Abstract
Proofs from assumptions are amongst the most fundamental reasoning techniques. Yet the precise nature of assumptions is still an open topic. One of the most prominent conceptions is the placeholder view of assumptions generally associated with natural deduction for intuitionistic propositional logic. It views assumptions essentially as holes in proofs, either to be filled with closed proofs of the corresponding propositions via substitution or withdrawn as a side effect of some rule, thus in effect making them an auxiliary notion subservient to proper propositions. The Curry–Howard correspondence is typically viewed as a formal counterpart of this conception. I will argue against this position and show that even though the Curry–Howard correspondence typically accommodates the placeholder view of assumptions, it is rather a matter of choice, not a necessity, and that another more assumption-friendly view can be adopted.
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Notes
Assumption (hypothesis, supposition) withdrawing rules first appeared in the works of Gentzen (1935) and Jaśkowski (1934). It is important to emphasize that we are not really interested in the implication introduction rule itself only insofar as it is an assumption withdrawing rule. Other assumption withdrawing rules from intuitionistic propositional logic such as, e.g., disjunction elimination rule or negation introduction rule, would be suitable for our analysis as well. I only choose to start with the implication introduction rule for its familiarity. I will return to this issue later in Sect. 2.
Schroeder-Heister (2016) describes this as a two-layer system. Note that, strictly speaking, the assumptions are not really withdrawn, they are rather incorporated into the propositional level in the form of an antecedent.
However, it is worth noting that even with Frege the situation is not straighforward, as one of the reviewers pointed out. Recently, Schroeder-Heister (2016, p. 257, footnote) noted that there is a “hidden two-layer system” of assumptions and assertions in the background of Frege’s system as well. See also Schroeder-Heister (2014).
See, e.g., Sørensen and Urzyczyn (2006).
As one of the reviewers correctly noted, this general form of inference rules can be achieved in standard natural deduction with sequent-style (also known as logistic) presentation as well. I agree, however, it would still rely on the same Curry–Howard correspondence and thus kept viewing assumptions as placeholders, which I wish to avoid. In other words, I want to move from \(x : A \vdash b(x) : B\), where x is a ground for A and b(x) is a ground for B, towards \(f : A \vdash B\), where f is a ground for the whole derivation from A to B. For more, see below.
See, e.g., Gentzen’s system NLK, discussed in von Plato (2012).
The other reason is that formulating the Curry–Howard correspondence for these systems is more complicated and much less explored (see, e.g., Geuvers and Nederpelt 2004).
I will talk more about the differences between \(\lambda x . b(x) : A \supset B\) and \(f : A \Rightarrow B\) later in Sect. 1.3.1. For now, it suffices to say that \(\lambda x . b(x)\) is a function in the sense that it is an element of some cartesian product type, while f is a function in a more primitive sense as an unsaturated entity awaiting arguments (see, e.g., Nordström et al. 1990, p. 49). Furthermore, the move from \(A \vdash B\) towards \(A \Rightarrow B\) should not be conflated with the move from a n-level turnstile \(A \vdash ^n B\) to a \(n+1\)-level turnstile \(\vdash ^{n+1} (A \vdash ^n B)\) considered, e.g., by Došen (1980) (I thank an anonymous reviewer for bringing this to my attention.) Conceptually, the main difference is that Došen’s iterating turnstiles grow infinitely “upwards”, while the idea behind \(A \Rightarrow B\) is rather that we are going “downwards” to the more fundamental notion of function behind \(A \vdash B\).
To borrow terminology from category theory, assumptions are now source objects of the whole derivation/function, with proper propositions being the target objects.
As one of the reviewers correctly pointed out, derivation composition is, strictly speaking, not a basic rule, but a property of natural deduction derivations. Thus the above rule should be rather viewed as an optional rule justified by the corresponding theorem proving the desired property (see, e.g., Theorem 8.1.4 in Negri et al. 2001, p. 171.)
We are interested in getting rid of the placeholder view of assumptions by capturing assumptions as arguments of functions which represent the corresponding derivations from these assumptions. That does not mean that every derivation has to be explainable via functions (of course, it can be done if we assume non-emptiness of all antecedents).
There are, of course, other possible ways to formalize it. If we weren’t limited to the Curry–Howard correspondence based systems, the most straightforward choice would probably be category theory, or more precisely categorial proof theory (see Došen 2016; Došen and Petrić 2004), where \(f : A \Rightarrow B\) would be interpreted simply as a morphism f from A to B.
Sets are to be understood in Martin-Löf’s constructive sense.
To simplify presentation, I omit the corresponding introduction rule for equal elements of this type.
See Martin-Löf (1984).
I would like to thank one of the reviewers for this remark.
I would like to thank one of the reviewers for drawing my attention to this issue. The reviewer also suggested analysing the rule of bivalence also called the rule of dilemma or the rule of excluded middle (see, e.g., Tennant 1978, p. 48 or Negri et al. 2001, p. 12; recently also adopted by D’Agostino et al. (2020) as the only assumption discharging rule) as an example of a classical rule. However, I chose the reductio ad absurdum rule because it is generally a more familiar rule and it lends itself more naturally to the functional interpretation in the style of the Curry–Howard correspondence.
Note that from the viewpoint of the lower-order presentation of constructive type theory, case is essentially just an auxiliary symbol and not a proper object of the type theory (the same can be said about \(\lambda \), \(\Pi \), etc.). This is not the case with higher-order presentation where case can be regarded as a higher-order function d of type \((A : prop)(B : prop)(C : (A \vee B)prop)(c : A \vee B)((x : A)C(i(A,B,x)))((y : B)C(j(A,B,y)))C(c)\) where \(i : (A : prop)(B : prop)(A)(A \vee B)\) corresponds to \(\mathbf{inl} \) (analogously for j) and \(\vee : (prop)(prop)prop\).
It probably goes without saying, but we are now leaving the strictly intuitionistic principles behind the original Curry–Howard correspondence.
Fore more, see Gabbay and de Queiroz (1992, p. 1345).
A question raised by one of reviewers.
The first derivation of the proof above is a one-step derivation the conclusion of which is the same as its only assumption. This is considered as a legitimated derivation (see, e.g., Hindley and Seldin 1986, p. 261).
Although I have discussed vacuous discharge only in connection with implication introduction rule, these observations can be analogously applied to other assumption withdrawing rules as well. For example, the general form for disjunction elimination would be \(\langle \langle \Gamma _1 , A \vee B \rangle , \langle \Gamma _2 , C \rangle , \langle \Gamma _3 , C \rangle , \langle \Delta , C \rangle \rangle \) where \(\Gamma _1 \cup (\Gamma _2 - A ) \cup (\Gamma _3 - B ) \) (see Prawitz 1965), which can be captured as \(((\Gamma _1) A \vee B)((\Gamma _2) C)((\Gamma _3) C) (\Delta )C \).
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I would like to thank the two anonymous referees for providing insightful comments which helped to improve this paper.
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Pezlar, I. The placeholder view of assumptions and the Curry–Howard correspondence. Synthese 198, 10109–10125 (2021). https://doi.org/10.1007/s11229-020-02706-z
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DOI: https://doi.org/10.1007/s11229-020-02706-z