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A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Abstract. We will present twelve diﬀerent mixed metainferential consequence relations.

Each one of them is speciﬁed using two diﬀerent inferential Tarskian or non-Tarskian conse-

quence relations: K3,LP,ST or TS. We will show that it is possible to obtain a Tarskian

logic with non-Tarskian inferential logics, but also a non-Tarskian logic with Tarskian infer-

ential logics. Moreover, we will show how some of these metainferential logics works better

than the corresponding inferential rivals. Finally, we will show how these logics prove that

it is not enough to work with inferences as pairs of sets of formulas to obtain a contractive

logic.

Keywords: Logic-Metainferences-Metainferential Validity-Substructural Logics-Empty

Logics

1. Introduction

What is a logic? A logic is usually deﬁned as a language plus a consequence relation. For the

sake of simplicity, we will focus on propositional logics. Thus, 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). Though the answer to the question about what is, exactly, a consequence relation,

may be tricky, there is some consensus about what counts as a Tarskian consequence relation.

Deﬁnition 1.1. ATarskian 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) Γ⊧Aif A∈Γ(Reﬂexivity)

(2) If Γ⊧Aand Γ⊆Γ′, then Γ′⊧A(Monotonicity)

(3) If ∆⊧Aand Γ⊧Bfor every B∈∆, then Γ⊧A(Cut)

Additionally, a (Tarskian) consequence relation ⊧is substitution-invariant whenever if Γ⊧A,

and σis a substitution on FOR(L), then {σ(B) ∣ B∈Γ}⊧σ(A).

1

2 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Deﬁnition 1.2. A Tarskian logic over a propositional language Lis an ordered pair (FOR(L),⊧

), where ⊧is a substitution-invariant Tarskian consequence

Throughout the years many scholars have argued that the Tarskian conception of logic is

quite narrow. For example, Shoesmith and Smiley [29], Avron [2] and Scott [26] claimed that

the Tarskian account should be generalized to a logic having multiple consequences;1and Avron

[2] and Gabbay [15] have argued that the condition of Monotonicity should be relaxed; whereas it

can be inferred that, derivatively, Malinowski [19] and Frankowski [11] argued for a generalization

or liberalization which allows logics to drop Reﬂexivity and/or Cut.

These modiﬁcations, in turn, can be made sense of by noticing a shift in the nature of

the collection of formulae featured in the consequence relation. Thus, for example, instead of

treating logical consequence to hold between (sets of) formulae, it may hold between labelled

formulae, sequences of formulae (where order matters), multisets of formulae (where repetition

matters), etc. In particular, the focus could be on the relationship between inferences. Those

inferences that relate traditional inferences are called metainferences.2

Deﬁnition 1.5. An inference or sequent on Lis an ordered pair (Γ,∆), where Γ,∆⊆F O R(L)

(written Γ⇒∆). SEQ0(L)is the set of all inferences or sequents on L.

Deﬁnition 1.6 ([9]).A meta-inference or meta-sequent on Lis an ordered pair (Γ, A), where

Γ⊆SEQ0(L)and A∈SEQ0(L)(written Γ⇒1A). SEQ1(L)is the set of all meta-inferences

or meta-sequents on L.3

1This is a multi-conclusion presentation of these notions. As we will be working in a multi-conclusion setting,

when we talk about a Tarskian consequence relation, or a Tarskian logic, we will have these things in mind:

Deﬁnition 1.3. ATarskian 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) Γ, A ⊧A, ∆if A∈Γ(Reﬂexivity)

(2) If Γ⊧∆,Γ⊆Γ′and ∆⊆∆′, then Γ′⊧∆′(Monotonicity)

(3) If Γ⊧∆, A and Γ′, A ⊧∆′, then Γ,Γ′⊧∆,∆′(Cut)

Moreover, a (Tarskian) consequence relation ⊧is substitution-invariant whenever happens that, if Γ⊧∆, and σ

is a substitution on F OR(L), then {σ(A) ∣ A∈Γ}⊧{σ(B) ∣ B∈∆}.

Deﬁnition 1.4. A Tarskian logic over a propositional language Lis an ordered pair (F O R(L),⊧), where ⊧is

a substitution-invariant Tarskian consequence relation.

2Though an inference could be single or multi-conclusion, in the rest of the paper we will just consider multi-

conclusions inferences, for the sake of simplicity.

3For reasons of simplicity, we choose a single-conclusion presentation of metainferences. But this does not mean

that we have philosophically signiﬁcant reasons to reject a multi-conclusion setting for metainferences. Moreover,

we will also be working with a ﬁnite set of premises. Nevertheless, nothing essential hinges on this decision.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 3

We will say, accordingly, that from the following the one on the left is an inference, whereas

the one on the right is a meta-inference

A, B ⇒A∧B

⇒A⇒B

⇒A∧B

and, indeed, according to the following deﬁnitions adapted from Avron [1], both are valid in e.g.

Gentzen’s sequent calculus LK for classical logic. This will be more clear when we deﬁne the

corresponding notions of validity.

Now, going back to the proposed shifts from the ontology of the Tarskian account of logical

consequence, Avron suggested in [1] 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 —and, most

prominently, with substructural sequent calculi.

All these generalizations linked to the notion of logical consequence suggest new proposals

about what a logic is. Is it just the set of its valid inferences? Are the metainferences that a

logic validate just a byproduct of what the logic is –e.g., a set of valid inferences? This is not an

unproblematic thesis as it seems at ﬁrst sight. In recent papers, Cobreros, Egré, Ripley and van

Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble

caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach,

which leads them to consider the logic ST used to later build a non-transitive theory of truth,

ST+.4Throughout the authors’ works, the non-transitive project has proved that it has indeed

many attractive features. One of the most fundamental ones is something that might come as

a surprise, namely that it gives up Cut without thereby abandoning Classical Logic –CL–, for

ST and CL coincide at the inferential level –e.g., they have the same set of valid inferences.5

We will not pursue the debate about whether or not ST is a classical logic here.6Never-

theless, one of the things we will do is present a logic called TS/ST.TS/ST is a logic for

metainferences, in the sense that its consequence relation is deﬁned for metainferences, and not

4Non-transitive approaches to logical consequence were discussed, previously, in many works—to which the

authors refer in their papers. Some of these are due to Strawson (as referred in [34] and [28]), Tennant [32], [33],

Weir [35], Cook [8] and Frankowski [11]. It should be highlighted, though, that the application of this logical

approach to paradoxical phenomena is original of Cobreros, Egré, Ripley and van Rooij.

5Moreover, though Cut is not a basic rule in the sequent calculi for ST –e.g., without the Cut rule–, it is

admissible in it. Neverthless, Cut is not locally valid in ST. We will talk extensivelly about local metainferential

validity in the chapters to come.

6For more about this debate, see [3] and [9].

4 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

directly for inferences –as ST,LP and most logics. TS/ST not only validates every classically

valid inference, as ST does, but also recovers every classically valid metainference.

This move opens the door for a whole family of metainferential consequence relations, where

the standard for premises does not coincide with the standard for conclusions –though both of

them are given by some inferential consequence relation. They are, as [5] call them, a kind

of mixed and impure consequence relations. But while [5] only refer to traditional inferential

consequence relations, as ST or TS, we will present diﬀerent metainferential consequence rela-

tions where the standard for premises and conclusions are given by four inferential consequence

relations with well-known three-valued semantics: ST,TS –both of them substructural conse-

quence relations–, K3 and LP—which are Tarskian consequence relations. We will present the

twelve diﬀerent mixed consequence relations that can be build with them. The propositional

logics build with those consequences relations have some unexpected features. For example, we

will show how one of them –ST/TS– is a truly empty logic, making valid even less things than

TS, a logic without valid inferences. Moreover, we will show how to get Tarskian metainfer-

ential logics build with non-Tarskian inferential logics, and also how to obtain non-Tarskian

metainferential logics starting with Tarskian inferential building-blocks. We will compare these

diﬀerent logics regarding the set of structural metainferential schemes they validate, and present

some general facts about them. Finally, we will see that these logics illuminate an interesting

fact about inferences and Contraction: that it is not enough to work with sets of formulae to

guarantee that Contraction becames valid.

The paper is structured as follows. In Section 2 we present the notion of metainferential

validity we will be working with. We will also give a brief semantic presentation of the four

inferential logics we will be working with: LP,K3,ST and TS. In Section 3 we will present

twelve mixed metainferential consequence relations, and some interesting properties they have.

In particular, we will focus on whether or not the validate some main structural metainferential

schmes, like Cut, Identity, Contraction and Exchange. Finally, in Section 4 we provide some

concluding remarks.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 5

2. Metainferences, metainferential validity and two substructural logics

Before moving on, it will be necessary to make some clariﬁcations. In particular, we will

state what we mean by metainferential validity. Moreover, we will introduce with some detail

the four basic inferential logics that are the tools to specify the metainferential logics we will be

introducing: K3,LP,ST and TS.

2.1. Metainferential validity. A metainference, thus, is a pair (Γ, A), where Γis a (ﬁnite)

set of inferences and Ais a particular inference. Every metainference will have the following

structure:

Γ1⊧∆1, ..., Γn⊧∆n

Σ⊧Π

We will call every Γi⊧∆i, a premise of the metainference, while Σ⊧Πwill be its conclusion.

An immediate question that pops up is the following: when is a metainference Γ1⊧∆1, ...Γn⊧

∆n≫Σ⊧Πvalid in a (propositional) logic L? Here is a plausible answer:

Deﬁnition 2.1. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in (a propositional logic)

Lif and only if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to L, then vconﬁrms

Σ⊧Πaccording to L.

A valuation vsatisﬁes –or conﬁrms– an inference Γ⊧∆in a speciﬁc logic Lif and only if v

is not a counterexample of Γ⊧∆’s validity in L.

This way to characterized the notion of metainferential validity is known as the ‘local con-

ception of metainferential validity’(Dicher and Paoli [9]).7As it stands, the deﬁnition speciﬁes

what it takes for a particular metainference to be valid in a speciﬁc logic. Nevertheless, it can

–and will– be used to speciﬁed when a metainferential scheme is valid in. In a nutshell, a scheme

is valid in Lif and only if every instance of it is valid (in L).

7For more about this notion, and the diﬀerence between a local and a global notion of metainferential validity,

see [9]. A similar distinction was previously introduced by Lloyd Humberstone, in [18].

6 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Our target logics have consequence relations for metainferences. To understand exactly how

they work, we need to introduce ﬁrst four inferential consequence relations: the Tarskians LP

and K3, and the non-Tarskian –or substructural–TS and ST.8

2.2. LP and K3: two Tarskian consequence relations. We will present a propositional

version of these logics, starting with LP. The propositional logic LP can be then understood as

a propositional language FOR(L)with the matrices associated to the 3-element Kleene algebra,

and a consequence relation understood as preservation of designated values. In the case of LP,

the designated values are 1,1

2.

Deﬁnition 2.2. The 3-element Kleene algebra is the structure

K=⟨{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}⟩

where the functions f¬

K, f ∧

K, f ∨

Kare as follows

f¬

K

1 0

1

2

1

2

1 0

f∧

K11

20

1 1 1

20

1

2

1

2

1

20

0 0 0 0

f∨

K11

20

1 1 1 1

1

211

2

1

2

0 1 1

20

Moreover, the functions →and ↔are deﬁnable via the usual deﬁnitions.

The valuation functions are 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: the functions given by the 3-element Kleene algebra. In the case of LP, we can

deﬁne what an LP-valid inference is in the following straightforward manner. (Notice that,

below, ⊧LP is a substitution-invariant consequence relation.)

Deﬁnition 2.3. A valuation vsatisﬁes an inference Γ⇒∆in LP (written v⊧LP Γ⇒∆) if

and only if, if v[Γ]⊆{1,1

2}, then v(A)∈{1,1

2}, for some A∈∆. An inference Γ⇒∆is LP-valid

(written ⊧LP Γ⇒∆) if and only if v⊧LP Γ⇒∆, for all valuations v.

In a similar vein, K3 can be then understood as a propositional language FOR(L)with

the matrices associated to the 3-element Kleene algebra that we have already presented, and a

8Those readers that are already familiar with these logics may safely skip the following two subsections.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 7

consequence relation understood as preservation of designated values. In the case of K3, the

only designated value is 1.

We are now ready to deﬁne K3-validity. Once again, ⊧K3 is a substitution-invariant conse-

quence relation.

Deﬁnition 2.4. A valuation vsatisﬁes an inference Γ⇒∆in K3 (written v⊧K3 Γ⇒∆) if

and only if if v[Γ]⊆{1}, then v(A)∈{1}, for some A∈∆. An inference Γ⇒∆is K3-valid

(written ⊧K3 Γ⇒∆) if and only if v⊧K3 Γ⇒∆, for all valuations v.

2.3. ST and TS: two substructural consequence relations. ST and TS are label as

substructural because at least one structural feature of a Tarskian consequence relation is given

up by them. ST abandons Cut, while TS drops Reﬂexivity.

The logic ST can be portrayed as a p-logic, as devised by Frankowski in [11] as a means to

characterize logical systems where valid derivations are such that the degree of strength of the

conclusions can be smaller than strength of the premises.9

Deﬁnition 2.5 ([11]).Ap-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) Γ⊧∆if for some A∈∆,A∈Γ(Reﬂexivity)

(2) If Γ⊧∆and Γ⊆Γ′, then Γ′⊧∆(Monotonicity)

Additionally, a p-consequence relation ⊧is substitution-invariant whenever if Γ⊧∆, and σis

a substitution on FOR(L), then {σ(B) ∣ B∈Γ}⊧σ(A)-for some A∈∆.

Deﬁnition 2.6 ([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 can be connected to p-matrices. ST can be represented as a p-matrix

logic associated to the 3-element Kleene algebra.

Deﬁnition 2.7 ([12]).For La propositional language, an L-p-matrix is a structure ⟨V,D+,D−,O⟩,

such that ⟨V,O⟩is an algebra of the same similarity type as L, with universe Vand a set of

operations O, where D+,D−⊆Vand D+⊆D−.

9For an extensive presentation of ST, see [6], [23], [25] and [7].

8 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Deﬁnition 2.8 ([6]).A 3-valued ST-matrix is a p-matrix

MST =⟨{1,1

2,0},{1},{1,1

2},{f¬

K, f ∧

K, f ∨

K}⟩

such that ⟨{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}⟩ is the 3-element Kleene algebra.

Now, as is common practice, semantic structures such as p-matrices induce consequence

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. Below, ⊧Mis

a substitution-invariant p-consequence relation, whence (FOR(L),⊧M)is a a p-logic.

Deﬁnition 2.9. For Map-matrix, an M-valuation vsatisﬁes a sequent or inference Γ⇒∆

(written v⊧MΓ⇒∆) if and only if if v[Γ]⊆D+, then v(A)∈D−, for some A∈∆. A

sequent or inference Γ⇒∆is M-valid (written ⊧MΓ⇒∆) if and only if v⊧MΓ⇒∆, for all

M-valuations v.

A more straightforward characterization of ST’s validity is the following one:10

⊧ST Γ⇒∆if and only if for every valuation v,

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

if v(B)∈{1}for all B∈Γ

then v(A)∈{1,1

2}for some A∈∆

10A third way to present ST’s validity requires talking about strict and tolerant satisfaction or truth. 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}.

Then, a valuation vsatisﬁes an inference Γ⇒∆if and only if, if vstrictly satisﬁes every B∈Γ, then vtolerantly

satisﬁes at least one A∈∆. (This is why ST is called Strict-Tolerant.) Finally, an inference from Γto ∆is

valid if and only if for every valuation v, if vsatisﬁes strictly every B∈Γ, then vsatisﬁes tolerantly some A∈∆.

Nevertheless, it is worth mentioning that this is not the only way ST’s supporters explain their position. They

prefer to talk about strict and tolerant assertion rather than talking about strict and tolerant satisfaction, or

strict and tolerant truth. As [10] explains, the reason why they use the idea of strict and tolerant assertion

instead of any of the last two (pair of) notions, is to avoid revenge paradoxes related to the notions of ‘strictly

true’ and ‘strictly false’ in the context of truth-theories based on ST.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 9

Another interesting generalization of Tarskian consequence relations is the notion of q-

consequence relation, due to Malinowski [19].11

Deﬁnition 2.10 ([19]).Aq-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) If Γ⊧∆and Γ⊆Γ′, then Γ′⊧∆(Monotonicity)

(2) Γ∪{A∣Γ⊧A}⊧∆if and only if Γ⊧∆(Quasi-closure)

Deﬁnition 2.11 ([19]).Aq-logic over a propositional language Lis an ordered pair (FOR(L),⊧

), where ⊧is a substitution-invariant q-consequence relation.

Deﬁnition 2.12 ([19]).For La propositional language, an L-q-matrix is a structure ⟨V,D+,D−,O⟩,

such that ⟨V,O⟩is an algebra of the same similarity type as L, with universe Vand a set of

operations O, where D+,D−⊆Vand D+∩D−=∅.

The 3-valued q-matrix logics associated to the 3-element Kleene algebra that will be dealing

with in our ongoing investigation is the logic TS.

Deﬁnition 2.13 ([6], [20]).A 3-valued TS-matrix is a q-matrix

MTS =⟨{1,1

2,0},{1},{0},{f¬

K, f ∧

K, f ∨

K}⟩

such that ⟨{1,1

2,0},{f¬

K, f ∧

K, f ∨

K}⟩ is the 3-element Kleene algebra.

TS is discussed by e.g., Cobreros, Ripley, Egré and van Rooij in [6], and also by Chemla, Egré

and Spector in [5] in the context of the more general discussion of what represents a ‘proper’

consequence relation between formulae. Moreover, it was also discussed by Malinowski in [20]

as a tool to model empirical inference with the aid of the 3-valued Kleene algebra, and more

recently was stressed by Rohan French in [13], in connection with the paradoxes of self-reference.

11Wansing and Shramko oﬀer in [30] a way to read these two kinds of logics. While a 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, the relation of q-logic is devised to qualify as valid derivations of true sentences from

non-refuted premises (understood as hypotheses).

10 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Now we need to clarify how q-matrix logics validate or invalidate inferences —following, e.g.

[31, p. 196]. ⊧Mis a substitution-invariant q-consequence relation, whence (FOR(L),⊧M)is

aq-logic.

Deﬁnition 2.14. For Maq-matrix, an M-valuation vsatisﬁes a sequent or inference Γ⇒A

(written v⊧MΓ⇒A) if and only if if v[Γ]∩D−=∅, then v(A)∈D+.

For Maq-matrix, an inference Γ⇒∆is M-valid (written ⊧MΓ⇒∆) if and only if

v⊧MΓ⇒∆, for all M-valuations v.

The following is a more straightforward characterization of TS’s inferential validity:

⊧TS Γ⇒∆if and only if for every valuation v,

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

if v(B)∈{1,1

2}for all B∈Γ

then v(A)∈{1}for some A∈∆

From these deﬁnitions of ST and TS’s validities, the following facts follow.

Fact 2.15 ([6]).TS is a non-reﬂexive, and thus a substructural, logic.

Fact 2.16 ([6]).ST is a non-transitive, and thus a substructural, logic.

Fact 2.17 ([13]).TS has no valid inferences or sequents.

Fact 2.18 ([16], [23]).ST and classical propositional logic CL have the same set of valid

inferences or sequents.12

3. Twelve mixed metainferential consequence relations

We will now present twelve mixed metainferential consequence relations. More precisely,

we will present diﬀerent metainferential consequence relations L1/L2, where L1and L2are

diﬀerent inferential consequences relations. Here, L1represents the standard that the premises

of a sound argument should meet, while L2stands for the canon for the conclusion. These

impure metainferential consequence relations are built around LP,K3,ST and TS. They are

12[4], [9] and [22] have shown that —through some suitable translation— the set of valid inferences in LP

coincides with the set of valid meta-inferences in ST. Moreover, [13] have 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 meta-inferences in TS.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 11

impure –as is explained in [5]– because the standard for premises is diﬀerent than the standard

for conclusions.

These consequence relations are TS/ST,ST/TS,LP/K3,K3/LP,ST/LP,ST/K3,TS/LP,

TS/K3,LP/ST,LP/TS,K3/ST and K3/TS, as Figure 1shows.

L1/L2ST TS LP K3

ST ST ST/TS ST/LP ST/K3

TS TS/ST TS TS/LP TS/K3

LP LP/ST LP/TS LP LP/K3

K3 K3/ST K3/TS K3/LP K3

Figure 1. Twelve metainferential consequence relations

3.1. TS/ST, a logic for classically valid metainferences. We will now present TS/ST, a

logic that not only validates every classically valid inference –as ST does–, but also validates

every classically valid metainference.13 This is possible because TS/ST’s consequence relation

embraces a feature of the inferential consequence relation ST, but applies it to the metainfer-

ential level. Just to remember, an inference is valid in ST if and only if, for every valuation

v, if the premises satisfy certain –demanding– standard, then the conclusion meet some less

demanding canon. In particular, an inference is valid in ST if and only if, if for every premise

A,v(A)=1–e.g., if vstrictly satisﬁes every premise–, then, for some conclusion B,v(B)=1,1

2

–e.g., vtolerantly satisﬁes a conclusion. Conversely, if vstrictly satisﬁes every premise, but does

not tolerantly satisﬁes no conclusion, then vis a counterexample to the validity of that inference

in ST.

We will adopt a similar norm for TS/ST’s notion of validity , but we will apply it to the

metainferential level. Thus, TS/ST’s standard for the premises will be more demanding than

its standard for the conclusion.

Deﬁnition 3.1. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in TS/ST if and only if,

for every valuation v, if every Γi⊧∆iis satisﬁed by vaccording to TS, then vconﬁrms Σ⊧Π

in ST.

13As we have already mentioned, we will present single-conclusion metainferential logics, TS/ST being the

ﬁrst of them. 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 may be safely extended to multi-conclusion versions

of these logics, which also admit arguments with inﬁnite premises.

12 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Before proving our main result, it is worth noticing two facts. The ﬁrst one relates ST and

the semantic consequence relation of classical propositional logic, CL. The second one relates

TS and CL. But even before, we should pause to clarify how –the propositional– CL behaves.

The valuations that deﬁne CL’s consequence relation are bivalent, exclusive and exhaustive:

for every valuation vand every formula A, either v(A)=1or v(A)=0, but not both. An

inference Γ⊧∆is valid in CL if and only if, for every valuation v, either v(γ)=0(for some

γ∈Γ), or v(δ)=1(for some δ∈∆). Similarly, a valuation vis a counterexample to Γ⊧∆in

CL if and only if for every γ∈Γ,v(γ)=1, and for every δ∈∆,v(δ)=0.

It is important to stress the following two facts: a valuation vis a counterexample in ST to

an inference Γ⊧∆if and only if vis a counterexample to that inference in CL. Remember

that vis a counterexample to Γ⊧∆in either of those two logics if and only if, for every γ∈Γ,

v(γ)=1, and for every δ∈∆,v(δ)=0. Similarly, vsatisﬁes an inference Γ⊧∆in TS if and

only if vsatisﬁes that inference in CL. Recall that vsatisﬁes Γ⊧∆in either of these logics if

and only if either v(γ)=0, for some γ∈Γ, or, for some δ∈∆,v(δ)=1.

Now we can introduce the main result for this logic. It establishes that a metainference is

valid in CL if and only if it is valid in TS/ST.

Theorem 3.2. The Collapse Result (For every metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Π)

Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in CL if and only if Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid

in TS/ST.

Proof. Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis invalid in CL if and only if for some vand every i

(1≤i≤n), vsatisﬁes every Γi⊧∆iaccording to CL and vdoes not conﬁrm Σ⊧Πin CL, if

and only if, for every Γi⊧∆i,v(γ)=0or v(δ)=1, for some γ∈Γior some δ∈∆i, and for every

σ∈Σand every π∈Π,v(σ)=1and v(π)=0, if and only if vsatisﬁes every Γi⊧∆iaccording to

TS and vdoes not satisﬁes Σ⊧Πaccording to ST, if and only if Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Π

is invalid in TS/ST.

The following is another way to understand this result: a metainference is invalid in CL if

and only if there is a valuation vthat satisﬁes every premise and does not satisfy the conclusion.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 13

But a valuation vsatisﬁes a premise of a metainference –e.g., an inference– according to CL

if and only if, either vgives value 0to some premise of the inference, or gives value 1to some

conclusion. And that is precisely what it takes for vto satisfy that premise according to TS.

Moreover, a valuation vdoes not satisfy the conclusion of the metainference according to CL if

and only if vgives value 1to every premise of the conclusion, and value 0to every conclusion

of the conclusion. And that is just what it takes for vto be a counterexample to the validity of

it in ST.

It is usual to consider sentences as degenerate cases of inferences (e.g., with an empty set of

premises). In a similar vein, inferences can be interpreted as degenerate cases of metainferences

(e.g., with an empty set of premises). Thus, every classically valid inference will be valid in

TS/ST, and vice-versa. A degenerate case of a metainference –e.g., an inference– is valid in

TS/ST if and only if every valuation satisﬁes the conclusion according to the standard for

conclusions –e.g., according to ST. And ST recovers every classically valid inference.

As every classically metainference is valid in TS/ST, so is every structural metainferential

scheme, including the most well known of them: Cut, Contraction, Identity, Weakening and

Exchange. We will talk in detail about each of them, because we will present some logics where

some –or all– of these metainferences fail. Though we would not explore it here, it is worth

noticing that it is possible to expand TS/ST’s language with a transparent truth predicate.

The resulting theory, TS/ST+, will be satisﬁable. In a nutshell, TS/ST+’s satisﬁability follows

from the fact that the truth predicate can be interpreted as a ﬁxed-point of a jump operator

over the Strong Kleene scheme, that is the ones used by TS/ST’s –and TS/ST+– models. In

fact, every consequence relation that we will be exploring shares this feature with TS/ST, for

the same reasons.

One last interesting feature of TS/ST is that it is a fully Tarskian logic, though the means

used to characterized it are non-Tarskian –or substructural– theories: ST and TS. We will

see later that it is also possible to go the other way, e.g., from Tarskian inferential logics, to

non-Tarskian metainferential consequence relations.

Before exploring that path, we will present a close relative of TS/ST:ST/TS.

14 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

3.2. ST/TS: An empty (metainferential) logic. ST/TS is a logic that invalidates not

only every inference –as TS does–, but also every metainference.ST/TS shares with TS

a similar characteristic. An inference is valid in TS if and only, for every valuation v, if v

satisﬁes the premises according to certain standard, then vmeets the conclusion according to

some other more demanding standard. ST/TS works in a similar way, but in a metainferential

setting. Thus, ST/TS’s standard for premises will be less demanding than its criterion for the

conclusion.

Deﬁnition 3.3. A metainference Γ1⊧A1, ...Γn⊧An≫Σ⊧Bis valid in ST/TS if and only if,

for every valuation v, if every Γi⊧Aiis satisﬁed by vaccording to ST, then vconﬁrms Σ⊧B

according to TS.

Thus, ST/TS is a metainferentially empty logic.

Fact 3.4. No metainference Γ1⊧A1, ...Γn⊧An≫Σ⊧Bis valid in ST/TS.

Proof. Consider a valuation vsuch that, for every propositional letter pthat appears in Γ1⊧

A1, ...Γn⊧An≫Σ⊧B,v(p)=1

2. As matrices for the logical constants are Strong Kleene ones,

vgives the value 1

2to every formula Ain the metainference. Thus, every premise Γi⇒Aiis

satisﬁed by vaccording to ST, but vdoes not satisfy the conclusion Σ⇒Baccording to TS.

As no metainference is valid in ST/TS, neither are the structural most well-known metain-

ferential schemes: Cut, Contraction, Identity, Weakening and Exchange.

3.3. LP/K3: one way to go from Tarskian to Non-Tarskian logics. TS/ST is a fully

Tarskian logic, despite being characterized through non-Tarskian –or substructural– theories.

The following logic goes in the opposite direction. LP/K3 is a non-Tarskian metainferential

consequence relation characterized using two Tarskian logics: LP and K3.

Deﬁnition 3.5. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in LP/K3 if and only

if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to LP, then vconﬁrms Σ⊧Π

according to K3. Thus, Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in LP/K3 if and only if, for every

valuation v, (i) either there is a Γi⊧∆isuch that v(Γi)=1,1

2and v(∆i)=0, (ii) or v(Σ)=0,1

2,

(iii) or v(Π)=1.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 15

LP/K3 is non-Tarskian –or substructural– because not every structural metainference is valid

in it. In particular, Cut fails in LP/K3.

Fact 3.6. Cut is invalid in LP/K3.

Γ, A ⇒∆ Γ′⇒∆′, A

Cut Γ,Γ′⇒∆,∆′

Proof. Consider an instance of Cut without logical constants—e.g., where every formula in the

metainferece is a propositional letter. The valuation vsuch that v(γ)=v(γ′)=1, for every

γ∈Γ, and γ′∈Γ′,v(δ)=v(δ′)=v(A)=1

2, for every δ∈∆,δ′∈∆′, is a counterexample to Cut’s

validity.

At this point, it might be interesting to consider a structural metainference that have not

receive much attention. We will call it Meta-Identity. We will also see how it fails in this logic.

Moreover, Weakening, Contraction and Exchange are also invalid in LP/K3—and the failure

of Meta-Identity at least partially explains their failure.

Fact 3.7. Meta-Identity, Weakening, Contraction and Exchange are invalid in LP/K3—even

though we are working with sets of formulas, and not with multisets or sequences.

Γ⇒∆

Meta-Identity Γ⇒∆

Proof. Consider an instance of Meta-Identity without logical constants. The valuation vsuch

that v(γ)=1, for every γ∈Γ, and v(δ)=1

2, for every δ∈∆, satisﬁes the premise according to

LP, but does not meet the conclusion in K3.

Γ⇒∆

Weakening Γ,Γ′⇒∆,∆′

Proof. Consider an instance of Weakening without logical constants, and a valuation vsuch

that v(γ)=v(γ′)=1, for every γ∈Γ,γ′∈Γ′,v(δ)=v(δ′)=1

2, for every δ∈∆,δ′∈∆′, satisﬁes

the premise according to LP, but does not meet the conclusion in K3.

16 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Γ, A, A ⇒∆

Contraction Γ, A ⇒∆

Proof. Consider an instance of Contraction without logical constants. The valuation vsuch

that v(γ)=1, for every γ∈Γ,v(δ)=1

2, for every δ∈∆, and v(A)=1, is a counterexample to it

in LP/K3.

Γ, A, B ⇒∆

Exchange Γ, B, A ⇒∆

Proof. Consider an instance of Exchange without logical constants. The valuation vsuch that

v(γ)=1, for every γ∈Γ,v(δ)=1

2, for every δ∈∆, and v(A)=v(B)=1, is a counterexample to

it in LP/K3.

One interesting way to evaluate these results is the following. As every instance of Meta-

Identity is also an instance of the other metainferential schemes –e.g., Weakening, Contraction

and Exchange–, then the failure of (an instance of) Meta-Identity is enough to guarantee the

failure of the rest of the previously mentioned metainferential schemes.

3.4. K3/LP: another way to go from Tarskian to Non-Tarskian logics. LP/K3 is not

the only non-Tarskian metainferential consequence relation characterized using LP and K3.

K3/LP is another member of this group.

Deﬁnition 3.8. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in K3/LP if and only

if, for every valuation v, if vconﬁrms every Γi⊧∆iaccording to K3, then vsatisﬁes Σ⊧Π

according to LP, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1and v(∆i)=0,1

2,

(ii) or v(Σ)=0, (iii) or v(Π)=1,1

2.

K3/LP, in a way, is another strongly non-Tarskian, in the sense that most structural metain-

ferences are invalid in it. In fact, even if Identity is valid (because it is valid in LP –and K3/LP’s

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 17

standard for inferences is just LP’s criterion for them),14 Cut, Meta-Identity, Weakening, Con-

traction and Exchange fail in it.

Fact 3.9. Cut is invalid in K3/LP.

Proof. Consider an instance of Cut without logical constants. The valuation vsuch that v(γ)=

v(γ′)=v(A)=1

2, for every γ∈Γ,γ′∈Γ′,v(δ)=v(δ′)=0, for every δ∈∆,δ′∈∆′, is a

counterexample to Cut’s validity in K3/LP.

Fact 3.10. Meta-Identity, Weakening, Contraction and Exchange are invalid in K3/LP -even

though we are working with sets of formulas, and not with multisets or sequences..

Proof. As we have already pointed out, if Meta-Identity is invalid, the rest of them are also

invalid. And to prove that Meta-Identity is invalid, it is enough to consider an instance of

Meta-Identity without logical constants. The valuation vsuch that v(γ)=,1

2, for every γ∈Γ,

and v(δ)=0, for every δ∈∆, satisﬁes the premise according to K3, but does not meet the

conclusion in LP.

3.5. ST/LP and ST/K3.Our next two logics have the non-Tarskian ST as the standard for

premises. Both of them are substructural logics. Every structural metainferential scheme –but

Identity, which is valid in both logics, since it is valid in both LP and K3– are invalid in the

two logics we will be introducing in this subsection.

Deﬁnition 3.11. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in ST/LP if and only

if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to ST, then vconﬁrms Σ⊧Π

according to LP, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1and v(∆i)=0,

(ii) or v(Σ)=0, (iii) or v(Π)=1

2,1.

Fact 3.12. Cut is invalid in ST/LP.

14Similarly, Identity is valid in LP/K3 because it is valid in K3.

18 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Proof. Consider an instance of Cut where every formula in the metainference is a propositional

variable. The valuation vsuch that v(γ)=v(γ′)=1, for every γ∈Γ,γ′∈Γ′,v(δ)=v(δ′)=0,

for every δ∈∆,δ′∈∆′, and v(A)=1

2, is a counterexample to it in ST/LP.

Fact 3.13. Meta-Identity, Weakening, Contraction and Exchange invalid in ST/LP.

Proof. Once again, for this it is enough to prove that Meta-Identity is invalid. Thus, consider

an instance of Meta-Identity without logical constants. The valuation vsuch that v(γ)=1

2, for

every γ∈Γ,v(δ)=0, and for every δ∈∆, is a counterexample to Meta-Identity in ST/LP.

Deﬁnition 3.14. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in ST/SK if and only

if, for every valuation v, if vconﬁrms every Γi⊧∆iaccording to ST, then vsatisﬁes Σ⊧Π

according to K3, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1and v(∆i)=0,

(ii) or v(Σ)=0,1

2, (iii) or v(Π)=1.

Fact 3.15. Cut, Meta-Identity, Weakening, Contraction and Exchange are invalid in ST/K3

-even though we are working with sets of formulas, and not with multisets or sequences..

Proof. The proof that Cut is invalid is the same as the one for the ST/LP’s case. Regarding

the others, consider an instance of Meta-Identity without logical constants. The valuation v

such that v(γ)=1, for every γ∈Γ, and v(δ)=1

2, for every δ∈∆, is a counterexample to

Meta-Identity in ST/K3.

3.6. TS/LP and TS/K3.Our next two logics have the non-Tarskian TS as a standard for

premises. Both of them –e.g., the one that has LP as its standard for conclusions, and the one

in which K3 does that job– are fully Tarskian logics: every structural metainferential scheme

is valid in them. (Once again, Identity is valid because it is valid in both LP and K3.)

Deﬁnition 3.16. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in TS/LP if and

only if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to TS, then vconﬁrms

Σ⊧Πaccording to LP, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1,1

2and

v(∆i)=1

2,0, (ii) or v(Σ)=0, (iii) or v(Π)=1,1

2.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 19

Deﬁnition 3.17. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in TS/SK if and

only if, for every valuation v, if vconﬁrms every Γi⊧∆iaccording to TS, then vsatisﬁes

Σ⊧Πaccording to K3, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1,1

2and

v(∆i)=1

2,0, (ii) or v(Σ)=0,1

2, (iii) or v(Π)=1.

Fact 3.18. Cut, Meta-Identity, Weakening, Contraction and Exchange are valid in both TS/LP

and TS/K3, which are, then, a fully structural logics.

Proof. We leave them as an exercise for the reader. (It is not hard to realize that, in each of

those cases, if a valuation vdoes not satisfy the conclusion according to LP –or K3–, then it

does not conﬁrm at least one conclusion in TS.)

3.7. K3/ST and K3/TS.The next two logics have the Tarskian K3 as a standard for premises.

The one that has ST as the standard for conclusions is a fully Tarskian logic. But the one with

TS as a criterion for conclusions, invalidates every structural metainferential scheme.

Deﬁnition 3.19. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in K3/ST if and only

if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to K3, then vconﬁrms Σ⊧Π

according to ST, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1and v(∆i)=1

2,0,

(ii) or v(Σ)=0,1

2, (iii) or v(Π)=1,1

2.

Fact 3.20. Cut, Meta-Identity, Identity, Weakening, Contraction and Exchange are valid in

K3/ST, which is, then, a fully structural logic.

Proof. We leave them as an exercise for the reader.

Deﬁnition 3.21. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in K3/TS if and only

if, for every valuation v, if vconﬁrms every Γi⊧∆iaccording to K3, then vsatisﬁes Σ⊧Π

according to TS, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1and v(∆i)=1

2,0,

(ii) or v(Σ)=0, (iii) or v(Π)=1.

Fact 3.22. Cut, Meta-Identity, Identity, Weakening, Contraction and Exchange are invalid in

K3/TS.

20 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

Proof. We will leave the proof that Cut is invalid as an exercise to the reader. Regarding the

others, consider an instance of Meta-Identity without logical constants. The valuation vsuch

that v(γ)=1

2, for every γ∈Γ, and v(δ)=1

2, for every δ∈∆, is a counterexample to Meta-Identity

in K3/TS, and, therefore, to the other structural metainferences.

3.8. LP/ST and LP/TS.The next two logics have the Tarskian LP as a standard for premises.

The one that has ST as the standard for conclusions is a fully Tarskian logic. But the one with

TS as the norm for conclusions, invalidates every structural metainferential scheme.

Deﬁnition 3.23. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in LP/ST if and only

if, for every valuation v, if vconﬁrms every Γi⊧∆iaccording to LP, then vsatisﬁes Σ⊧Π

according to ST, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1,1

2and v(∆i)=0,

(ii) or v(Σ)=0,1

2, (iii) or v(Π)=1,1

2.

Fact 3.24. Cut, Meta-Identity, Identity, Weakening, Contraction and Exchange are valid in

LP/ST, which is, then, a fully structural logic.

Proof. We leave them as an exercise for the reader.

Deﬁnition 3.25. A metainference Γ1⊧∆1, ...Γn⊧∆n≫Σ⊧Πis valid in LP/TS if and only

if, for every valuation v, if vsatisﬁes every Γi⊧∆iaccording to LP, then vconﬁrms Σ⊧Π

according to TS, if and only if, (i) either there is a Γi⊧∆isuch that v(Γi)=1,1

2and v(∆i)=0,

(ii) or v(Σ)=0, (iii) or v(Π)=1.

Fact 3.26. Cut, Meta-Identity, Identity, Weakening, Contraction and Exchange are invalid in

LP/TS.

Proof. We will leave the proof that Cut is invalid as an exercise to the reader. Regarding the

others, consider an instance of Meta-Identity without logical constants. The valuation vsuch

that v(γ)=1

2, for every γ∈Γ, and v(δ)=1

2, for every δ∈∆, is a counterexample to Meta-Identity

in this logic, and, thus, to the other structural metainferential schemes.

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 21

Metainferences TS/ST ST/TS K3/LP LP/K3 LP/ST LP/TS

CUT Yes No No No Yes No

IDENTITY Yes No Yes Yes Yes No

META-IDENTITY Yes No No No Yes No

WEAKENING Yes No No No Yes No

CONTRACTION Yes No No No Yes No

EXCHANGE Yes No No No Yes No

Figure 2. Comparison table

Metainferences K3/ST K3/TS ST/LP ST/K3 TS/LP TS/K3

CUT Yes No No No Yes Yes

IDENTITY Yes No Yes Yes Yes Yes

META-IDENTITY Yes No No No Yes Yes

WEAKENING Yes No No No Yes Yes

CONTRACTION Yes No No No Yes Yes

EXCHANGE Yes No No No Yes Yes

Figure 3. Comparison table

3.9. A summary. Figures 2and 3summarize the results that we have presented so far:

Moreover, we can partially order these diﬀerent logics considering the diﬀerent strength they

have. The measure of the strength is the metainfences they prove. The one that prove more

things is TS/ST. Every classically valid metainference is valid in it. Regarding the rest of the

logics, we have focus on the limited set of structural metainferential schemes that we have exten-

sively talked about, e.g., Cut Identity, Meta-Identity, Weakening, Contraction and Exchange.

The order between these logics—regarding these structural a schemes—, then, is strict.15

●1: TS/ST

●2: TS/LP,TS/K3,LP/ST,K3/ST

1

2

3

4

●3: K3/LP,LP/K3,ST/LP,ST/K3

15There is still some work to be done here. For example, it is not the case that the four logics of the second group—

e.g., TS/LP,TS/K3,LP/ST and K3/ST— validates exactly the same metainferences. In fact, for example,

TS/LP and TS/K3 are incomparable, as K3 and LP are incomparable at the inferential level. Nevertheless,

we will leave the exploration of the exact relations between these logics for future work.

22 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

●4: ST/TS,LP/TS,K3/TS

Before moving on, we would like to dig a little bit deeper in a pattern that these mixed

metainferential consequence relations follow that we have already mentioned. If Meta-Identity

is invalid, then Contraction,Exchange and Weakening will also be invalid, since every instance

of the ﬁrst one (in this setting) is an instance of the last ones. And it does not need much to

invalidate Meta-Identity. Take two inferential logics, L1and L2. If there is an inference Γ⇒∆

and one valuation vsuch that vsatisﬁes Γ⇒∆in L1but not in L2, then Meta-Identity will be

invalid in L1/L2. Nevertheless, it is not necessary that L1is stronger than L2. In fact, they may

even be incomparable and Meta-Identity might still be invalid. (If they are, then Meta-Identity

will also became invalid in L2/L1.)

4. Conclusion: about metainferential logics

We have presented twelve diﬀerent mixed metainferential consequence relations. Each one of

them is speciﬁed using two diﬀerent inferential Tarskian or non-Tarskian (e.g., substructurals)

consequence relations: K3,LP,ST or TS. The standard for premises and consequence is

diﬀerent in each case.

Five of these logics –e.g., TS/ST,TS/LP,TS/K3,LP/ST,K3/ST– are Tarskian logics

-e.g., they validate Cut, Identity and Weakening. Moreover, they also validate Meta-Identity,

Contraction and Exchange. Nevertheless, they are build with substructural inferential conse-

quence relations. In fact, TS/ST is built entirely with substructural inferential consequence

relations.

Moreover, TS/ST collapses with CL at the metainferential level. Do the four other conse-

quence relations share this feature? In fact, they do not. It is not diﬃcult to prove that TS/LP

and TS/K3 do not validate every classically valid metainference. Neither LP nor K3 have the

same set of valid inferences than classical logic –in fact, they are sublogics of CL. Thus, it is

enough to consider any metainference with an empty set of premises, and a conclusion valid in

CL but invalid in LP/K3. For example, the following metainference will be valid in CL, but

invalid in TS/LP:

⊘⇒⊘

A, A →B⇒B

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 23

Whereas the following metainference is invalid in TS/K3, but is nevertheless valid in CL:

⊘⇒⊘

⊘⇒A→A

The cases of LP/ST and K3/ST are a little bit trickier. While it is true that there are

valuations that satisfy some metainferences in CL (or in TS/ST), but not in LP/ST and

K3/ST, this does not automatically means that the set of valid metainferences is actually

diﬀerent in these cases. But in fact, they are. In particular, the set of valid metainferences of

LP/ST and K3/ST is included in the set of valid CL’s (and TS/ST’s) metainferences. It is

not hard to realize that, though every possible counterexample in CL is also a counterexample

in both LP/ST and K3/ST, these last logics in fact have more possible counterexamples. And

some of those valuations in fact invalidate some classically valid metainferences. The following

one is an instance of a metainference valid in CL but invalid in LP/ST.

⊘⇒A∧ ¬A

B⇒C

Though no valuation will satisfy its premise according to CL, a valuation vsuch that v(A)=

1

2, v(B)=1and v(C)=0, conﬁrms the premise, but does not satisfy the conclusion in LP/ST.

The following metainferential scheme is valid in CL, but not in K3/ST.

¬(A∧ ¬A)⇒⊘

B⇒C

Once again, while no valuation will conﬁrm the premise in CL, a valuation vsuch that

v(A)=1

2, v(B)=1and v(C)=0will conﬁrm the premise, but not the conclusion, in K3/ST.

On the opposite side of the spectrum, ST/TS,LP/TS and K3/TS are completely substruc-

tural logics -e.g., not even one of the structural metainferential schemes that we have talked

about is valid in them.

Another interesting result is the following: K3/LP and LP/K3 are substructural logics build

entirely with inferential Tarskian logics. Thus, they are, in a way, the reverse of TS/ST.

But maybe the most surprising result that this exploration reveals is that it is not true that

using sets –instead of multisets or sequences– is enough to obtain a consequence relation that

warrants the validity of Contraction. For example, K3/LP is a (metainferential) consequence

24 A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS

relation that understands inferences as pairs of sets of formulas. Nevertheless, Contraction is

not valid in it.

All these facts reveal a partial answer to a question that can be raised against this project:

why should we cared about metainferential consequence relations? Why are they interesting,

and worth spending time on?

One partial answer is that, on the one hand, they seems to work ‘better’ than any inferential

consequence relation that we have explored in this paper. TS/ST provides one clear example.

Non-classical theories of truth pursue two conﬂicting desiderata. On the one hand, they search

for a paradox-free transparent truth predicate. On the other hand, they want to retain as

much classical logic as possible. This conﬂict is recently examined in [17]. There, Hjortland

claimed that ‘nonclassical theories try to recapture classical principles in special cases. This is

a form of damage control’ ([17], p. 1). Hjortland calls this desideratum ‘the maxim of minimal

mutilation.’ Thus, though it might be argued that ST seems to do much better than the

others inferential non-classical solutions to paradoxes –precisely because it resolves paradoxes

while ‘mutilating’ less classical logic than the other non-classical theories–, TS/ST seems to

work even better than ST.TS/ST retains every classically valid inference, as ST does, but,

moreover, it recovers every classically valid metainference –while ST loses Cut (and many others

classically valid metainferences).

Another example of how metainferential consequence relations work ‘better’ than inferential

logics is provided by ST/TS.TS is a logic that has no valid inferences. Nevertheless, TS is

informative about the metainferences –e.g., TS validates some, but not all, metainferences. But

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. ST/TS

is such logic. Nevertheless, this does not mean that ST/TS is ‘as empty’ as a logic can be. But

surely it is ‘emptier’ than TS.

We have already spot another answer to the question about why is it worth exploring metain-

ferential consequence relations: they force us to revise some unquestioned claims that we have

suscribed. For example, that using sets –instead of multisets or sequences– is enough to obtain

a consequence relation that warrants the validity of Contraction. We have saw that K3/LP

is a (metainferential) consequence relation that uses sets to specify inferences, but invalidates

A FAMILY OF METAINFERENTIAL LOGICS-FEDERICO PAILOS 25

Contraction. Moreover, we have shown a model-theoretic way to give up Contraction. Non-

contractive consequence relations, for example, the ones explored in [21], [36], [37] and [27],

exploit the fact that in many contexts it is important to distinguish between two occurrences of

a sentence and just one occurrence of it to give non-trivial solutions to paradoxes. on simulta-

neously Notwithstanding, those approaches are mainly proof-theoretical. It is not an easy thing

to provide extensional –and philosophically relevant– semantics for them. But K3/LP seems

to provide us with that kind of semantics—not available, as far as we know, for traditional,

inferential non-contractive approaches.

Nevertheless, there still is plenty work to do in this ﬁeld. For example, it seems not easy

to ﬁgure out how to compound inferential logics that have diﬀerent many-valued semantics.

For example, how should CL/TS look like? As [14] have established, there is no two-valued

presentation of TS. There are, though, three-valued presentations of CL. One possibility is to

specify a three-valued semantics for CL/TS. Notwithstanding, this kind of solution might face

the question of whether, in this case, that three-valued characterization of CL counts as a truly

classical logic or not.16 We leave this exploration for future work.

5. acknowledgments

Word Count: 9551

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