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## Abstract

We will present a three-valued consequence relation for metainferences, called MI, defined in terms of ST and TS, two well known inferential consequence relations. MI has no valid metainference, and thus no valid inference either. Notwithstanding , MI is informative about the meta-metainferential level. Afterwards, we will introduce a hierarchy of consequence relations MI n for metainferences of level n (for 1 ≤ n < ω). Every metainference of level n or less is invalid in each MI n. Nevertheless, each MI n is informative about the metainferences of higher levels. Finally, we will present a logic L empty , based on the hierarchy of logics MI n , that is genuinely empty, in the sense that it has no valid metainference of any level whatsoever.
Empty Logics
Federico Matias Pailos
Abstract
We will present a three-valued consequence relation for metainferences, called
MI, deﬁned in terms of ST and TS, two well known inferential consequence rela-
tions. MI has no valid metainference, and thus no valid inference either. Notwith-
standing, MI is informative about the meta-metainferential level. Afterwards, we
will introduce a hierarchy of consequence relations MInfor metainferences of level
n(for 1 n < ω). Every metainference of level nor less is invalid in each MIn.
Nevertheless, each MInis informative about the metainferences of higher levels.
Finally, we will present a logic Lempty, based on the hierarchy of logics MIn, that
is genuinely empty, in the sense that it has no valid metainference of any level
whatsoever.
KEYWORDS: Logic, Metainferences, Metainferential Validity, Substructural
Logics, Empty Logic
1 Introduction
A logic is ordinarily presented as a consequence relation applied to a particular lan-
guage1. Informally, a consequence relation is a standard that an inference may, or may
not, live up to. The following quote by Rohan French assesses this issue.
Suppose that you... thought that logic was only the study of logical truths,
of determining which formulas are tautologies... Then one would be moved
to say that Strong Kleene logic -—the logic according to which a sentence is
a logical truth if it gets the value 1 on every valuation which assigns values
in the set {1, i, 0}to the propositional atoms and calculates the truth-values
of components... is not a logic... Of course, as we well know, while Strong
Kleene has no tautologies it does have many valid arguments... What I want
to propose here is that we make a similar move... While [the non-reﬂexive
sequent calculi] LKR does not have any valid sequents, it does have a great
many valid metasequents, higher-order sequents which have sets of sequents
as their premises and sequents as their conclusions. That is to say, while
LKR and other irreﬂexive systems have no valid inferences they do have a
great many valid metainferences. ([12]: p. 120.)
1As Priest notices in [17], logic is an ambiguous term. It can both refer to the theory of an investi-
gation, or to the subject matter of that survey. Throughout this work, we will use the word logic only
to refer to a logical theory
1
the semantic counterpart of metasequents.3In that quote, French suggests that being
informative about metainferences is enough for a consequence relation to specify a logic
(e.g., given a language). In fact, LKR–e.g., LKR without the truth rules– has no
valid inference.4But if consequence relations don’t need to validate inferences to specify
a logic, do they really need to validate some metainferences? Could there be a logic
without valid metainferences?
We will give a positive answer to this question, and prove our point with an example.
Such logic, called MI, includes a consequence relation, a certain standard, that metain-
ferences, may or may not meet. In fact, no metainference will satisfy that standard.
Thus, no metainference will be valid in MI.
Notwithstanding, it seems reasonable to expect that a respectable logic establishes
that some things are right –or valid– and others are wrong -or invalid. This means that
such logic is informative about sentences, inferences or metainferences. More precisely,
a logic Lis informative about some set of things Sif and only if, for at least some
aS,Lvalidates a, and for at least some bS,Linvalidates b. Fortunately, we will
see that MI is informative about meta-metainferences, or (as we will also call them)
metainferences of level 2 -e.g., informally, inferences that have sets of metainferences as
premises, and a metainference as a conclusion.
Moreover, the path that French describes can be extended indeﬁnitely. It starts
with the Strong Kleene logic K3, a logic without tautologies, and follows with LKR,
a logic without tautologies and without valid inferences that is sound and complete with
respect to the non-reﬂexive semantic logic TS.5The logic we will introduced, MI, not
only has no valid inferences, but also has no valid metainferences. We will see in what
precise sense MI looks a lot like TS.
So far, we have talked about metainferences as inferences from a set of inferences to
a particular inference. We have also talked about metametainferences as inferences that
have a set of metainferences as premises and a metainference as a conclusion. Thus,
it will be extremely advantageous to have a comprehensive label for all these things.
Thus, we will talk about metainferences as a general term applying to all those things,
and assign to each one of them a particular level. What we have called metainferences,
then, will be, strictly speaking, metainferences of level 1, or metainferences1.Meta-
2The non-reﬂexive logic LKR is the result of dropping the Identity axiom-scheme from a standard
presentation of Gentzen’s multiple-conclusion sequent calculus for classical logic, G1c, and adding the
bottom-up (e.g, invertible) versions of G1c’s operational rules, plus two rules for the truth predicate.
In this paper we will not be dealing with truth theories, but only with logics. Thus, out primary interest
will not be LKR, but the calculus obtained from LKR by dropping the rules for the truth predicate.
For a presentation of G1c, see [25]: p. 52).
3Metainferences can be obtained from metasequents by replacing the turnstile of the sequent calculus
for a double turnstile that indicates a semantic consequence relation. In Section 2 we will present a
formal deﬁnition of them.
4Though if propositional constants for truth values –with a suitable set of new rules for them– are
added, then the new system will have valid sequents. Every logic we will talk about in the rest of the
paper is based on a language that lacks these truth-value constants.
5We will explore TS in detail in the next Section.
2
metainferences will be, then, metainferences of level 2, or metainferences2, and so
on.
We will prove that, for every level nof metainferences, there is a consequence relation
that invalidates every sentence, inference and metainference of level nor less. Moreover,
that logic is informative about metainferences of higher levels.
This paper is structured as follows. In Section 2 we introduce the distinction between
inferences and metainferences, along with the notion of metainferential validity. We will
also present two substructural logics, ST and TS, that will be used to deﬁne the logic
MI. In Section 3 we present MI, a metainferential consequence relation that invalidates
every metainference (of level 1). In Section 4 we introduced MI2, a consequence relation
for metainference of level 2. No meta-metainference is valid in MI2. This result is
pushed one step forward in Section 5. There, we deﬁne a hierarchy of logics MIn. Each
MIninvalidates every metainference of level nor less. In Section 6, we introduce an
empty logic Lempty.Lempty is deﬁned with the resources provide by the hierarchy of
logics MIn. No metainference of any level is valid in Lempty. Finally, in Section 6,
we oﬀer some concluding remarks, and point to some directions in which the present
explorations can be further developed.
2 Metainferences, metainferential validity and two sub-
structural logics
In the next section, we will introduce the target consequence relation for metainferences,
MI.MI invalidates every sentence, inference and metainference of level 1. Nevertheless,
it is informative about metainferences of higher levels. But MI is not a standard
consequence relation. It does not relate sets of formulas of a language L, but sets of
inferences. But before introducing MI, it will be necessary to give some deﬁnitions.
In particular, we will need to make a clear distinction between standard inferences (as
relations between sets of sentences) and metainferences. Moreover, we must clarify
what we mean by metainferential validity. Finally, we will introduce in some detail two
important substructural logics: ST and TS.
2.1 Inferences and Metainferences
To understand and carry on our investigation, it will be essential to have an accurate
To analyze these matters, it will be useful to ﬁx some terminology. Let Lbe a
propositional language, such that FOR(L) is the absolutely free algebra of formulae
of L, whose universe we denote by F OR(L). As usual, we will let Γ,,and other
Greek capital letters represent ﬁnite sets of formulae. Finally, A,B,Cand Drepresent
formulae.6
6For the sake of simplicity, we will present single-conclusion logics. Moreover, the consequence
relation will be deﬁned for inferences with a ﬁnite set of premises. Nevertheless, we hope that the
results that we will introduce can be safely extended to multi-conclusion versions of these logics, that
3
Deﬁnition 2.1. A Tarskian consequence relation over a propositional language Lis
a relation (F OR(L)) ×F OR(L)obeying the following conditions for all A
F OR(L)and for all Γ,F OR(L):
1. ΓAif AΓ(Reﬂexivity)
2. If ΓAand ΓΓ, then ΓA(Monotonicity)
3. If Aand ΓBfor every B, then ΓA(Cut)
Additionally, a (Tarskian) consequence relation is substitution-invariant whenever if
ΓA, and σis a substitution on FOR(L), then {σ(B)|BΓ}σ(A).
Deﬁnition 2.2. A Tarskian logic over a propositional language Lis an ordered pair
(FOR(L),), where is a substitution-invariant Tarskian consequence relation.
Notwithstanding, many philosophers have argued against the Tarskian conception
of logic. For example, [22], [2] and [21] argue for the generalization of the Tarskian
account multiple consequences. [2] and [13] give reasons for relaxing the Monotonicity
condition. Moreover, [15] and [11] argued for a generalization or liberalization that
allows logics to drop Reﬂexivity, Cut, or both of them. In this context is where logics
like the non-reﬂexive TS of the non-transitive ST where introduced.
These modiﬁcations, in turn, bring on the possibility of a shift in the nature of the
collection of formulae featured in the consequence relation. Thus, instead of treating
logical consequence to hold between (sets of) formulae, it may hold between labeled for-
mulae, multisets of formulae (where repetition matters), sequences of formulae (where
order makes a diﬀerence), etc. As we already mention, our target logic, MI, explicitly
moves from understanding logical consequence as a relation between collections of for-
mulae, to conceiving it as a relation between collections of inferences. Here, we should
understand inferences as an ordered pair whose ﬁrst member is a set (or multiset, or
sequence, etc.) of formulae, and a set of formulae as its second term. A metainference,
then, is an ordered pair whose ﬁrst member is a set (or multiset, etc.) of inferences, and
a single inference as its second term. This is a relatively standard move in the literature.
[1] , ﬁrst, and [4] , afterwards, discuss a generalization of the Tarksian account that al-
lows moving to logical consequence relations that do not hold only between collections
of formulae, but between objects of other nature.7Thus, more precisely:
Deﬁnition 2.3. A inference on Lis an ordered pair , A), where ΓF OR(L)and
AF OR(L)(written ΓA). SEQ0(L)is the set of all inferences or sequents on L.
Deﬁnition 2.4 ([8]).A metainference on Lis an ordered pair , A), where Γ
SEQ0(L)and ASEQ0(L)(written Γ1A). SEQ1(L)is the set of all metainfer-
ences on L.
admit also arguments with inﬁnite premises.
7Moreover, [1] suggested that the idea that logical consequence can be said to hold of relata other
than formulae is very reasonable to those used to sequent calculus systems —and, most prominently,
with substructural sequent calculi.
4
We will say, accordingly, that the following one on the left is a inference, whereas
the one on the right is a metainference
ΓA
BΣC
ΓA
Therefore, we will distinguish two diﬀerent kinds of consequence relations: the in-
ferential and the metainferential.MI –our target logic– is a consequence relation of this
last type. In the next subsection we will give a precise characterization of this notion.
2.2 Metainferential validity
Put it informally, a metainference (or, as we will also call it, a metainference of level
1), is a pair of sets of inferences –as its ﬁrst member– and an inference –as its second
member. Every metainference will have the following structure:
Γ11, ...Γnn
ΣΠ
We will call every Γii, a premise of the metainference, while Σ Π will be its
conclusion.
Thus, a metainference is a inference-like structure. Thus, it makes sense to ask when
will a metainference Γ11, ...ΓnnΣΠ be valid in a particular propositional
logic L.
Deﬁnition 2.5. A metainference Γ11, ...ΓnnΣΠis valid in (a proposi-
tional logic) Lif and only if, for every valuation v, if vsatisﬁes every Γiiaccording
to L, then vsatisﬁes ΣΠaccording to L.
A valuation vsatisﬁes or conﬁrms a inference Γ ∆ in a speciﬁc logic Lif and only
if vis not a counterexample to Γ ∆’s validity in L.
Dicher and Paoli [8] have named this a local conception of metainferential validity.8
As it stands, the deﬁnition speciﬁes a standard of validity for metainferences –e.g.,
instances of metainferential schemes– in a particular logic. Nevertheless, it can –and
will– be also used to speciﬁed when a metainferential scheme is valid. In a nutshell, a
scheme is valid in Lif and only if every instance of it is valid.
Our target logic, MI, is a consequence relation for metainferences. But to under-
stand exactly how it works, we need ﬁrst to introduce two inferential but non-Tarskian
consequence relations: the non-reﬂexive TS and the non-transitive ST.9
see [8].
9Those readers that are already familiar with these logics may safely skip this part.
5
2.3 ST and TS: two substructural consequence relations
ST and TS are label as substructural because they give up at least one structural
feature of a Tarskian consequence relation. More speciﬁcally, ST drops Cut, while TS
abandons Reﬂexivity.
The logic ST can be portrayed as a p-logic. It was devised by Frankowski in [10] as
a means to characterize logical systems where valid derivations are such that the degree
of strength of the conclusions can be smaller than that of the premises.10
Deﬁnition 2.6 ([11]).Ap-consequence relation over a propositional language Lis
a relation (F OR(L))× ⊆ (F OR(L)) obeying the following conditions for all
AF OR(L)and for all Γ,F OR(L):11
1. ΓAif AΓ(Reﬂexivity)
2. If ΓAand ΓΓ, then ΓA(Monotonicity)
Additionally, a p-consequence relation is substitution-invariant whenever if ΓA,
and σis a substitution on FOR(L), then {σ(B)|BΓ}σ(A).
Deﬁnition 2.7 ([11]).Ap-logic over a propositional language Lis an ordered pair
(FOR(L),), where is a substitution-invariant p-consequence relation.
In general, p-logics (q-logics) can be connected to p-matrices (q-matrices). ST can
be represented as a p-matrix logic associated to the 3-element Kleene algebra.
Deﬁnition 2.8 ([11]).For La propositional language, an L-p-matrix is a structure
hV,D+,D,Oi, such that hV ,Oi is an algebra of the same similarity type as L, with
universe Vand a set of operations O, where D+,D⊆ V and D+⊆ D.
Deﬁnition 2.9. The 3-element Kleene algebra is the structure
K=h{1,1
2,0},{f¬
K, f
K, f
K}i
where the functions f¬
K, f
K, f
Kare as follows
f¬
K
1 0
1
2
1
2
10
f
K11
20
1 1 1
20
1
2
1
2
1
20
00 0 0
f
K11
20
1 1 1 1
1
211
2
1
2
011
20
Moreover, the functions and are deﬁnable via the usual deﬁnitions.
Deﬁnition 2.10 ([19]).A 3-valued ST-matrix is a p-matrix
MST =h{1,1
2,0},{1},{1,1
2},{f¬
K, f
K, f
K}i
such that h{1,1
2,0},{f¬
K, f
K, f
K}i is the 3-element Kleene algebra.
10For an extensive presentation of ST, see but also [6], [7] , [19] and [20].
11Once again, we will give a single-conclusion presentation of these logics.
6
Now, as is common practice, semantic structures such as p-matrices induce con-
sequence relations and, therefore, logics, with the help of valuation functions, e.g.,
homomorphisms from FOR(L) to the set of truth-values of the semantic structure
in question—in this case, the set {1,1
2,0}. Valuations are extended from propositional
variables to complex formulae with the help of the truth-functions for the connectives; in
this case the functions given by the 3-element Kleene algebra. In our particular case, we
can deﬁne what a valid inference or sequent is in any p-matrix logic—and, therefore, in
ST—in the following straightforward manner. Notice that, below, Mis a substitution-
invariant p-consequence relation, whence (FOR(L),M) is a a p-logic. In addition to
that, when some p-logic Lis induced by p- matrix M, we may interchangeably refer to
Mas L.
Deﬁnition 2.11. For Map-matrix, an M-valuation vsatisﬁes a sequent or inference
ΓA(written vMΓA) if and only if, if v[Γ] ⊆ D+, then v(A)∈ D. A sequent
or inference ΓAis M-valid (written MΓA) if and only if vMΓA, for all
M-valuations v.
But a more straightforward characterization of ST’s validity is the following one:
Deﬁnition 2.12. ST ΓAif and only if for every valuation v, if v(B)
{1}for all BΓ, then v(A)∈ {1,1
2}
Another12 13 interesting generalization of Tarskian consequence relations is the no-
tion of q-consequence relation, due to [15].14
Deﬁnition 2.14 ([15]).Aq-consequence relation over a propositional language Lis
a relation (F OR(L))× ⊆ (F OR(L)) obeying the following conditions for all
A, B F OR(L)and for all Γ,F O R(L):
12This deﬁnition can be adjusted to a multi-conclusion setting:
Deﬁnition 2.13. ST Γif and only if for every valuation v, if v(B)∈ {1}for all BΓ, then
for all A, v (A)∈ {1,1
2}
13Moreover, there is a third way to present ST’s validity, that requires to talk about strict and tolerant
satisfaction or truth. (In fact, ST stands for Strict-Tolerant, just as TS stands for Tolerant-Strict.) A
valuation vsatisﬁes tolerantly a formula Aif and only if v(A)∈ {1,1
2}, and satisﬁes it strictly if and
only if v(A)∈ {1}. A valuation vsatisﬁes an inference Γ Ain ST if and only if, if vstrictly satisﬁes
every BΓ, then vtolerantly satisﬁes A. Finally, an inference from Γ to Ais valid if and only if, for
every valuation v, if vsatisﬁes strictly every BΓ, then vsatisﬁes tolerantly A. Nevertheless, it is
worth mentioning that ST’s supporters themselves prefer to talk of strict and tolerant assertion rather
than talking about strict and tolerant satisfaction, or strict and tolerant truth. As [9] explains, the
reason why they use the idea of strict and tolerant assertion instead of either of the last two (pair of)
notions, is to avoid revenges paradoxes related to the notions of strictly true and strictly false.
14[23] claims that the relation of q-logic is devised to qualify as valid derivations of true sentences from
non-refuted premises (understood as hypotheses), whereas the notion of p-logic is devised to qualify as
valid derivations of conclusions whose degree of strength (understood as the conviction in its truth) is
smaller than that of the premises.
7
1. If ΓAand Γ, then A(Monotonicity)
2. Γ∪ {A|ΓA}Bif and only if ΓB(Quasi-closure)
Deﬁnition 2.15 ([15]).Aq-logic over a propositional language Lis an ordered pair
(FOR(L),), where is a substitution-invariant q-consequence relation.
Notice that while q-logics may fail to validate Reﬂexivity, p-logics might fail to
validate Cut and, thus, if they do, both are non-Tarskian or substructural logics.
Deﬁnition 2.16 ([15]).For La propositional language, an L-q-matrix is a structure
hV,D+,D,Oi, such that hV ,Oi is an algebra of the same similarity type as L, with
universe Vand a set of operations O, where D+,D⊆ V and D+∩ D=.
A 3-valued q-matrix logics associated to the 3-element Kleene algebra that we would
like to present in connection to our ongoing investigation is the logic TS.
Deﬁnition 2.17 ([19] , [16]).A 3-valued TS-matrix is a q-matrix
MTS =h{1,1
2,0},{1},{0},{f¬
K, f
K, f
K}i
such that h{1,1
2,0},{f¬
K, f
K, f
K}i is the 3-element Kleene algebra.
TS is discussed by e.g. [6], and also by [5] in the context of the more general dis-
cussion of what represents a proper consequence relation between formulae. Moreover,
it was also discussed by [15] as a tool to model empirical inference with the aid of the
3-valued Kleene algebra. More recently, it was also stressed by Rohan French in [12] ,
in connection with the paradoxes of self-reference.
Now we need to clarify how q-matrix logics validate or invalidate both inferences and
metainferences —following e.g. [24] . Notice that, below, Mis a substitution-invariant
q-consequence relation, whence (FOR(L),M) is a q-logic. In addition to that, when
some q-logic Lis induced by a q-matrix M, we may interchangeably refer to Mas L.
Deﬁnition 2.18. For Maq-matrix, an M-valuation vsatisﬁes a sequent or inference
ΓA(written vMΓA) if and only if, if v[Γ] ∩ D=, then v(A)∈ D+. For
Maq-matrix, an inference ΓAis M-valid (written MΓA) if and only if
vMΓA, for all M-valuations v.
Deﬁnition 2.19. For Maq-matrix, an M-valuation vsatisﬁes a metainference Γ1
A(written vMΓ1A) if and only if, if vMB, for all BΓ, then vMA. A
metainference Γ1Ais M-valid (written MΓ1A) if and only if, if vMB, for
all BΓ, then vMA, for all M-valuations v.
The following is a simpler characterization of TS’s inferential validity:
Deﬁnition 2.20. TS ΓAif and only if for every valuation v, if v(B)∈ {1,1
2}
for all BΓ, then v(A)∈ {1}
8
From these deﬁnitions15 of ST and TS’s validities, the following facts follow.
Fact 2.22 ([7]). TS is a non-reﬂexive, and thus a substructural, logic.
Fact 2.23 ([7]). ST is a non-transitive, and thus a substructural, logic.
Fact 2.24 ([12]). TS has no valid inferences or sequents.
Fact 2.25 ([14], [19], [20]). ST and classical propositional logic CL have the same set
of valid inferences or sequents.
It is worth remembering that TS is sound and complete with respect to LKRinv
–e.g. LKR plus inversible rules for the operational part of the calculi, minus the truth
rules.16
3 MI, a logic for metainferences
We will now present MI, a logic that invalidates not only every inference –as TS does–,
but also every metainference (of level 1). To achieve this goal, MI’s consequence rela-
tion behaves pretty much as the inferential consequence relation TS does. Remember
that an inference is valid in TS if and only if, for every valuation v, if the premises
satisfy certain standard, then the conclusion meet some more demanding standard. In
particular, Γ Bis satisﬁed by a valuation if and only if, if for every premise AΓ,
v(A) = 1,1
2–e.g., if vtolerantly satisﬁes every premise–, then v(B) = 1 –e.g., vstrictly
satisﬁes the conclusion. Conversely, if vtolerantly satisﬁes every premise, but does not
strictly conﬁrms the conclusion, then vis a counterexample to the validity of that in-
ference. As an example, Identity AA– is not valid in TS because there is at least
some formula Asuch that for some valuation v,v(A) = 1
2.
MI will work in a similar way, but in the metainferential level. Thus, MI’s standard
for the premises will be less demanding than the one for the conclusion.
Deﬁnition 3.1. A metainference Γ1A1, ...ΓnAnΣBis valid in MI if and
only if, for every valuation v, if every ΓiAiis satisﬁed by vaccording to ST, then v
satisﬁes ΣBaccording to TS.
Fact 3.2. Therefore, no metainference Γ1A1, ...ΓnAnΣBis valid in MI.
15This deﬁnition can also be adjusted to a multi-conclusion setting:
Deﬁnition 2.21. TS Γif and only if for every valuation v, if v(B)∈ {1,1
2}for all BΓ,
then for all A, v (A)∈ {1,1
2}
16About these logics we shall mention that in [3] , [8] and [18] it is shown that —through some
suitable translation— the set of valid inferences in LP coincides with the set of valid metainferences
in ST, while in [12] it is conjectured that —again, through some suitable translation— the set of valid
inferences in K3, e.g., Strong Kleene logic, coincides with the set of valid metainferences in TS.
9
Proof. Consider a valuation vsuch that, for every propositional letter pthat appears
in Γ1A1, ...ΓnAnΣB,v(p) = 1
2. As MI matrices for the logical constants
are the ones of K3,vgives the value 1
2to every formula Ain the metainference. Thus,
every premise ΓiAiis satisﬁed by vaccording to ST, but vdoes not conﬁrm the
conclusion Σ Baccording to TS.
Therefore, MI does not validate any metainference whatsoever, including those with
an empty set of premises –e.g., the inferences. Still, this does not mean that MI does
not validate metainferences of higher levels. Moreover, MI is informative about the
next metainferential level: MI validates some, but not every, meta-metainference –e.g.,
metainferences of level 2.
Ameta-metainference is an inference with the following structure: ΓΓ1, ..., ΓΓn
∆∆. Each ΓΓi, and also ∆∆, is a metainference, and represents a consequence
relation between metainferences.
Deﬁnition 3.3. A meta-metainference ΓΓ1, ..., ΓΓn∆∆ is valid in MI if and only
if, for every valuation v, if if vsatisﬁes every ΓΓiaccording to MI, then vsatisﬁes ∆∆
according to MI.
Deﬁnition 3.4. Thus, a meta-metainference ΓΓ1, ..., ΓΓn∆∆ is valid in MI if
and only if, for every valuation v, if [for every ΓΓi, if vsatisﬁes every premise of ΓΓi
according to ST, then vsatisﬁes ΓΓi’s conclusion according to TS], then [if vsatisﬁes
every premise of ∆∆ according to ST, then vsatisﬁes ∆∆’s conclusion according to
TS].
Remark 3.5. More generally, if L1is an inferential consequence relation, a metain-
ference of level nis valid according to L1if and only if, for every valuation v, if every
premise is satisﬁed by vaccording to L1, then its conclusion is satisﬁed by vaccording
to L1. And for every i < j, if L2is a metainferential consequence relation of level i,
a metainference of level jis valid according to L2if and only if, for every valuation v,
if every premise is satisﬁed by vaccording to L2, then its conclusion is satisﬁed by v
according to L2.
With this deﬁnition at hand, we can prove the informativeness of MI.
Proof. We will give an example of a valid meta-metainference ΓΓ1, ..., ΓΓn∆∆,
and also an example of a invalid meta-metainference ΓΓ1, ..., ΓΓn∗ ≫ ∆∆, both
according to MI. Meta-metainference (1) is an MI’s invalid meta-metainference, while
metainferences (2) and (3) are examples of a valid meta-metainferences in MI.
AA
AA
(1) AA
AB
AA
AA
(2) AA
AA
10
AA
AA
(3) AA
AA∨ ¬A
To see why (1) is invalid it is enough to consider an instance of it such that every
formula that appears in the metainference is atomic. A valuation vsuch that v(A) = 1
and v(B) = 0 will do the rest of the job. Notice that vsatisﬁes the premise of the
meta-metainference according to MI. And this happens because vsatisﬁes AA
according to ST and also in TS. Moreover, vsatisﬁes the premise of the conclusion
of the meta-metainference, because vsatisﬁes AAaccording to ST. But vdoes
not satisfy the conclusion of the conclusion of the meta-metainference in MI, because
vdoes not conﬁrm that inference according to TS.
To evaluate what happen with (2) and (3), we need to consider three diﬀerent cases:
(i) valuations vwhere v(A) = 0, (ii) valuations vwhere v(A) = 1
2, and (iii) valuations v
where v(A) = 1. In sub-cases (i) and (iii), vconﬁrms the conclusion of the conclusion
of the metainference (e.g., AA, in (2), and AA∨ ¬A, in (3)) in TS, and thus
satisﬁes the meta-metainference itself according to MI. In (ii), vdoes not satisfy the
premise of the meta-metainference, because vsatisﬁes the premise of the premise in ST,
but does not satisfy the conclusion of the premise according to TS. Thus, vconﬁrms
the meta-metainference according to MI.
4 One step forward: MI2
So far, we know that TS goes one step forward K3, because it not only invalidates
every formula, but also has no valid inference whatsoever. We have also shown that MI
goes beyond TS, because it also invalidates every metainference (of level 1). It seems
natural to wonder whether MI is the limit, or if there is some consequence relation that
not only invalidates every metainference, but also every meta-metainference.
We will present a consequence relation that works exactly like that. The logic that
we will call MI2validates no meta-metainference, but nevertheless is informative about
the meta-meta-metainferential level –e.g., about metainferences of level 3. As MI is
a consequence relation for metainferences –or metainferences of level 1–, MI2is a
consequence relation for meta-metainferences –or metainferences of level 2.
Ameta-metainference ΓΓ1∆∆1, ...ΓΓn∆∆nΣΣ ΠΠ, then, is an inference
that has a set of metainferences as premises, and a metainference as its conclusion. Each
ΓΓi∆∆iis a premise of the meta-metainference, while ΣΣ ΠΠ is the conclusion.
The following is another –and more detailed– way to present the structure of meta-
11
metainferences, that will be used in the rest of the Section.
Γ1
11
1, ...Γ1
j1
j, ...
Σ1
1Π1
1
Γn1
1n1
1, ...Γn1
kn1
k
Σn1
1Πn1
1
Γn
1n
1, ...Γn
jn
j
Σn
1Πn
1
The metainferences above the double line are the premises of the meta-metainference,
while the metainference below the double line is the conclusion. We need now to intro-
duce a new notion. A stage of a metainference of any level nis a line in the metainfer-
ence. Thus, an inference has only one stage, a metainference –of level 1– has two stages
–e.g., one corresponding to the premises and one corresponding to the conclusion–, and a
meta-metainference has four stages –e.g., one for the premises of each premise, a second
one for the conclusions of the premises, a third one for the premises of the conclusion,
and a fourth one for the conclusion of the conclusion. More generally, a metainference
of level nhas 2nstages. Let us see, then, how MI2behaves.
Deﬁnition 4.1. A meta-metainference ΓΓ1, ..., ΓΓn∆∆ is valid in MI2if and only
if, for every valuation v, if vsatisﬁes every ΓΓiaccording to ST, then vsatisﬁes ∆∆
according to MI.
Fact 4.2. Therefore, no meta-metainference ΓΓ1∆∆1, ...ΓΓn∆∆nΣΣ ΠΠ
is valid in MI2.
Proof. Consider a valuation vsuch that, for every propositional letter pthat appears
in ΓΓ1∆∆1, ...ΓΓn∆∆nΣΣ ΠΠ, v(p) = 1
2. As MI2matrices for the
logical constants are K3’s ones, vwill give the value 1
2to every formula Ain the meta-
metainference. Thus, every premise ΓΓi∆∆iwill be satisﬁed by vaccording to ST,
but vwill not satisfy the conclusion ΣΣ ΠΠ according to MI.
Nevertheless, MI2is informative about the next metainferential level. MI2makes
valid some, but not every, meta-meta-metainference –e.g. metainferences of level 3. A
meta-meta-metainference is an inference with the following structure: ΓΓΓ1, ..., ΓΓΓn
∆∆∆. ∆∆∆ and every ΓΓΓiare meta-metainferences, and represents a consequence
relation between meta-metainferences.
Deﬁnition 4.3. A meta-meta-metainference of level 3 ΓΓΓ1, ..., ΓΓΓn∆∆∆ is valid
in MI2if and only if, for every valuation v, if vsatisﬁes every ΓΓΓiaccording to MI2,
then vsatisﬁes ∆∆∆ according to MI2.
Fact 4.4. It is not hard to show that MI2is informative about the meta-meta-metainferences.
Proof. There is at least one MIii’s valid meta-meta-metainference ΓΓΓ1, ..., ΓΓΓn
∆∆∆, and at least one MI2’s invalid meta-meta-metainference ΓΓΓ1, ..., ΓΓΓn∗ ≫
∆∆∆. We will give one example of each kind.
12
AA
AA
AA
AA
(*) AA
AA
AA
AA
AA
AA
AA
AB
(**) AA
AA
AA
AC
The meta-meta-metainferencential scheme (*) will be valid in MI2. Let ΓΓΓ be
any meta-metainference whatsoever. Thus, for every valuation v, either vsatisﬁes ΓΓΓ
according to MI2, or it does not. If it does, then vconﬁrms (*)’s conclusion according
to MI2, and then satisﬁes (*) itself. If it does not, then vdoes not satisﬁes (*)’s premise,
and then conﬁrms (*) according to MI2.
The meta-meta-metainferential scheme (**) will be invalid in MI2. Take a valuation
vsuch that v(A) = v(B) = 1 and v(C) = 0. vwill satisfy (**)’s premise, but will not
satisfy (**)’s conclusion according to MI2.
5 A general result: a hierarchy of logics MIn
In the previous section we have shown that MI is not the limit of the sequence of logics
that begins with K3–e.g., a logic that has no valid sentences– and follows with TS
–e.g., a logic that has no valid inferences. Though MI invalidates every metainference,
MI2also invalidates every metainference of level 2. Nevertheless, MI2is not the limit
of these kind of logics either. In fact, the sequent can be indeﬁnitively extended in
diﬀerent ways. We will sketched what seems to us a natural way to do it.
Fact 5.1. General Result
For any level nof metainferences, there is a consequence relation MIn for metain-
ferences of level n such that there are no valid MIn’s sentences, inferences an metain-
ferences of any level less or equal than n. Nevertheless, any MIn is informative about
metainferences of level n+ 1.
Let TS=MI0and MI=MI1. Both of them invalidate metainferences of level nor
less, but nevertheless they are informative about the next metainferential level. But we
still don’t know how logics MIn–for 2 < n– might look like. We will present a general
deﬁnition of logics MIn, for any nsuch that 2 n.
Deﬁnition 5.2. Let Γ1, ...Γjand be metainferences of level n. For any n, a metain-
ference of level n+ 1,Γ1, ...Γjis valid in MIn+1if and only if, for every valuation
v, if vsatisﬁes every Γ1, ...Γnin MIn1, then vsatisﬁes according to MIn.
13
Once the hierarchy is establish, the ﬁrst thing we need to prove is that each MIn
invalidates every metainference of level n.17
Proof. For any n, let Γ1, ...Γj∆ be a metainference of level n. Let vbe a valuation
such that, for every propositional letter pin Γ1, ...Γj∆, v(p) = 1
2. Then vis a
counterexample to Γ1, ...Γn∆’s validity in MIn+1.
The proof the previous assertion will be an induction on the level nof the logics
MIn. The inductive hypothesis says that for any metainference Γ1, ...Γn∆ of level
n, a valuation vsuch that for every propositional letter pin Γ1, ...Γn∆, v(p) = 1
2, is
a counterexample to the validity according of the metainference in MIn.
Base case: If n= 0, then MIn+1=MI1=MI. We have already shown that a
valuation vthat assigns the value 1
2to every propositional letter pin a metainference
of level 1, Γ1, ...Γn∆, is a counterexample to the metainference’s validity in MI.
Inductive step: If 1 < n, we need to prove that vwill not satisfy any metainference of
level n+ 1 in MIn+1. Take any metainference of level n+ 1, Γ1, ...Γn∆. By inductive
hypothesis, vwill not satisfy ∆ –a metainference of level n– according to MIn. We still
need to prove that vsatisﬁes every Γi(1 in) according to MIn1. Remember
that each Γiis a metainference of level n. Thus, each of its premises is a metainference
of level n1. By inductive hypothesis, vdoes not satisfy them according to MIn1.
Thus, vsatisﬁes each Γi(because it does not satisfy any Γi’s premises) according to
MIn1.
We will now prove that every MInis informative about the metainferences of level
n+ 1. As an example of a valid metainference of level n+ 1, take what may be called an
instance of Meta-Identity of level n+ 1, e.g., a metainference ΣΣ ΣΣ, where ΣΣ is a
metainference of level n. For any valuation v, either vsatisﬁes ΣΣ according to MIn,
or it doesn’t. If it does, then it satisﬁes the conclusion of ΣΣ ΣΣ, and thus satisﬁes
ΣΣ ΣΣ itself. And if it doesn’t, then it does not satisﬁes the premise of ΣΣ ΣΣ,
and so it satisﬁes the metainference itself.
The following one is an example of a metainference of level n+ 1 that is invalid
in MIn. Consider a metainference of level n+ 1, ΓΓ ∆∆, where both ΓΓ and
∆∆ are diﬀerent instances of Meta-Identity of level n. Moreover, at each stage of the
metainference, there will be only one inference, an instance of Identity, e.g., a case of
AA. For the sake of simplicity, let Abe a propositional letter, and a diﬀerent one in
each case. Thus, the only inference that will appear in ΓΓ will be, for example, pp;
17There are other ways to build the hierarchy. The following one is an alternative that may not be as
elegant as the one we have presented, but, nevertheless, achieves the same goals. (We leave the proofs
of these results as exercises for the reader.) The following one is another general deﬁnition of another
hierarchy of logics MIn
Deﬁnition 5.3. Let Γ1, ...Γnand be metainferences of level n. For any n, a metainference of level
n+ 1,Γ1, ...Γnis valid in MI
n+1if and only if, for every valuation v, if every Γ1, ...Γnis satisﬁed
by vaccording to ST, then vconﬁrms according to MI
n.
14
and the only inference that will be part of ∆∆ will be qq. Take a valuation vsuch
that v(p) = 1 and v(q) = 1
2. That valuation will satisfy ΓΓ according to MIn, but
would not satisfy ∆∆ according to MIn.
5.1 An interesting result about the hierarchy, and a few open ques-
tions
Some questions arise at this point. For example, what is the exact relation between
an MInand the rest of the logics in the hierarchy? How do the diﬀerent MInbehave
with respect to the metainferences of a level higher than n+ 1? We will present one
intriguing result.
An interesting question is how two diﬀerent MIns, MIjand MIk(where j < k),
behave with respect to metainferences of a level higher than k. Do they validate exactly
the same metainferences? Is one included in the other (and which one)? Are they
incomparable?
Fact 5.4. Here is a partial answer. For any two diﬀerent MIns, MIjand MIk, such
that k=j+ 1,MIjand MIkare incomparable with respect to metainferences of level
k+ 1.
Proof. By induction on the level nof the metainferential consequence relation MIn.
Base case: n= 0. This is a question about how TS and MI behave with respect to
meta-metainferences (e.g., metainferences of level 2).
Therefore, there must be two meta-metainferences (1) and (2) such that: (i) (1) is
valid in TS, but invalid in MI, and (ii) (2) is valid in MI and invalid in TS. We will
begin with the proof of (i).
BB
BB
(1) AA
AA
As every valuation satisﬁes its conclusion, (1) is a valid TS meta-metainference. For
every valuation v, either v(A) = 1
2, or v(A) = 0, orv(A) = 1. If v(A) = 1
2, then vdoes
not satisfy the premise of the conclusion of the meta-metainference (e.g. AA) in TS.
Thus, vconﬁrms the conclusion of the metainference, and so satisﬁes (1). If v(A) = 0,1,
vsatisﬁes the conclusion of the conclusion in TS. Therefore, vconﬁrms the conclusion
of the meta-metainference, and thus the meta-metainference itself. Nevertheless, (1) is
an invalid MI meta-metainference. A valuation vsuch that v(A) = 1
2and v(B) = 0,1
satisﬁes the premise of the meta-metainference, but does not conﬁrms the conclusion
according to MI.
15
AA
AA
(2) BBC
AA
(2) is a valid MI’s meta-metainference. For every valuation v, either v(A) = 1
2,
or v(A) = 0, orv(A) = 1. If v(A) = 1
2, then vsatisﬁes the premise of the premise
of the meta-metainference (e.g., AA) according to ST, but does not conﬁrms the
conclusion of the premise of the meta-metainference (e.g. AA) in TS. Thus, vdoes
not satisfy the premise of the meta-metainference according to MI, and thus satisﬁes
the meta-metainference itself in MI. If v(A) = 0,1, vsatisﬁes the conclusion of the
conclusion of the meta-metainference according to TS. Thus, vconﬁrms the conclusion
of the meta-metainference, and therefore (2) itself.
Nevertheless, (2) is an invalid TS’s meta-metainference. A valuation vsuch that
v(A) = 1
2and v(B) = 0,1 conﬁrms the premise of the meta-metainference, but does not
satisfy the conclusion, both according to TS.
Inductive case: n > 0. Take any MInand MIn+1, such that n > 0. We will present
two metainferences of level n+ 2, (1*) and (2*), such that: (i) (1*) is valid in MIn, but
invalid in MIn+1, and (ii) (2*) is valid in MIn+1and MIn.
∆∆
∆∆
(1*) ΓΓ
ΓΓ
ΓΓ and ∆∆ are metainferences of level n. To simplify things, let every sequent that
belongs to them be an instance of Identity, but a diﬀerent one in each case. Thus, let
AAbe the only inference that is part of ΓΓ, and let BBbe the only inference in
∆∆. Aand Bare diﬀerent formulas, and to make things even more easy to handle, let’s
take them as (diﬀerent) propositional letters. ΓΓ and ∆∆ will have, in each case, 2n
stages, and each one will have only a single occurrence of the inference AA/BB).
(1*) is a valid MInmeta-metainference. Every valuation vsatisﬁes the conclusion
–e.g., a metainference of level n+ 1– in MIn, because, either vsatisﬁes the conclusion,
or does not satisfy the premise. Thus, every valuation vconﬁrms the metainference
according to MIn. Therefore, vconﬁrms (1*) according to MIn.
Nevertheless, (1*) is an invalid MIn+1metainference. Consider a valuation vsuch
that v(p) = 1
2for every propositional letter pthat is part of A, and v(q) = 1 to every
propositional letter qthat is part of B. The valuation vwill satisfy the premise, but
will not satisfy the conclusion. Thus, it will be a counterexample to the validity of (1*)
in MIn+1.
16
ΓΓ
ΓΓ
(2*) ∆∆
ΓΓ
(2*) is a valid MIn+1’s meta-metainference. The premise of (2*) is a metainference
of level n+ 1. Thus, a valuation vthat gives value 1
2to every propositional letter
pin Awill be a counterexample in MIn+1. Therefore, as vis a counterexample to
(2*)’s premise, it also conﬁrms (2*) itself according to MIn+1. Valuations vsuch that
v(p) = 0,1, satisfy ΓΓ according to MIn, and thus satisfy (2*)’s conclusion according
to MIn+1.
Nevertheless, (2*) is an invalid MIninference. Consider a valuation vsuch that
v(p) = 1
2and v(q) = 1. That valuation does not satisfy the premise of (2*)’s premise
according to MIn, and thus conﬁrms (2*)’s premise itself. Nevertheless, that valuation
does not satisfy the conclusion of (2*)’s conclusion according to MIn, but conﬁrms its
premise according to that same consequence relation. Then, vdoes not satisfy (2*)’s
conclusion itself in MIn. But, as vsatisﬁes (2*)’s premise, it is a counterexample in
MIn.
Still, this does not tell us nothing about how two diﬀerent MIns, MIjand MIk
–where j < k, but not necessarily k=j+ 1– behave about metainferences of a level
k+ 1. Moreover, we still don’t know which is the exact nature of the relation between
any two logics MIjand MIk–where j < k– with respect to metainferences of a level l
–where k < l, but not necessarily l=k+ 1. These are, at this point, open questions.
6 A truly empty logic
So far, what we have is a hierarchy of logics MIn, where each MIninvalidates every
metainference of level nor less –including every inference and every sentence. Is it
possible to follow this path until we found what might be called a truly empty logic
–e.g., a logic that invalidates any sentence, inference and metainference of any level
whatsoever?
It is not an easy thing to characterize such logic. For example, let EL1to be
a consequence relation with just two values, 1 and 0. EL1’s consequence relation is
understood classically as preservation of value 1 from premises to conclusion. What is
peculiar about it is that the only proper valuations are those that assign value 0 to every
formula. Thus, EL1will have no valid sentence. Still, any inference (with a non-empty
set of premises) Γ ∆ will be valid in it, because no valuation vwill give value 1 to
each premise. Nevertheless, a slight modiﬁcation of EL1’s consequence relation might
do the work. Let EL2be as EL1, but with the following consequence relation: an
inference Γ ∆ is valid in EL2if and only if, for every valuation v,v(γ) = 1 –for every
γΓ– and for some δ∆, v(δ) = 1. As the only valuations are the ones of EL1, there
will be no valid inference, because no valuation will give value 1 to any formula. Still,
17
every metainference of level 1 (with a non-empty set of premises) will be valid, because
no valuation will satisfy the premises according to EL1.
We may go on modifying these logics in order to invalidate more and more metainfer-
ences. Ultimately, if we are lucky, what we might get is a series of consequence relations
that invalidates every metainference of level nor less, for every level n. Nevertheless,
these consequence relations seems a little bit too artiﬁcial to be really interesting. More-
over, we already have a similar hierarchy of logics, generated in a recursive way: the
hierarchy of MIns.
The next question we face is if we can build an empty logic from the hierarchy. In
a nutshell, we can. Let’s call an inference Σ Π a metainference of level 0. Moreover,
let Γ1, ...Γj∆ be a metainference of level n(1 n < ω). We will call the target
logic, Lempty.
Deﬁnition 6.1. A metainference of level n(for any level n,0n < ω)Γ1, ...Γj
is valid in Lempty if and only if Γ1, ...Γjis valid in MIn, for every MIn. Thus,
Lempty is a truly empty logic, because no sentence, inference and metainference of level
n–for any level n– will be valid in Lempty.
Proof. Take any metainference whatsoever. That metainference will have a particular
level j. Then, that metainference will be invalid in MIj. Thus, it will be invalid in
Lempty.
7 Conclusion
TS is a logic that has no valid inferences. Nevertheless, TS is informative about the
metainferences (of level 1). It is a fair question to ask if TS is as empty as a logic
can be. In particular, could there be a logic without valid metainferences? We have
given a positive answer to this question. MI is that logic. A metainference is valid
in MI if and only if, for every valuation v, if vsatisﬁes every premise according to
ST, then vsatisfy the conclusion according to TS. This move is very similar to the
one that justify substructural logics like ST or TS. In those cases, the standards that
premises and conclusions of a sound inference should satisfy, are diﬀerent. The same
thing happens with MI. The main diﬀerence is that the sound inference, in this case,
is, strictly speaking, a metainference.
But MI is not the limit either. We have present a consequence relation for metain-
ferences of level 2, called MI2, which has no valid meta-metainferences, but is, nev-
ertheless, informative about the metainferences of level 3. Moreover, it is possible to
specify, for any level nof metainferences, a consequence relation MInfor metainferences
of level n, such that no metainferences of level nor less in valid in it. Nevertheless,
every MI is informative about the metainferences of level n+ 1.
Thus, every such MInwill validate some metainference –in particular, metainfer-
ences of level higher than n. No logic MIn, then, is really empty. Is there a truly
empty logic? Though it is not easy to design such a consequence relation, it is posible
18
to specify a truly empty logic using the hierarchy of logics MIn. A metainference of
level n, for any level n, will be valid in that logic, Lempty, if and only if it is valid in
every MIn. As any metainference has a particular level j, that metainference is invalid
in MIj. Thus, it will be invalid in Lempty. Therefore, Lempty is a truly empty logic.
At this point, there are many open questions about the hierarchy we have presented.
For example, what is the exact nature of the relation between two members of the
hierarchy, MIjand MIk–where jk– with respect to metainferences of level higher
thatn k? Is one included in the other? Are they incomparable? Moreover, is it possible
to design sequent calculi for these logics? How should such a calculus look like? We
have present a partial answer to the ﬁrst of these question in Section 5. But we have
remain silent with respect to the rest of them. These questions have no easy answers,
and we will leave them for future work.
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