, and an assumption may be discharged in the cases specified by the rules.

Now the important bit is that the linked lecture notes always construct the trees starting bottom-up, first writing down the rule application that licences the discharge, and then immediately placing the related assumption and the label, and indicating it as discharged with a horizontal line, because at that point in the stepwise proof construction procedure, the discharging rule application already took place.
However, if we look at the subtree that ends in $B$, and doesn't include the $\subset I$, then locally seen, in this snippet of the tree, $A$ is undischarged, and that's where we may substitute. What the author presumably means is that we may not substitute for occurrences of $A$ that already have been discharged in some other way in that part of the tree, for example if we had some $\neg I$ on $A$ that would discharge it in the part leading to $B$ before the $\subset I$. So to determine whether $A$ is undischarged and may be substituted we consider a subpart of the tree and imagine the $\subset I$ away, ignoring the order in which we put it together and that we only put $A$ because we already had $\subset I$ written.

Still, it is unusal I think to keep the discharge annotation (the horizontal line) around after placing another derivation on top of it in the context of a tree transformation, so if you want to be sure you could consider contacting the author and asking for clarification.

Natalie Clarius
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I shall provide an interim, but I hope, workable enough answer, later to be replaced with a permanent one.

Discharging a hypothesis is essentially to relieve a formula of its function as a hypothesis by making it a constituent (subformula) conclusion of a proof/subproof. Note that a hypothesis never occurring as a constituent of a conclusion would be a spurious one.

Consider the following table from Wikipedia (later I shall replace it with a MathJax one). We see three inference rules (implication introduction, negation introduction, and proof by contradiction) that discharge hypotheses:

Natural deduction propositional rules

The line with the label 'u' on its left (because the proof might employ more than one hypothesis) separates the categorical part from the hypothetical, signalling that the part below the line has a hypothesis is at work; it's the author's choice.

In the text, the judgement 'A true' is the hypothesis and the line as explained. As for the substitution of $\mathcal{D}$, a "hasty" analogy can be drawn for the sake of quick explanation to the following case:

Suppose we have two lines

(1) Assume x is 2

(2) The total is x + 3

(3) 'The total is 5' true

The subproof $\mathcal{D}$ works like the lines (1) and (2), hence we can substitute it anywhere "'The total is 5' true" occurs.

Tankut Beygu
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