Yll \surBuzoku111The author has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 101007627.
A Proof-theoretic Semantics for
Intuitionistic Linear Logic
Abstract
The approach taken by Gheorghiu, Gu and Pym in their paper on giving a Base-extension Semantics for Intuitionistic Multiplicative Linear Logic is an interesting adaptation of the work of Sandqvist for IPL to the substructural setting. What is particularly interesting is how naturally the move to the substructural setting provided a semantics for the multiplicative fragment of intuitionistic linear logic. Whilst ultimately the Gheorghiu, Gu and Pym used their foundations to provide a semantics for bunched implication logic, it begs the question, what of the rest of intuitionistic linear logic? In this paper, I present just such a semantics. This is particularly of interest as this logic has as a connective the bang, a modal connective. Capturing the inferentialist content of formulas marked with this connective is particularly challenging and a discussion is dedicated to this at the end of the paper.
keywords:
Proof-theoretic Semantics, Base-extension Semantics, Substructural Logic, Intuitionistic linear logic, exponentials, modalities, additivity1 Introduction
Providing a proof-theoretic semantics for intuitionistic linear logic is an interesting problem, not least because, to date, we have very few concrete results on how to express modalities within a P-tS framework, but also because proof-theoretic considerations of substructural logics are relatively non-existant. With regards to the latter point, the ground work for this line of investigation was really set up by Gheorghiu, Gu and Pym in their papers [1, 2] for intuitionistic multiplicative linear logic and bunched implication logic respectively. With regards to the former point, discussions in works such as that of Kürbis [3] give useful and interesting philosophical insights into what might be expected of a proof-theoretic semantics for a logic with modalities but it is clear that due to a lack of almost any such results, giving semantics for such logics is a hard problem. The work by Eckhardt and Pym in [4] is the first case, as far as I am aware, where a sound and complete proof-theoretic semantics was provided for a logic with a modality. The approach taken within deals in terms of Kripke-like relations on bases and uses these relations to give a categorisation of . This approach works well in the modal logic case but it is harder to see how one may do something similar in the substructural framework we are considering. A possible approach would be to go via the Kripke semantics of Hodas and Miller for a fragment of intuitionistic linear logic [5], though it is presently unknown if it is possible to extend such a result to the whole logic. In this paper however, we take a completely different approach, one that is much closer to the work of Sandqvist in [6] and Gheorghiu, Gu and Pym in [1, 2], instead giving a sound and complete semantics relative to a derivability relation, for the whole logic.
We begin now with a brief overview of the semantics presented in this paper and give a short discussion of the considerations that one needs to take to obtain such a semantics. As in [6], our notion of base is still the same, with hypothesis discharging rules defined on basic sentences only. Such rules are called basic rules and are elements of the base. By a basic sentence, we mean a propositional sentence devoid of any logical content such as “The sky is blue”. That is to say, basic sentences are atomic propositional sentences. A basic derivability relation, relative to a set of basic rules is defined, again, similarly to those in [6, 1, 2]. However, basic rules now contain an additional layer of abstraction which is necessary to express the additional structurality required for our basic derivability relation to work in the substructural setting considered in this paper. I am not sure how serious an issue a reader may find the extension made to the definition of a basic rule, however, it is important to note that this is not an attempt at adding any primitive logical structure to basic rules. It is instead a pure expression of structurality; precisely something that should be reflected in basic rules of a system whose relation of basic derivability aims to give meaning to a substructural logic. Let us proceed with an example that highlights this structurality. Let , , , , , be basic sentences. Suppose we have a rule that says given a derivation from to and to we can infer . We can represent this with the inference figure:
[] []
Now suppose, we want to use this rule in a derivation, but we have instead derivations from to and from to . We should still be able to obtain using this rule but doing so presents us with a choice: What do we do with the additional ’s? The proposed additional structure allows us to encode in the rule, precisely the answer to this question. In this case we have but two sensible options; as we are accepting in the presence of these ’s we either take one or take both. Since we want to reject the premise that weakened hypotheses suffice in derivations, the rule has to fix one of these two possibilities. For argument’s sake, were we to take one only, then we have a derivation from to in our system. In more complex rules, some derivations may be such that if they are possible in the presence of the same multiset of basic sentences, then the conclusion is obtained in the presence of only one copy of that multiset of basic sentences, plus all the additional multisets. The simplest case, that we collect all individual multisets from each individual derivation, is the case in the atomic system presented in [1].
With respect to the logical structure and our basic setup, we appeal to the following guiding principle, which is at the heart of our inferentialist reasoning:
.
Given a sequent in the logic, , one’s willingness to assert the sequent occurs just in case one is willing to assert as a consequence of asserting .
We make this principle precise by introducting a second consequence relation now on formulas of the logic. This relation is a conservative extension of the basic consequence relation at the heart of our relation of basic derivability. This new consequence relation is indexed by a base of atomic rules and a multiset of basic sentences (multiset, to ensure correct multiplicities of atoms). Whilst the purpose of the base is perhaps clear (see, for example [7, 8, 9, 10, 6]), in that it keeps note of which basic rules we can use to make basic derivations with, the purpose of the multiset is maybe less immediately obvious. As identified in [2, 1], its purpose is to keep track of the movement of data in a derivation. This can be better understood by considering what “data” actually means in this setting. One possible interpretation, which is the approach taken in this paper, is to identify which basic sentences must be used as hypotheses in basic derivations. It is now natural to ask what precisely we expect to infer from such that follows. In our setup, this amounts to accepting that any pairs of rules and atoms that suffice to assert , must also suffice to prove . This approach, due to Gheorghiu, Gu and Pym [1], can be seen as an extension of a similar derivability relation that was discussed by Sandqvist in [10] and de Campos Sanz et al. in [9], but now generalised to the substructural setting. There are two important points to be made about assertions with this consequence relation:
-
•
Firstly, if we wish to make a proper proof-theoretic justification of a sequent, asserting it against a particular base and multiset is not possible. This is because ultimately, any inferentialist reasoning cannot a priori justifiably fix any proof system to reason in. We must always consider all possible extensions. This is well described in [11].
-
•
Secondly, for some sequents, it is possible to assert them in all bases with, a priori, no atoms required. This will be taken as the criterion of validity of a sequent.
It must be noted however that relative to this consequence relation, treatment of any connective which is not inherently “resource sensitive” becomes rather troublesome. Since ILL has such a connective, the exponential (), we will see that what it means to infer from a hypothesis is fundamentally different to what it means to infer from a formula , and indeed further different still, to what it means to be able to derive a formula . I believe that this adjustment to what it means to make such inferences is the sine qua non of this system giving a semantics for ILL.
2 Overview of Intuitionistic Linear Logic
Intuitionistic linear logic is the intuitionistic fragment of linear logic, a logic introduced by J.-Y. Girard in his 1987 paper “Linear Logic” [12]. A good overview of linear logic can be found in [13], and similarly for intuitionistic linear logic in [14]. In short, linear logic can be thought of as a classical logic whose characteristic feature is that it disallows the general use of the structural rules of weakening and contraction in proofs. As a consequence, the usual connectives of classical logic now split into two groups, the additives and multiplicatives. Each classical connective thus becomes two separate connectives, for example, classical conjunction gets split into an additive conjunction and a multiplicative conjunction . Along with a notion of an involutive negation, which gives rise to a notion of a de Morgan duality, what gives linear logic it’s power is its ability to reintroduce structurality in a completely controlled way. Structurality is now tied to two modalities, which govern whether a formula is allowed to be weakened/contracted in the antecedent/succedent repsectively. These modalities thus allow for the embedding of both classical and intuitionistic reasoning within the logic. This makes linear logic particularly appealing to use as a tool to study the fine mechanics of classical and intuitionistic proofs.
In this paper we focus on the intuitionistic fragment of linear logic. This fragment is generally obtained by removing the involutive negation, the multiplicative disjunction and the right hand side structural modality, thus breaking the symmetry in the connectives and removing the ability of the logic to prove classical theorems. Unsurprisingly, another way to think of this is that one takes a normal two-sided sequent calculus for classical logic, such as that in [15], and force the condition that all sequents are single conclusion only. The result of doing so is that now any connectives which allow for multiple conclusions, must now be either redefined or disallowed. Of course, such justifications dont make sense in single-sided sequent calculi, such as those presented in [12, 16]. In any case, we now fix what it means to be a fomula in Intuitionistic linear logic and what we mean by a sequent in ILL:
Definition 2.1.
Formulas of ILL are defined inductively as follows, where is a set of propositional atoms:
Definition 2.2 (Sequent).
A sequent is a pair , where is a multiset of ILL formulas and is a single ILL formula.
Example 2.3.
Two example sequents include and .
Before we move on, we should discuss a little bit about the modality of ILL. This connective, , called “of-course” or “bang”, enables formulas marked by this connective to exhibit structurality when in the antecedent of a sequent. Its function in the succedent is quite different however, in that such formulas generally exhibit S4 box-like behaviour. By this, it is meant that the only way to introduce a formula into an argument, is if every formula in the antecedent is also of the form .
Intuitionistic linear logic is rich in the number of proof systems that it has. Sequent calculi exist but interestingly, there are also natural deduction systems for intuitionistic linear logic too. These natural deduction systems usually appeal to being presented in sequent caluclus form for ease of making explicit the hypothesis distribution as natural deduction is generally a poor tool when it comes to keeping track of structurality. The work of Negri and von Plato [17] shows that in natural deduction for IPL and CPL, hypothesis discharge can be made to correspond to weakening and contraction in sequent calculus proofs for particular sequent calculi. Intuitionistic linear logic however has the problem that because of the explicitness of the structurality of the , natural deduction systems need to reflect this behaviour explicitly. Several attempts have been made, such as those presented in [18, 19, 20, 21, 22]. Though all of these, for the most part, agree on the forms for the rules governing the majority of the connectives of the logic, they differ on how they treat the bang. For example, the works of Wadler [22] and Mints [20] use additional structures added to the sequent to express the rules governing the in a more implicit way, to be in spirit, closer to the natural deduction systems used by classical and intuitionistic logic. We will opt for a more explicit system, such as that given in [14], opting to change the form of the dereliction rule, from the simple form as is presented in the paper, to a form closer to a generalised elimination rule, noting that the author themselves makes note of the fact that both systems are equivalent, and makes their choice in their work for simplicity.
Definition 2.4.
The derivability relation is defined inductively as follows, where capital greek letters are multisets of ILL formulas and lower case greek letters are ILL formulas:
Ax
-I
-I
-I
-I
-
-I
-E
-E
-E
-
-E
-E
-Promotion
-Dereliction
-Weakening
-Contraction
Note: The rules (-Promotion), (-I) and (-E) hold for all .
3 Substructural Basic Derivability
We begin our treatment by setting up some notation. We start by fixing a countably infinite set of propositional atoms . This set of propositional atoms is intended to represent a set of basic sentences, though unless stated otherwise, the term “atoms” will be used throughout this paper to refer to basic sentences. Unless otherwise stated, lower case latin letters are used to denote propositional atoms and upper case latin letters are used to denote finite multisets of propositional atoms. The empty multiset is denoted . The sum of two multisets and is denoted as . Lower case greek letters are used to denote formulas and upper case greek letters are used to denote finite multisets thereof. A multiset may be, for example, denoted as though the brackets will be dropped in all cases except where confusion might arise. In such cases we would write , where each should be considered a singleton multiset. Finally, throughout this work, the term atomic multiset is taken to mean multiset of propositional atoms.
Definition 3.1 (Atomic rules).
Atomic rules take the following form:
where
-
•
Each is an atomic multiset.
-
•
Each and is an atomic proposition.
-
•
Each is a pair called an atomic sequent.
-
•
Each multiset is called an (additive) atomic box.
An atomic rule may also be so written as to save space:
Such a rule may be read as saying: Given derivations of from relative to some multisets , one infers relative to all , discharging from the derivations in question, the premiss atomic multisets .
The length of each additive atomic box will always be written as where denotes the additive atomic box in question. If each additive atomic box contains only one element, then the rule may be said to be multiplicative. Else, it may be said to be additive. Finally, in some atomic rules it may be the case that some of the atomic sequences are outside of any atomic box. These atomic sequents will be written outside any atomic boxes in the rule. Such a rule might look like this:
In this case, the rule should be interpreted as follows: Given any derivation of from relative to some multiset , and a derivation of from , one infers relative to , discharging from the derivations in question, the atomic multisets and . From this, the reading of atomic sequents outside atomic boxes should be clear; namely, any atomic sequent that occurs outside an atomic box, must be “self-contained”, i.e. not occuring relative to any atomic multiset . We see that rules which contain atomic sequents that occur outside any atomic boxes are a subset of rules which occur in atomic boxes. Thus, in what follows, unless explicitly noted, we talk in terms of general atomic rules, i.e. atomic rules in which all atomic sequents occur in an atomic box.
Definition 3.2 (Base).
A base is a set of atomic rules.
Definition 3.3 (Derivability in a base).
The relation of derivability in a base , denoted as , is a relation indexed by the base between an atomic multiset and an individual atomic proposition, defined inductively according to the following two clauses:
- (Ref)
-
.
- (App)
-
Given that and atomic multisets such that the following hold:
Then .
To clarify, if one wishes to use a rule with atomic sequents outside of any atomic boxes, such as the rule in an atomic derivation, then for the ((App)) clause to hold, it must be the case that and that for any such that and , then will hold.
Proposition 3.4 (Monotonicity of ).
If then for all we also have that .
This proposition holds as follows:
Proof 3.5.
Supposing the hypothesis, then holds in one of two ways.
Lemma 3.6 (Cut admissibility for ).
The following are equivalent for arbitrary atomic multisets , atom , and base , where we assume :
-
1.
.
-
2.
For every , atomic multisets where , then .
Proof 3.7.
We begin by proving that (2) implies (1). For this, we begin by taking and to be for each . Since all hold by ((Ref)), it thus follows from (2) that which is nothing more than .
To now show (1) implies (2), we need to consider how is derived; that is, we proceed by induction, considering the cases when the derivation holds due to ((Ref)), our base case, and ((App)) separately.
-
-
holds by ((Ref)). This gives us that , giving . There are thus two cases to consider, depending on which of and is .
-
Case 1:
and . In this case, the statement of (2) becomes, for every and atomic multiset where , then . This holds trivially.
- Case 2:
We are now left to show that (1) implies (2) according to ((App)). We show this by induction on .
-
Case 1:
-
-
holds by ((App)).
Let that and that . Note that some of the may be empty. Suppose further that ((App)) applies by:(1) (2) The hypothesis gives that we have multisets such that hold. Now we assume that each and that for and apply the induction hypothesis to each formula in (2) giving:
(3) Since the atomic rule (1) is in then by ((App)) we get that
Which by rearranging the multisets gives
which is nothing more than
as desired, completing the induction.
4 Semantics
We now define the key relation of our semantics, that of support. In what follows, when a multiset is prefixed by a it is taken to mean that every formula in that multiset has the as top-level connective.
Definition 4.1 (Support).
The relation of support, denoted as , is a relation indexed by a base and an atomic multiset , defined inductively as follows:
- (At)
-
iff .
- ()
-
iff for every base , atomic multiset , and atom , if then .
- ()
-
iff for every base , atomic multiset , and atom , if , then .
- ()
-
iff .
- ()
-
iff and .
- ()
-
iff for every , atomic multiset and atom such that and hold, then .
- ()
-
iff , for all .
- ()
-
iff implies , for all .
- ()
-
iff there are atomic multisets and such that and that and .
- ()
-
iff for any such that , atomic multisets and any , if for any such that , (if then then .
- (Inf)
-
For being a nonempty multiset, iff for all , atomic multisets , if and then .
We read as saying supports an inference from to relative to some multiset . This multiset is called the “atomic context”.
To verify that the inductive definition of is well founded we define the following notion of the degree of a formula:
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•
To atoms , we assign a degree of .
-
•
To the constants , and assign a degree of .
-
•
To each formula , , and , assign the degree the sum of the degrees of and plus .
-
•
To , assign the degree of plus .
Now we see that for all the definitional clauses in Definition 4.1 we have that the formula being defined is always of greater degree than any formula in its definition, thus verifying the claim.
Before we discuss what we mean by validity in this setting, we claim that our relation has a notion of monotonicity associated with it. We will make extensive use of this property in this paper.
Proposition 4.2 (Monotonicity of ).
If then for all , we have that holds.
The proof follows immediately from Lemma 3.4 and the inductive clauses of Definition 4.1. Note however, that the support relation is monotone only with respect to the base, not with respect to the context. Were it so, it would correspond to having a notion of weakening and contraction in the context. Given that our support relation is monotone with respect to bases, we can therefore define the validity of a sequent.
Definition 4.3 (Validity).
A sequent is said to be valid if . This is denoted as .
This notion of validity might seem odd at first as one might want to say that a sequent is valid if and only if for all bases that holds, however, in light of the fact that support is monotonic with respect to bases, this is equivalent to saying that our sequent is supported in the smallest base which is included in every base; namely, the empty base.
Before moving on, it is worth nothing that the clause for (()), can be simplified as so:
This is an immediate consequence of ((Inf)). We will make use of this form of the definition in the following sections. Given this setup, we have the following properties.
Theorem 4.4 (Soundness and completeness for atomic systems).
iff .
Proof 4.5.
If , then the result holds immediately by ((At)). So consider . Proceeding from left to right, we begin by immediately applying Lemma 3.6 to the hypothesis which gives us that for all and atomic multisets such that for , that we have . By ((At)), this means that . Since we have that for then by ((At)) we therefore have that for . Thus by ((Inf)) we conclude that . Finally, because ((Inf)), ((At)) and Lemma 3.6 are bi-implications, that therefore completes the proof.
In general, we would like the following lemma to hold.
Lemma 4.6.
Given and , then it holds that .
The rest of this section will be dedicated to showing this lemma holds, as well as several weaker forms of the statement, which will be necessary in assisting us in proving soundness. It is clear that if contains no formulas with as a top level connective, then the lemma holds immediately according to ((Inf)). The case when this is not so requires more careful treatment, as we will see. We begin with the assisting lemmas first, followed by a lemma about “cutting” on formulas , after which comes the proof of the above lemma.
Lemma 4.7.
Given and then holds.
In the proof, we only consider the additive connectives. For the multiplicative connectives, I refer the reader to [1].
Proof 4.8.
We proceed by proving by induction on the structure of .
-
-
. By (()), we see that we need to show that and .
-
-
. Spelling out the conclusion of the lemma gives that that we want to show that for all atomic multisets and atoms such that and implies that . To do that it suffces to show the following two things:
-
–
. This holds by monotonicity from the first hypothesis.
-
–
. To show this we suppose that we have for all and atomic multisets that . Thus, since we get that . Unfolding the definition of (()) gives that we have that for all , atomic multisets and atoms such that and implies and in particular when that . Thus we conclude that , as desired.
To finish the argument, we unfold according to (()) which gives that for all , atomic multisets and atoms such that implies that . In particular this holds when and . Thus we conclude that , as desired.
-
–
-
-
. To show this we start by unpacking which gives that we have that for all atomic and that . Further unpacking according to (()) gives that for all , atomic multisets and atoms , if then . Since by hypothesis we have for all and that , we can conclude that for all and all which equivalently gives as desired.
-
-
. Unfolding the conclusion gives that supposing that for all bases , atomic multisets and atoms , such that we want to show that . To this end, we prove the following:
-
–
. This holds by monotonicity from the first hypothesis.
-
–
. To show this, we start from the second hypothesis which gives by ((Inf)), for all bases and atomic multisets such that , then . This, by (()), is equivalent to considering further all extensions , atomic multisets and atoms such that if then . By our additional hypothesis and monotonicity, if we consider the case when and , we obtain therefore , which by ((Inf)) gives as desired.
To finish the argument, we note that the first point is equivalent to saying for all , atomic multisets , and atoms , if then we obtain . By the second point, setting and we thus obtain as desired.
-
–
Lemma 4.9.
Given and then holds.
Again, in the proof, we only consider the additive connectives. For the multiplicative connectives, I once more refer the reader to [1].
Proof 4.10.
We proceed by proving by induction on the structure of .
- -
-
-
. Unfolding the conclusion gives that for all , atomic multisets and atoms if and then . Thus given such an and we want to show . To show this, we do the following:
-
-
. This follows by monotonicity.
- -
-
-
. The proof of this case is identical to the previous case.
We thus have sufficient grounds to use the second hypothesis to conclude as desired.
-
-
-
-
. Unfolding the second hypothesis gives us that for all atoms and we have . Using the first hypothesis we get that this implies that for all and that . Thus we conclude , as desired.
-
-
. Unfolding the conclusion gives that supposing that for all bases , atomic multisets and atoms , such that we want to show that . To this end, we prove the following:
-
–
. This holds by monotonicity from the first hypothesis.
-
–
. To show this, we start from the second hypothesis which by (()) is equivalent to considering further all extensions , atomic multisets and atoms such that if then . By our additional hypothesis, we consider the case when , thus giving as desired.
To finish the argument, we note that the first point is equivalent to saying for all , atomic multisets , and atoms , if then we obtain . By the second point, setting and we thus obtain as desired.
-
–
Lemma 4.11.
Given that , and all hold then .
In this case we consider the base case, one multiplicative and one additive case. The other cases follow similarly.
Proof 4.12.
We proceed by induction on the structure of .
-
-
, for atomic . The second and third hypotheses combined give sufficient conditions to conclude from the first hypothesis and (()) that .
- -
-
-
. Unfolding the conclusion gives that we want to show that for all , atomic mulitsets , atoms such that then we can conclude . To do this, we need to show three things:
-
–
. This follows by monotonicity.
-
–
. To show this, suppose we have that for all and atomic multisets such that . Then we have by the second hypothesis that . Thus by the definition of (()) we have that for all , atomic multisets and atomic if then . In particular, this holds when and when , so we get , and thus, we conclude that .
-
–
. Follows similarly to the previous case.
Thus, from the first point, we have that for all , atomic multisets and atoms , if and then . Thus, by considering when and , we get that as desired.
-
–
All other cases follow similarly, thus concluding the lemma.
As mentioned, when it comes to (()), it is no longer the case that by ((Inf)), we are able to cut. That doesn’t mean that we are not able to cut on formulas with (()) as top level connective, as the following lemma shows us.
Lemma 4.13.
Given and then .
Proof 4.14.
We proceed by induction on the structure of . We show three cases, one multiplicative, one additive and the base case to highlight the different aspects of the induction.
-
-
for some . In this case our second hypothesis says and we want to show that . The first hypothesis is equivalent to the statement that for all , atomic multisets and atoms , if then . Thus, in the case when , and , we can use this to obtain our desired conclusion, namely that .
-
-
. In this case, it is equivalent to show that from our original hypotheses and from the hypothesis that for all bases atomic multisets and atoms that , that holds. To do that we need to show that . To show this, we first consider our second hypothesis , which holds in by monotonicity. By ((Inf)), this is equivalent to considering for all such that if then . The conclusion here is equivalent to considering all extensions , atomic multisets and atoms , if then . By monotonicity, and in particular at and our additional hypothesis gives that . Thus, by ((Inf)), we have , as desired. To finish this proof off, we note that our first point, by (()) is equivalent to considering all , atomic multisets and atoms , such that if then . In particular, when and we obtain our desired conslusion .
-
-
. In this case, the second hypothesis gives two hypotheses, and . By the inductive hypothesis, it then follows that and . It then follows that as desired.
All other cases follow similarly.
We thus obtain the following corollary.
Corollary 4.15.
Given and then .
This corollary is an immediate consequence of Lemma 4.13, thus putting us in a prime position to prove Lemma 4.6.
Proof 4.16 (Proof (Lemma 4.6)).
Given , we suppose , where neither multiset is empty. Furthermore, suppose . That is to say there is a partition of such that and . Under these hypotheses, we want to show that . Starting from which we now write as , by ((Inf)), this is equivalent to considering all bases and atomic multisets such that if and then . This is equivalent to considering all bases and atomic multisets such that if then, by monotonicity for all extensions , (if then ). This, by ((Inf)) is equivalent to considering all bases and atomic multisets such that if then . By Corollary 4.15, this thus implies that we have that for all bases and atomic multisets such that if then . Since, we are considering all bases and all multisets such that , then in particular, we are considering the case that and , thus giving us the result , as desired.
Thus, our semantics behaves as expected when cutting on any formulas, including formulas with (()) as their top level connective. We however also have a slightly weaker form of cut on such formulas, which is a consequence of the interplay between (()) and ((Inf)).
Lemma 4.17.
Given and for all if then .
Proof 4.18.
Corollary 4.19.
Given then .
Proof 4.20.
We start by considering all bases where is supported. Thus, our statement becomes, under this quantifier, given then . This is equivalent to saying given and that for all bases , atomic multisets and atoms such that , we want to show that holds. This immediately follows by monotonicity (Proposition 4.2) and Lemma 4.17.
Lemma 4.21.
Given then .
Proof 4.22.
This follows immediately by ((Inf)).
Lemma 4.23.
Given then .
Proof 4.24.
To prove this, it is equivalent to prove that given for all , atomic multisets and atoms , such that implies and for all , atomic multisets and atoms , such that , that we show . Expanding gives us that for all if then . We see that, in particular at and when this implication holds true since we are given that by hypothesis. Thus, we obtain that , as desired.
Finally, it is worth mentioning the following points related to the multiplicative unit. These are to be expected. We won’t make use of many of these results in what follows, except for the second point, as this allows us to prove soundness of the weakening rule:
-
•
By expanding the definition of , one sees that is valid.
-
•
Consequently, it is the case that for any , we have that . Going right to left, it holds by Lemma 4.9. Going right to left, it holds immediately since we cut with .
-
•
Finally, we have that if and hold, then also holds, again by cut.
We are now ready to prove the main results of this paper, that this semnatics is indeed sound and complete for ILL.
5 Soundness
Theorem 5.1.
If then .
Proof 5.2.
By the inductive definition of , it suffices to prove the following:
- (Ax)
-
- (I)
-
If then .
- (E)
-
If and then .
- (I)
-
If and then .
- (E)
-
If and then .
- (I)
-
- (E)
-
If and then .
- (I)
-
If and then .
- (E)
-
If then and .
- (I)
-
If or then .
- (E)
-
If and and then .
- (I)
-
If then .
- (E)
-
If and then .
- (Promotion)
-
If and then .
- (Dereliction)
-
If , and then holds.
- (Weakening)
-
If and then .
- (Contraction)
-
If and then .
We now proceed through the cases, noting that ((Promotion)), ((I)) and ((E)) hold for all .
- -
- -
- -
- -
- -
- -
-
-
((E)) Suppose some and and bases such that and . Then we have that and . It now suffices to show that . This follows immediately by induction over the structure of .
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((Promotion)) We start by fixing an arbitrary . Then, by Corollary 4.19, we have that the second hypothesis gives that . Now, by ((Inf)), if we consider all bases and multisets such that and say , then it follows that . Thus, by Corollary 4.15, we obtain that , which by ((Inf)) gives our desired result .
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((Dereliction)) It suffices to show that given and that holds. We know that the second hypothesis, by Lemma 4.21, implies that . This, together with the hypothesis that , by Lemma 4.13, gives , as desired.
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((Weakening)) It suffices to show, given and that . By Lemma 4.23, we have that the first hypothesis implies . Similarly, the second hypothesis implies . Thus, by cutting on we obtain the desired result.
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((Contraction)) It suffices to show that given and we can obtain . By ((Inf)), we observe that the second hypothesis is equivalent to . From here, by Corollary 4.15, we obtain , as desired.
This completes the proof of all items.
6 Completeness
Theorem 6.1.
If then .
The goal of this section is to prove that the semantics are complete with respect to ILL. The proof presented is very similar to the proof of completeness in [1] for IMLL and [6] for IPL and consequently, the cases for IMLL are again noted but not included. The strategy goes as follows:
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1.
Define two functions which allow us to uniquely map formulas to atoms and vice-versa. Finding such atoms is always possible as our set of atoms is countable.
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2.
Show that any valid judgement remains valid when flattened using the previously defined functions.
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3.
Construct a base whose rules simulate the natural deduction rules of ILL.
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4.
Show that every derivation in this simulation base is equivalent to a corresponding derivation in ILL.
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5.
Show we can go from a valid judgement to a derivation in ILL.
For this section we will assume that we are working on the sequent and that is the set of all subformulas in .
Definition 6.2 (Flattening and naturalising functions).
We define a class of functions called flattening functions as:
Each flattening function has an inverse, , called the naturalising function and is defined as:
Such functions distribute over elements of multisets i.e. if then and .
One might wish to question the existence of such flattening functions. Indeed, their existence follows immediately from the injectivity of as one may pick any atom in to assign to a formula, so long as that atom has not been already assigned a formula or is not already an atom in . Since is countable, we always have enough atoms with which to do this assignment. The existence of a corresponding for any given is immediate. For the rest of this work, we assume a generic such as the particulars of the assignments are irrelevent to our treatment.
We now wish to construct a base using such a flattening function, in a similar fashion to that found in [6] to simulate the rules of natural deduction for ILL.
Definition 6.3 (Simulation Base ).
Given a flattening function , we define the base to contain the following rules, where and (including for any ) are universally quantified over all elements of and is universally quantified over all elements of :
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2.
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3.
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4.
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5.
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6.
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7.
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8.
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9.
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10.
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11.
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12.
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For all ,
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14.
For all ,
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15.
For all , .
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16.
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17.
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18.
Note that by defining a base , we must have fixed a particular flattening function which gives a corresponding inverse . Where is considered in the following section, this will be implicit.
We are now ready to prove the key lemmas that will give us the tools with which to prove completeness. We begin by making the following observations about in relation to (()), (()) and (()). This extends similar observations made in [1] about (()), (()) and (()).
Proposition 6.4.
The following hold for arbitrary bases and atomic multiset , for arbitrary and in :
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iff
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2.
iff for all , atomic multisets and atomic , such that and then,
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iff for every base , atomic multiset , and atom , if for every base such that implies then .
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4.
iff for all atomic multisets and atoms we have .
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5.
iff implies .
Proof 6.5.
Let us fix arbitrary base and atomic multiset .
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1.
Showing both sides separately:
- •
- •
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2.
Once again, showing both sides separately:
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•
Right to left: Given that for all , atomic multisets and atomic , such that and then, then we want to show that . In particular our hypotheses hold in the case where and and . Thus our hypothesis gives us that:
We show the case for the first relation as the second follows similarly. To apply ((App)) with the rule (10) we need to show that we can derive from some context . Since by ((Ref)) we have that then we simply put giving us the correct precondition to use the rule, namely . Thus we get by ((App)) that . Thus we conclude that as desired.
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3.
We have again two directions to consider.
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•
Going right to left, we observe that rule (15) holds for . In this case, we degenerate into the case where the inference, implies is valid. Thus, by restricting our attention to and , by using ((App)) with rule (15) with , we conclude , which in particular holds when , giving our desired conclusion.
- 4.
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5.
Finally we are left to show that iff for all , implies . Going left to right, we are immediately done. Going right to left, start by noting our hypothesis is always true. Another fact that is always true is that given any atomic multiset , it is the case that holds for each by ((Ref)). Therefore, by ((App)) using rule (14), we have that as desired.
This completes the proof of all statements.
Lemma 6.6.
For every , and atomic multiset , we have that iff .
Proof 6.7.
We begin by fixing an arbitrary , atomic multiset and prove by induction on the structure of . The cases where the topmost connective of is (()) and (()) and where is atomic and (()) are considered in [1]. So we consider the cases where the topmost connective of is (()), (()), (()) and where is (()) and (()). We take these cases in turn:
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is of the form . Then follows immediately by Proposition 6.4.
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Finally, is of the form . Then also follows immediately by Proposition 6.4.
This concludes our induction on and thus our proof of Lemma 6.6.
Lemma 6.8.
For every atomic multiset and atom , if then .
Proof 6.9.
We begin by noting that by the definition of it suffices to show the following:
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If and for and then .
The first point holds true by the (Ax) rule of our natural deduction system. The second point requires analysis of the rules in . Luckily, we know them all, and thus all that is left to show is that for each rule and given sufficient premiss, that we can conclude as expected. For rules (1) to (6), we again cite [1] as those cases are considered there. Thus we only consider rules (7) to (18) below:
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Suppose is in . Then we want to show that given and , we conclude that . By the definition of , our hypothesis is eqivalent to and which by (-I), gives that .
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8.
Suppose is in . Then we want to show that given we can conclude . By the definition of , our hypothesis is eqivalent to . Thus by (-E), we conclude crucially that .
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9.
Suppose is in . Then we want to show that given we can conclude . By the definition of , our hypothesis is eqivalent to . Thus by (-E), we conclude crucially that .
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10.
Suppose is in . Then we want to show that given that we can prove . By the definition of , our hypothesis is eqivalent to . Thus, by (-I) we conclude that .
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11.
Suppose is in . Then we want to show that given that we can prove . By the definition of , our hypothesis is eqivalent to . Thus, by (-I), we conclude that .
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12.
Suppose is in . Then we want to show that given:
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we can prove that .
By the definition of , our hypothesis is eqivalent to-
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.
Thus, by (-E), we conclude that .
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13.
Suppose for all , is in . Then we want to show that given for each and we can conclude for all atomic . By the definition of , this follows immediately by (-E).
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14.
Suppose for all , is in . Then we want to show that given for each , then for any . This holds by immediately by (-I).
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15.
Suppose for all , is in . Then we want to show that for any , given
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For that
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For that
that it follows that . By the definition of , our hypothesis is eqivalent to, for any
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For that
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For that
which by (-Promotion) gives that , as desired.
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•
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16.
Suppose is in . Then we want to show that given and then . By the definition of , our hypothesis gives that and . Then by (!-Dereliction) it follows that as desired.
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17.
Suppose is in . Then we want to show that given and then . By the definition of , our hypothesis gives that and . Then by (!-Weakening) it follows that as desired.
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18.
Suppose is in . Then we want to show that given and then . By the definition of , our hypothesis gives that and . Then by (!-Contraction) it follows that as desired.
This completes the case analysis for establishing Lemma 6.8.
We can now prove the statement of completeness, namely that if , then .
Proof 6.10 (Proof (Theorem 6.1)).
Given that , we begin by fixing some arbitrary base with atomic multiset such that (which give us the hypothesis for using ((Inf))). Since Lemma 6.6 is bidirectional, we have that and thus we get that . Therefore, it must hold that from which we get by ((Inf)). By Lemma 4.4, we get from that holds. Thus, by Lemma 6.8, we have that . Finally, by the definitions of and , we get can unflatten this equation giving which is nothing more than as desired.
7 Comments on the semantics
The work of Gheorghiu, Gu and Pym in [1] provided a solid framework to begin developing a proof-theoretic semantics for Intuitionistic Linear Logic. The extension to include the additives was, to some degree, a natural extension of their work. The real difficulty was in finding a clause for (()) and indeed, it turned out that what was needed was not only a clause of (()) but also an understanding of what it means to use a formula as hypothesis. This gives us a semantics which is considerably more complex than that for the multiplicative fragment. In this concluding section, I wish to dedicate some time to trying to understand the clauses for (()) and ((Inf)) better and to try and discuss how this relates to what we already know about Intuitionistic Propositional Logic. To better understand how (()) works, we will focus on two theorems of ILL:
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, for arbitrary many
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2.
if and only if
We consider these identities in turn. The first identity appears to be a complete violation of any “resource” reading of the logic. If one were to naïvely read the sequent, one would be convinced that should be interpreted as a “source” of formulas . This reading is a good intuition to have for what it means to infer from a formula , but doesn’t give any hints as to what it should be to infer . In fact, one can show that this identity is equivalent to for arbitrarily many (but non-zero) in the antecedent and in the succedent. Knowing this, we are justified in concluding that the formula when considered as hypothesis needs to behave much closer to how hypotheses in IPL behave; that is to say, that they are sensitive to structural notions that have generally been rendered inaccessible by our semantics. We observe, by unpacking the meaning of this sequent, that in some sense, this sensitivity is captured by our ((Inf)) clause. The sequent is equivalent to saying that for all bases such that if then it follows that . Let us show this explicitly. As a consequence of Lemma 4.7, we know that it is the case that is always valid. Since we have then we have also , so by Lemma 4.6 we obtain as desired.
So what is the obstruction to such an inference holding in the case of ? If we expand again by ((Inf)), we now have that if for all bases and multisets , such that if then it follows that . But this is clearly not the case! In fact, the best we can do is to obtain where is the number of ’s in the succedent. Thus, we see that here, the multiset is our “obstruction” to structurality. Until now we have not made explicit mention of what the role of the multiset of atoms is, but the answer is clear from the definition of ((At)); namely, the multiset of atoms carries the atoms required to end atomic deductions, which we may call terminating atoms. The issue as observed, is that formulas not marked with a (()) in the hypothesis of a sequent, may introduce rules which when supporting inference of the immediate subformula, require the use of terminating atoms. On the other hand, when considering a hypothesis by a formula marked with a (()) in the hypothesis, we must consider extensions by rules which are in some sense, self-contained, that is to say, extensions that support the immediate subformula without any terminating atoms and that these rules alone must suffice. This is an important point and indeed a significant departure from how the ((Inf)) rule might have been expected to behave. Nevertheless, it is a feature of the logic that we are studying that (()) affects inference so. Thus, the idea that extensions by rules which only give self-contained proofs of a formula being at the heart of the meaning of inferring from the formula , allows us to conclude that, at least in the present setup, substructurality is captured by considering bases with rules that allow derivations that require terminating atoms and demanding that all such atoms be used in atomic derivations (this is a consequence of the ((Ref)) rule).
We claimed above that in some sense, the statement is self-contained. Let us try and make this precise. If, for example, we consider the first identity in the atomic case, namely that , for arbitrary many , we see that our inference rule demands from us that any base such that is valid, must suffice to prove the arbitrary conjunction. We clearly see that the class of valid such bases is dictated by the fact that must hold. If we imagine what rules may therefore contain, we quickly conclude that it indeed needs to be the case that we are able to derive some atom(s), not necessarily , without the assistance of terminating atoms. In other words, we will need some axiom rules in the base, i.e. rules of the form for some atom .
So we may perhaps be convinced that (()) as hypothesis, should behave as described. We must now answer what it means for to hold. Why is it that we cannot simply say that iff and , as it is used ((Inf))? The answer, perhaps unsurprisingly, is that this, and similar clauses, fail to be sound. Precisely, clauses of this form fail to pass the case when in the elimination rule lemmas, i.e. Lemma 4.7, Lemma 4.9 and Lemma 4.11. The reason these fail worth investigating. Since the reason is always the same, so let us consider the case of Lemma 4.7. We have as hypotheses that
and we want to show that . In this setup, if we define iff and , we can show by induction on the structure of that obtains and that but we are left with showing that . We cannot, a priori, demand this so this lemma fails to hold, implying that the resultant semantics would fail to be sound. Therefore, this clause cannot be the defining clause of (()) leaving us to ask what the form of the clause for (()) should be? A quick way to answer this is to appeal to the second identity, i.e. that if and only if . Now that we understand how ((Inf)) works, this distributivity law allows us to get a good idea of what the definition of (()) should be. Expanding gives us that we want to be equivalent to considering all bases , atomic multisets and atoms such that if then . Let us consider the additive identity and set ; since we know from the logic that , if we identify these two formulae and then expand according to ((Inf)), we obtain our clause directly. This therefore suggests that a clause of the form iff for all bases , atomic multisets and atoms , if then should work, and indeed, as we have seen, it does. Thus again, we see that the key to obtaining this clause came from our understanding of ((Inf)). This plays hand in hand with our developing understanding of base-extension semantics, in that the meanings of the connectives seem to be heavily determined by the form of the ((Inf)) rule [23] and not only their clauses as one might have thought. This highlights that indeed the proof-theoretic meaning of a formula is more governed by the way it is used as hypothesis and not simply by its structural properties, or its “promotability”, as one might have thought, giving credence to the claim made in the introduction that the behaviour of as hypothesis is in fact the sine qua non of our treatment of (()).
To conclude, I would like to discuss the prospects for the embedding of the semantics for IPL in our semantics for ILL. We know from [16, 12, 14] that there are many possible translations of formulas from IPL to ILL, called Girard translations. We present a particular one below:
Definition 7.1.
The mapping can be defined as follows:
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1.
, where is a propositional atom
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2.
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3.
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4.
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Girard, in [16], says that the crux of the translation is the following: is intuitionistically provable if and only if is linearly provable, a relevant proof of which can be found in [14]. This translation ties in to our intuition that structurality in the hypothesis of a sequent is really properly represented by our treatment of (()) in ((Inf)). Since we have a sound and complete P-tS for IPL and now a sound and complete P-tS for ILL, we therefore know that we can always map any valid IPL sequent (using for the support relation of Sandqvist’s semantics in [6]) to a valid sequent in ILL of the form , and vice-versa. This result is interesting as it gives a way of analysing valid sequents of IPL in the framework we have setup for ILL which is certainly not without its quirks (for example, consider how disjunction maps over!). However, a natural question to ask would be how may one generalise this mapping? What if we were given a formula and base in which inference of the formula is supported i.e. , and wanted to try and understand it in the linear setting, i.e. to find a multiset and a base such that the support relation now holds? Whilst it is obvious that the formula is mappable directly, we are then stuck with how to obtain and , as rules and atomic derivability in the two semantics are quite differently behaved. At present, it is not clear to me how this mapping should be done, though I do believe such a mapping between the semantics of Sandqvist and ours for ILL is possible. However, I believe that instead, the correct approach to take if we are committed to this line of investigation, is to define a support relation for IPL that is in some sense much closer to ours for ILL, whose treatment of formulas is closer to our own and whose atomic derivability relation mirrors ours in how it uses rules of the base, and whose base rules may also include the additional structure that ours do. Whilst it can easily be shown that one can have a sound and complete P-tS for IPL which keeps track of atoms in much the same way as ours for ILL does, I have had no luck in finding a way of constructing such a mapping. I thus leave this problem open for further study.
Acknowledgements
I would like to thank Timo Eckhardt, Alex Gheorghiu, Tao Gu, Victor Nascimento, Elaine Pimentel, Katya Piotrovskaya and David Pym for our many discussions on P-tS for ILL, for all your suggestions and help with writing this manuscript. I am grateful to you all!
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