This paper explores the problem of evaluating meta-arguments within the framework of heterogeneous logics. I introduce two new heterogeneous logics, KLP and LK3, which are based on the interpreted languages of LP and K3. I then present three distinct approaches to meta-argument evaluation, each aimed at preserving, in different ways, the heterogeneity of the underlying logics. Finally, I show that the second approach allows me to distinguish these new logics from others such as TS and ST, offering a more nuanced framework to address meta-argument distinctions in heterogeneous settings.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1 Introduction
According to Humberstone [22], a logic L is heterogeneous iff the language used in the premises of its arguments differs from the one used in the conclusion. In the traditional model-theoretic conception of a logic, both premises and conclusions are formulated within the same language. Such an approach entails that premises and conclusions are evaluated under the same interpretations. In contrast, I take heterogeneous logics to allow the admissible interpretations associated with the premises and the conclusion to differ, since the language of the premises may have interpretations different from those for the language of the conclusion. This breaks with the traditional symmetrical conception of logical reasoning, as it permits arguments in which at least two distinct languages determine, at least partially, what counts as a valid argument.
The asymmetry of heterogeneous logics becomes particularly relevant, for instance, in modeling inferential processes that involve context-sensitive reasoning and translation of languages (see more in [23, 34, 35]). For example, Gillian Russell [35, p.316] defined the context-sensitive sentence PREM as follows:
[L]et ‘prem’ be an atomic sentence whose value is true when it features in the premises of an argument and false when it features in the conclusion (or in any other linguistic context).
Depending on the languages specified, PREM could fit this description. Suppose that I have a truth-preserving logical consequence relation over a language containing PREM. The argument “PREM, therefore PREM” would be invalid—thereby providing a counterexample to Identity, \(A\models _{\tiny {{\textbf {L}}}} A\). (See more in [18].)
Anzeige
On the other hand, according to Humberstone, such logics can also model arguments involving translations, for example: “The grass is green, therefore el pasto es verde”. Moreover, these heterogeneous logics allow for a formal and precise treatment of how meaning and inferential validity can change or be preserved when translating between different languages, which is essential for understanding and formalizing multilingual reasoning and contextual interpretation.
Heterogeneous logics are also useful in modeling languages that share the same evaluation conditions for their connectives but have differences in other semantic aspects, such as their admissible interpretations. Cases like these can be found in certain model-theoretic presentations of logics such as K3 and LP. Using a Dunn-style model-theoretic approach, I can present K3 and LP. Both logics share the same language, although their differences are determined by their collection of admissible interpretations. In K3, some formulas may be neither true nor false, while in LP, some formulas may be both true and false.
In this paper, I consider heterogeneous logics obtained by combining the interpreted languages of K3 and LP. Given the sparse, virtually inexistent literature on heterogeneous logics of this kind, I will show that there are at least three options to evaluate their meta-arguments. My aim is to answer the central question: how should the meta-arguments of heterogeneous logics be evaluated? This is significant within the philosophy of logic as it broadens our understanding of the nature of heterogeneous logics. The plan of this paper is as follows. First, I provide the formal background by presenting the logics LP, K3, ST, and TS. Second, I present two heterogeneous logics. Finally, I introduce three possible options to evaluate meta-arguments in such logics.
2 Background
Let \(\mathcal {L}\) be a formal language obtained in the usual way from a denumerable set PROP of propositional variables \(p_{1},\ldots , p_{n}\). I will use the first capital letters of the Latin alphabet, ‘A’, ‘B’, ‘C’, ... as arbitrary formulas of \(\mathcal {L}\), and the Greek letters ‘\(\Gamma \)’ and ‘\(\Delta \)’ to denote sets of formulas. An interpretation for \(\mathcal {L}\) is a relation \(\sigma \) between PROP and two truth values 1 and 0, where ‘1’ stands for truth and ‘0’ for falsity, such that a propositional variable \(p_i\) can be related to truth values in one of the following four ways:
\(p_i\) is true but not false, represented by ‘\(1\in \sigma (p_i)\) and \(0\notin \sigma (p_i)\)’; more briefly, \( \sigma (p_i)=\{1\}\)
\(p_i\) is true but also false, represented by ‘\(1\in \sigma (p_i)\) and \(0\in \sigma (p_i)\)’; more briefly, \( \sigma (p_i)=\{1, 0\}\)
\(p_i\) is neither true nor false, represented by ‘\(1\notin \sigma (p_i)\) and \(0\notin \sigma (p_i)\)’; more briefly, \( \sigma (p_i)= \)
\(p_i\) is false but not true, represented by ‘\(0\in \sigma (p_i)\) and \(1\notin \sigma (p_i)\)’; more briefly, \( \sigma (p_i)=\{0\}\)
I will extend the interpretations to valuations for all formulas in \(\mathcal {L}\) according to the following evaluation conditions:
Today, it is customary to think of a logic L as a collection of arguments. However, when I speak of arguments, I may refer to different collections, such as valid arguments, invalid arguments, anti-valid arguments, or even meta-arguments, meta-meta-arguments, and so on. (More on the notion of meta-argument below.) Among them, valid arguments are used to identify a logic, in such a way that two logics L1 and L2 are identical if and only if they have the same valid arguments.
Nonetheless, as it has become clear in recent years (see [4, 13, 15, 26, 36]), identifying a logic requires considering at least meta-arguments as well. Thus, I will say that a logic L is a collection of valid arguments and meta-arguments, and that two logics L1 and L2 are identical if and only if they have the same valid arguments and meta-arguments.1
An argument is an expression of the form \(\Gamma \models _{\tiny {{\textbf {L}}}} A\), where \(\models _{\tiny {{\textbf {L}}}}\) stands for a relation of logical consequence, and \(\Gamma \) is also known as a premise set and A is called ‘conclusion’.
Usually, we find that the definition of validity is as follows:
Definition 1.1
An argument is valid in L, \(\Gamma \models _{\tiny {{\textbf {L}}}} A\), iff, for all interpretation \(\sigma \), \(1\in \sigma (A)\) if \(1\in \sigma (B)\), for all \(B\in \Gamma \).
A meta-argument is an argument between arguments, and takes the following form:
Given the properties of meta-arguments, it is not necessary for logical consequence to be strictly identified with the notion of valid argument. That is, for there to be a meta-argument, it is enough that I infer one argument from other arguments. Just as with arguments, where truths or falsehoods can sometimes be inferred from falsehoods, in the realm of meta-arguments it is also possible to infer valid or invalid arguments from invalid arguments. In this way, I can distinguish several forms of meta-arguments, for example:
The following are the usual definitions of global and local validity:
Definition 1.2
A meta-argument is globally valid if and only if, whenever the meta-premises \(\Gamma _1 \models _{\tiny {{\textbf {L}}}} A_1, \ldots , \Gamma _n \models _{\tiny {{\textbf {L}}}} A_n\) are valid, the meta-conclusion \(\Delta \models _{\tiny {{\textbf {L}}}} B\) is also valid.
To define local validity, we must first define what it means for an argument to be satisfied:
Definition 1.3
An argument is satisfied if and only if there are an interpretation such that, whenever the premises are true in that interpretation, the conclusion is also true.
Definition 1.4
A meta-argument is locally valid if and only if the meta-conclusion is satisfied in every interpretation in which the meta-premises are satisfied.
***Digression
Although these definitions are considered standard for the task, the definitions of global and local validity for meta-arguments overlook the characterization I have provided regarding the variety of meta-arguments. To illustrate my point, consider the following meta-arguments. In what follows, I will use the symbol \(\otimes \) to represent conjunction in these meta-arguments.
The first is a form of Explosion for meta-arguments, according to which, in any logic that has at least one argument that is both valid and invalid, any argument can be proven valid.2 To evaluate this meta-argument, it is necessary for me to include an invalid argument among the meta-premises. Meanwhile, Non-Persistence seeks to prevent the first conclusion (B) being combined with a premise from which it followed. For this, I take the meta-conclusion of Non-Persistence to be invalid (see more in Zardini 2019).
If I evaluate both meta-arguments under Global validity, Mexa-Explosion turns out to be vacuously valid, since the requirement that all meta-premises be valid is not met due to the invalidity of the second meta-premise. Therefore, the meta-argument is valid by vacuity. As a result, this notion of Global validity cannot be used to rule out Mexa-Explosion, because it does not allow me to identify logics that are not Mexa-Explosive. The same happens with Persistence: assuming the meta-premises are valid and the meta-conclusion is not, Persistence will be a “valid” meta-argument when it is globally invalid.
One might think that the case of Non-Persistence could be reformulated through subtle adjustments to the traditional notion. Specifically, I could require that the meta-conclusion be valid and the property called Persistence (or given another name). This reformulation proves technically adroit for Non-Persistence. It reframes the condition by requiring validity of the meta-conclusion while introducing Persistence as the complementary property. Thus, a logic validates Non-Persistence precisely when it violates Persistence. While this approach resolves Non-Persistence effectively, it remains inapplicable to Mexa-Explosion.
An analogous situation arises when I evaluate these arguments under the definition of Local validity. The only difference is that, for a meta-argument to be locally valid in the case of Mexa-Explosion, the meta-premises must be both satisfiable and not satisfiable, while the meta-conclusion must be satisfiable. On the other hand, Persistence requires satisfiable meta-premises and an unsatisfiable meta-conclusion. This configuration exposes a fundamental limitation in Local Validity for evaluating such meta-arguments. Nevertheless, my proposed No-Persistence framework resolves this deficiency.
Since both existing definitions prove inadequate for capturing all meta-argumental phenomena (notably Meta-Explosion), I propose the following formal notation: An argument for L is evaluated through a relation between the collection of subsets of \(\mathcal {P}(Form) \times Form\) and the collections VAL (valid arguments) and INVAL (invalid arguments), such that an argument \(\Gamma \models _{{\textbf {L}}} A\) can be classified in exactly one of the following four ways, thereby generalizing the notation originally proposed in [12, p. 190]:
\(\Gamma \models _{\tiny {{\textbf {L}}}} A\) is valid but not invalid, represented by \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \in VAL\) and \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \notin INVAL\); more briefly: \(\Gamma \mathrel {\models _{L}^{\checkmark \!\!\circ }} A\).
\(\Gamma \models _{\tiny {{\textbf {L}}}} A\) is both valid and invalid, represented by \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \in VAL\) and \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \in INVAL\); more briefly: \(\Gamma \mathrel {\models _{L}^{\checkmark \!\!\times }} A\).
\(\Gamma \models _{\tiny {{\textbf {L}}}} A\) is neither valid nor invalid, represented by \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \notin VAL\) and \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \notin INVAL\); more briefly: \(\Gamma \mathrel {\models _{L}^{\circ \circ }} A\)
\(\Gamma \models _{\tiny {{\textbf {L}}}} A\) is invalid but not valid, represented by \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \in INVAL\) and \((\Gamma \models _{\tiny {{\textbf {L}}}} A) \notin VAL\); more briefly: \(\Gamma \mathrel {\models _{L}^{\circ \times }} A\).
This convention provides me with a clear distinction between both cases within a logic \(\textbf{L}\). To capture a more general definition, I let \(\Gamma \mathrel {\models _{L}^{\star \star }} A\) denote a consequence relation (which may be valid or invalid), where \(\star \) is a placeholder that can be replaced by:
\(\checkmark \) or \(\circ \) on the left
\(\times \) or \(\circ \) on the right
Thus, a general way to define a meta-argument is as follows:
The definition of global validity is then given as follows:
Definition 1.5
A meta-argument is globally valid if and only if, whenever the meta-premises \(\Gamma _1 \mathrel {\models _{L}^{\checkmark \!\!\star }} A_1, \ldots , \Gamma _n \mathrel {\models _{L}^{\checkmark \!\!\star }} A_n\) the meta-conclusion \(\Delta \mathrel {\models _{L}^{\checkmark \!\!\star }} B\).
As is evident, this notation is insufficient to generalize the definition of local validity. However, by replacing \(\checkmark \) with \(+\) (satisfaction) and \(\times \) with − (non-satisfaction), mutatis mutandis, we can extend the framework to capture local validity as follows:
Definition 1.6
A meta-argument is locally valid if and only if, if meta-premises \(\Gamma _1 \mathrel {\models _{L}^{+ \star }} A_1, \ldots , \Gamma _n \mathrel {\models _{L}^{+ \star }} A_n\), the meta-conclusion \(\Delta \mathrel {\models _{L}^{+ \star }} B\).
These definitions are more general than those employed in the related literature and allow me to evaluate different types of meta-arguments.3 In particular, they make it possible for me to assess any type of meta-argument without the need to restrict the analysis solely to those with valid or satisfiable premises.
End of digression***
Based on the language and definition of validity given in Definition 1.1, I can obtain a logic such as FDE. Given that my interest is to present the logics K3 and LP, I now show how this can be done within this theoretical-model framework. The logics LP and K3 can be obtained from the same framework by excluding, respectively, the interpretation that is neither true nor false ( ) and the one that is both true and false ({1,0}). Below, I present some properties of each logic.
K3 is a paracomplete logic since \(B \not \models _{\tiny {{\textbf {L}}}} A \vee \sim A\). It has no logical truths, but it preserves the same logical falsehoods as classical logic. This logic has been proposed for dealing with incomplete data. See more about this logic in [24]. On the other hand, LP is a paraconsistent logic since \(A \wedge \sim A \not \models _{\tiny {{\textbf {L}}}} B\). It also lacks logical falsehoods but shares the same logical truths as classical logic. Using LP, it is possible to handle inconsistent information without leading to logical triviality. See more about this logic in [28].
With these preliminaries, I now move into presenting the logics ST and TS. To do so, I consider two logical consequence relations: st-consequence (see more in [20], and [19]) and ts-consequence:
Definition 1.7
An argument is st-valid in L, \(\Gamma \models _{\tiny {{\textbf {L}}}} A\), iff, for all interpretations \(\sigma \), if \(1\in \sigma (B)\) and \(0\notin \sigma (B)\), for all \(B\in \Gamma \), then \(1\in \sigma (A)\) or \(0\notin \sigma (B)\).
Definition 1.8
An argument is ts-valid in L, \(\Gamma \models _{\tiny {{\textbf {L}}}} A\), iff, for all interpretations \(\sigma \), if \(1\in \sigma (B)\) or \(0\notin \sigma (B)\), for all \(B\in \Gamma \), then \(1\in \sigma (A)\) and \(0\notin \sigma (B)\).
These logical consequence relations have received considerable attention in recent years due to their applications in treating various semantic paradoxes. The st-consequence can be read informally as follows: if the premises are true but not false, the conclusion cannot be merely false (i.e., it must be true or not false). In contrast, the ts-consequence can be read informally as: if the premises are not only false (i.e., they are true or not false), the conclusion is true but not false. In general, with the appropriate ingredients, st-consequence relations tend to be non-transitive, whereas ts-consequence relations tend to be non-reflexive. These properties have enabled the blocking of certain arguments that would otherwise lead to results approaching triviality (see, for example, [3, 21, 31]). Furthermore, by keeping truth and falsehood distinct from non-truth and non-falsehood, these relations make room for non-Tarskian notions of logical consequence.
From the language I have introduced, together with the st-consequence relation, I obtain the logic ST; similarly, from the ts-consequence relation, I obtain the logic TS. These logics are typically presented using a semantics with three admissible interpretations, see for example [11]. Nevertheless, both presentations are coextensional.4 Below, I outline some properties of ST, followed by some properties of TS.
ST is a logic that shares both the logical truths and the valid arguments with classical logic. That is, every valid argument in ST is also valid in classical logic. However, depending on whether a local or global notion of validity is adopted, the two logics may differ in terms of their meta-arguments. As shown by [2, 4, 9], Transitivity is a meta-argument that serves this role. Transitivity has the following form:
\(\text {If} \ A \models _{\tiny {{\textbf {L}}}} B \ \text {and} \ B \models _{\tiny {{\textbf {L}}}} C, \ \text {then} \ A \models _{\tiny {{\textbf {L}}}} C\)
In ST, Transitivity is globally valid. Since this logic preserves the same valid arguments as classical logic, we cannot find a counterexample in which the meta-premises are valid but the meta-conclusion is not. Therefore, with the tools I have presented here, it is not possible to distinguish ST from classical logic in terms of global validity. Hence, when ST is evaluated with global validity, it is identical to classical logic. See more about this in [3, 8, 17].
In ST, Transitivity is not locally valid. This is because there is at least one interpretation in which \(A\) is only true, \(B\) is true and false, and \(C\) is only false. This interpretation satisfies the meta-premises but fails to satisfy the meta-conclusion. From this I draw an important insight: it is only by appealing to local validity at the meta-argumentative level that we can distinguish ST from classical logic.5
Finally, TS is a logic with no valid arguments; that is, regardless of the set of premises, all arguments are logically invalid. A consequence of having no valid arguments is that TS is a paranormal logic, being both paraconsistent and paracomplete. This is because, for a logic to be paraconsistent or paracomplete, it must invalidate a specific argument. Since TS invalidates all arguments, it also invalidates Explosion and Implosion. Furthermore, the set of invalid arguments is trivialized because every argument in TS is invalid.
The status of valid meta-arguments in TS varies depending on whether local or global validity is used. If I evaluate the meta-arguments of TS with global validity, all meta-arguments turn out to be valid. This is due to the interpretation of the conditional in the global definition of validity. Usually, the conditional is understood extensionally; that is, a conditional of the form ‘if...then...’ is equivalent to ‘not...or ...’, so the global validity definition is equivalent to: an argument is globally valid if and only if the premises are invalid or the conclusion is valid. Since in TS all arguments are invalid, all meta-arguments satisfy the first disjunction.6 Therefore, global validity leads to a trivialization of the collection of meta-arguments.
If we evaluate the meta-arguments of TS with local validity, then there are arguments like ‘If \(A \models _{\tiny {{\textbf {TS}}}} A\) then \(A \models _{\tiny {{\textbf {TS}}}} A\)’ that are locally valid and arguments like ‘If \(A \models _{\tiny {{\textbf {TS}}}} B\) then \(C \models _{\tiny {{\textbf {TS}}}} D\)’ that are not. The latter is not locally valid because there is an interpretation \(\sigma (A)=\{1\}\), \(\sigma (B)=\{1\}\), \(\sigma (C)=\{1\}\) and \(\sigma (D)=\{0\}\) in which the premises are fulfilled but the conclusion is not. In this way, the definition of local validity allows me to distinguish valid from invalid meta-arguments, thus identifying which arguments could be invalid in a logic like TS.
3 Heterogeneous Logics
In this section, I present two heterogeneous logics based on the interpreted languages of LP and K3. However, before doing so, I first introduce some relevant formal definitions. Let \(\mathcal {L1}_{\sigma {p}}\) and \(\mathcal {L1}_{\sigma {c}}\) be interpreted languages, each associated with a set of interpretations for the premises and the conclusion, respectively.
Definition 2.1
A logic L is heterogeneous iff for all argument of the form \(\Gamma \models A\), for all \(B\in \Gamma \), \(B\in \mathcal {L1}_{\sigma _{p}}\) and, \(A\in \mathcal {L2}_{\sigma _{c}}\); such that \(\mathcal {L1}_{\sigma _{p}}\ne \mathcal {L2}_{\sigma _{c}}\).
Consider two heterogeneous logics which we will call LK3 and KLP. The admissible sets of interpretations for LK3 are as follows:
For premises (as in LP) \(\{\{1\}, \{1,0\}, \{0\}\}\).
For the conclusion (as in K3) \(\{\{1\}, \{ \ \}, \{0\}\}\)
And the admissible sets of interpretations for KLP are as follows:
For premises (as in K3) \(\{\{1\}, \{ \ \}, \{0\}\}\).
For the conclusion (as in LP) \(\{\{1\}, \{1,0\}, \{0\}\}\)
Both LK3 and KLP will be evaluated according to the set of admissible interpretations of the languages that correspond to them, as follows:
I will now present the logic LK3, followed by the logic KLP, highlighting some of their properties.
3.1 LK3
An interpretation for premises, \(\sigma _{p}\), is a function between the set of propositional variables and the set of interpretations \(\{\{1\}, \{1,0\}, \{0\}\}\). An interpretation for the conclusion, \(\sigma _{c}\), is a function between the set of propositional variables and the set of interpretations \(\{\{1\}, \{ \ \}, \{0\}\}\). The evaluation conditions of the connectives are the same as those I presented in the previous section. The logical consequence relation for this logic is the usual one: a truth-preserving logical consequence relation.
Definition 2.2
Let \(\Gamma \) be a set of formulas. A is a logical consequence of \(\Gamma \) in LK3, \(\Gamma \models _{\tiny {{\textbf {LK3}}}} A\), if and only if, for every interpretation \(\sigma _{p}\) and \(\sigma _{c}\), \(1\in \sigma _{c}(A)\), if \(1\in \sigma _{p}(B)\) for every \(B\in \Gamma \).
The premises can be true, false, or both true and false, while the conclusion can be true, false, or neither true nor false. For example, suppose I am evaluating the argument \(A \models _{\tiny {{\textbf {LK3}}}} A\). All the possible admissible evaluations in this logic are represented in the following table:
The argument is logically invalid, since there is at least one interpretation in which the premises are true and the conclusion is not. Consider the following interpretation: \(\sigma _{p}(A)= \{1,0\}\) and \(\sigma _{c}(A)= \{ \ \}\). Since this interpretation will always be available regardless of the argument, all arguments are logically invalid in LK3. In other words, no matter the set of premises or the conclusion, there is no conclusion that logically follows from any set of premises in LK3.
3.2 KLP
Now for KLP, an interpretation for premises, \(\sigma _{p}\), is a function between the set of propositional variables and the set of interpretations \(\{\{1\}, \{ \ \}, \{0\}\}\). An interpretation for the conclusion, \(\sigma _{c}\), is a function between the set of propositional variables and the set of interpretations \(\{\{1\}, \{1,0\}, \{0\}\}\). The evaluation conditions of the connectives are the same as those I presented in the previous section. The logical consequence relation is the same as in LK3, that is, one of truth preservation.
Definition 2.3
Let \(\Gamma \) be a set of formulas. A is a logical consequence of \(\Gamma \) in KLP, \(\Gamma \models _{\tiny {{\textbf {KLP}}}} A\), if and only if, for every interpretation \(\sigma _{p}\) and \(\sigma _{c}\), \(1\in \sigma _{c}(A)\), if \(1\in \sigma _{p}(B)\) for every \(B\in \Gamma \).
The premises can be true, false, or neither true nor false, while the conclusion can be true, false, or both true and false. If I evaluate the same argument presented above, but now using KLP, the following table represents all the admissible evaluations in this logic:
The argument is logically valid in KLP, since whenever the premises are true, the conclusion is also true. Some valid arguments in this logic are the following:
When we work with model-theoretic presentations of the logics ST or TS, using three interpretations (for example, \(\{\{1\}, \{1,0\}, \{0\}\}\)), the way we evaluate premises in these heterogeneous logics can be related to how we assess mixed logical consequence relations. Consider, for instance, the set of admissible interpretations that preserve truth under the two standards of ST and TS:
The standard7s preserves the same truth interpretation as the one I used in my model-theoretic presentation of K3. Similarly, the standard t preserves the same truth interpretation as that employed in my model-theoretic presentation of LP. That is, when I evaluate an argument in the logic KLP, I use the interpreted language of K3 for the premises and that of LP for the conclusion. According to this correspondence, the logic ST should preserve truth from premises to conclusion in the same arguments. A similar situation arises in the case of LK3.
Considering the earlier discussion, I observe that the logic TS preserves truth in exactly the same arguments as LK3. Likewise, the logic ST preserves truth in exactly the same arguments as KLP. What I aim to establish here is that an argument is valid in TS if and only if it is valid in LK3. Since neither logic admits any valid arguments, this is easily seen in this case. On the other hand, an argument is valid in ST if and only if it is valid in KLP. Since the proof is straightforward, I leave it to the reader.8 However, this is not sufficient to claim that both logics are identical, since to do so they should also share the same valid meta-arguments.
Given the novelty of heterogeneous logics, there are still no clear proposals on how to evaluate meta-arguments within these logics. Philosophically, an even more interesting question is: what does it really mean for a meta-argument to be valid in a heterogeneous logic? Below, I will present some options for evaluating meta-arguments in these logics and the consequences that adopting each option would have in LK3 and KLP.
4 Meta-Arguments in Heterogeneous Logics
There are at least three ways in which we can evaluate meta-arguments in heterogeneous logics. These evaluation tools are independent of whether I assess the meta-arguments using the global or local definition of validity. I will now present two of these options and leave the third for the end of this section. In both cases, HL refers to a heterogeneous logic, and the labels I and II indicate that the formulas belong to different formal languages.
To facilitate organization and comparison, I have assigned each logic a subindex. These subindexes will be used throughout this section.
The subindex G stands for global validity.
The subindex L stands for local validity.
The number 1 or 2 indicates the use of Option 1 or Option 2, respectively.
The absence of a number indicates the logic is considered without specifying an option.
4.1 Option 1
Option 1 preserves the heterogeneity of arguments, meaning I evaluate each one heterogeneously. However, I assess meta-arguments traditionally, as in logics like ST and TS (see for example [4, 9, 27]). I will now examine how certain meta-arguments from LK3 and KLP behave under Option 1 and then under Option 2.
According to Option 1, meta-arguments must be evaluated as follows:
Both meta-arguments are globally valid. In fact, any meta-argument I evaluate globally using Option 1 will be valid. This occurs because, since there are no logically invalid arguments in LK3, all meta-arguments will be globally valid under Option 1. This situation is analogous to the case I previously presented with TS and the trivialization of the meta-argument collection. Therefore, with this option, LK3G1 is identical to TSG, since the same arguments and meta-arguments are valid in both logics.
Global Validity in KLP: Let us now examine how Option 1 works for meta-arguments in KLP using Option 1 (1.5). Since ST and KLP validate the same arguments, it is worth considering the following meta-argument:
\(\text {If} \ A \models _{\tiny {{\textbf {KLP}}}} B \ \text {and} \ B \models _{\tiny {{\textbf {KLP}}}} C, \ \text {then} \ A \models _{\tiny {{\textbf {KLP}}}} C\)
When I evaluate this argument using global validity and Option 1, the meta-argument will be valid in both logics. In fact, all meta-arguments that are valid in STG are also valid in KPLG1. Consequently, I can establish that under this option, KPLG1, classical logic, and STG are identical logics, as they validate the same arguments and meta-arguments.
Local Validity in LK3: Now, let’s proceed to evaluate the logics LK3 and KLP using Option 1 and the local validity definition (1.6). If we analyze the same arguments of LK3 as before, we find that arguments such as:
This turns out to be logically invalid because there is at least one interpretation in which the premises are true and the conclusion is non-true. Consider the interpretation \(\sigma _{p}(A)= \{1\}\), \(\sigma _{p}(c)= \{1\}\), \(\sigma _{c}(B)= \{1\}\), and \(\sigma _{c}(D)= \{0\}\). However, meta-arguments such as ‘If \(A \models _{\tiny {{\textbf {LK3}}}} A\) then \(A \models _{\tiny {{\textbf {LK3}}}} A\)’ are valid in LK3L1. In fact, every meta-argument that I find locally valid in LK3L1 is also valid in TSL. Therefore, LK3L1 and TSL are identical.
Local Validity in KLP: Now, when I consider the status of valid meta-arguments in KLP using Option 1 and local validity, I find that arguments such as Transitivity turn out to be invalid. Consider the interpretation \(\sigma _{p}(A)= \{1\}\), \(\sigma _{c}(B)= \{1,0\}\), \(\sigma _{p}(B)= \{ \ \}\), and \(\sigma _{c}(C)= \{0\}\). I observe that what is really doing the work is that B can be evaluated in two different languages. When it appears as a premise, it is evaluated as \(\{ \ \}\), but when it appears as a conclusion, it is evaluated as \(\{1,0\}\). In fact, under Option 1, every valid meta-argument in KLPL1 is also valid in STL. Therefore, KLPL1 and STL are identical.
Option 1 does not allow me to establish that LK3 and KLP are logics distinct from TS and ST, respectively. This is because the global definition of validity can only be assessed based on the arguments that hold in each logic. Since the valid arguments coincide across these logics, the meta-arguments coincide as well. As a result, I can only regard the heterogeneous logics LK3 and KLP as model-theoretic presentations of TS and ST, respectively. This result is not entirely negative. It shows that it is possible to provide a model-theoretic presentation of a heterogeneous logic that is probably identical to classical logic. However, given the focus of this paper, I leave the proof of this claim to the reader.
4.2 Option 2
In contrast, Option 2 evaluates meta-arguments heterogeneously but arguments homogeneously. That is, I assess arguments through a uniform criterion, while meta-arguments follow a differentiated evaluation. According to Option 2, meta-arguments must be evaluated as follows:
Under Option 2, we preserve the heterogeneity of the logics based on how their meta-premises and meta-conclusions are evaluated. From a formal perspective, meta-premises are premises, likewise, meta-conclusions are conclusions, and therefore I must evaluate them in their corresponding languages. If meta-premises are evaluated as premises, then they will only use interpretations \(\sigma _{p}\) and \(\sigma _{c}\), which corresponds to interpretation \(\sigma \). Consequently, every meta-premise of LK3 or KLP will be evaluated as in LP (K3), and every meta-conclusion will be evaluated as K3 (LP).
Let me begin with the evaluation of the logics LK3 and KLP using Option 2 and the definition of global validity given in (1.6).
Global Validity in LK3: Consider the same meta-arguments that are globally valid in TS, and let us proceed to evaluate them in LK3:
Since Identity is logically valid both in LP and in K3, I find that the first meta-argument will be valid. However, the situation with the second meta-argument is different. When I evaluated a meta-argument in LK3G1, every meta-argument turned out to be valid. That is, since all the meta-premises were invalid, all such meta-arguments were considered valid. However, under Option 2 and global validity, only those meta-arguments whose meta-premises are invalid in LP will be valid. Let me look at a couple of examples to understand this more clearly:
As the argument \(\models _{\tiny {{\textbf {LP}}}} \sim (A \rightarrow A)\) is invalid and argument \(\models _{\tiny {{\textbf {K3}}}} A \rightarrow A\) is invalid, I find that the meta-argument is logically valid. Up to this point, the result appears to coincide with what I observed under Option 1. However, the second meta-argument is logically invalid. This is because the premises hold in LP, but not in K3. This result is significant, as it allows me to assert that not all meta-arguments are valid in LK3G2. Since both logics differ in their sets of valid meta-arguments, I can conclude that LK3G2 is not identical to LK3G1 (nor to TSG).
Global Validity in KLP: Now, consider the same meta-arguments that are globally valid in ST, and let us proceed to evaluate them in KLP. To evaluate Transitivity, let us consider the following instance of the meta-argument:
If \(A \wedge \sim A \models _{\tiny {{\textbf {K3}}}} B,\ B \models _{\tiny {{\textbf {K3}}}} B\) then \(A \wedge \sim A \models _{\tiny {{\textbf {LP}}}} B\)
This argument is invalid in KLPG2. This is because the meta-premises \(A \wedge \sim A \models _{\tiny {{\textbf {K3}}}} B\) and \(B \models _{\tiny {{\textbf {K3}}}} B\) hold in K3, but the meta-conclusion \(A \wedge \sim A \models _{\tiny {{\textbf {LP}}}} B\) does not hold in LP. I consider the following interpretation \(\sigma _{c}(A)= \{1,0\}\) and \(\sigma _{c}(B)= \{0\}\), which makes the meta-conclusion invalid in LP. Therefore, KLPG2, is a non-transitive logic. This result differs from the one I previously presented for STG, since STG, under this criterion, is a transitive logic identical to classical logic. In contrast, as I have shown, KLPG2 and STG are not identical logics.
Local Validity in LK3: Now, I am left with evaluating LK3 and KLP using the local validity definition under Option 2. Let me first consider some meta-arguments from LK3, and then others from KLP. For example, I consider the following meta-argument of LK3, which is logically valid in TS (under local validity):
The meta-argument will be invalid in LK3L2, since there is an interpretation that satisfies the meta-premises but does not satisfies the meta-conclusion. It is enough for me to consider interpretation \(\sigma _{p}(A)= \{1\}\), \(\sigma _{p}(A)= \{1,0\}\), \(\sigma _{c}(A)= \{1\}\), and \(\sigma _{c}(A)= \{ \ \}\). That is, although it is valid in TSL, it is invalid in LK3L2. Moreover, I may be able to make a more general claim about arguments like:
\(\text {If } \models _{\tiny {{\textbf {LP}}}} A, \text { then } \models _{\tiny {{\textbf {K3}}}} A\)
This one meta-argument is invalid in LK3L2. However, I find Option 2 useful for distinguishing LK3L2 from TSL, but it is still not sufficient to differentiate it from LK3G2. This is because the last argument would also be globally invalid, given that the meta-premises hold in LP but do not hold in K3.
However, consider the following argument from LK3:
‘If \(A\rightarrow A \models _{\tiny {{\textbf {LP}}}} B \) then \(B \models _{\tiny {{\textbf {K3}}}} A\rightarrow A\)’
This meta-argument is valid in LK3G2. This is because the premises do not hold, and therefore I consider the meta-argument valid by vacuity. On the other hand, consider the following interpretation: \(\sigma _{p}(A)= \{1,0\}\), \(\sigma _{p}(B)= \{1\}\), \(\sigma _{c}(B)= \{1\}\), and \(\sigma _{c}(A)= \{ \ \}\). Under this interpretation, the meta-premises are satisfied, but the meta-conclusion is not. Since this meta-argument is invalid in LK3L2 and valid in LK3G2, I conclude that LK3G2 is not identical to LK3L2.
Local Validity in KLP: Finally, I evaluate the meta-arguments of KLP under local validity and Option 2. Recall that Transitivity is invalid in STL, although it is also invalid in KLPG1. The counterexample to Transitivity in Option 1, namely \(\sigma _{p}(A)= \{1\}\), \(\sigma _{p}(B)= \{1,0\}\), \(\sigma _{p}(C)= \{0\}\), \(\sigma _{c}(A)= \{1\}\) and \(\sigma _{c}(C)= \{0\}\), cannot be replicated under Option 2, as it would yield \(\sigma _{p}(A)= \{1\}\), \(\sigma _{p}(B)= \{ \ \}\), \(\sigma _{p}(C)= \{0\}\), \(\sigma _{c}(A)= \{1\}\) and \(\sigma _{c}(C)= \{0\}\) instead. Thus, no counterexample to Transitivity is possible in this case. Therefore, Transitivity is valid in KLPL2. However, now I consider the following argument:
$$\begin{aligned}&\text {If } \Gamma \models _{\tiny {{\textbf {L}}}} A, \text { then } \Gamma , B \models _{\tiny {{\textbf {L}}}} A. \end{aligned}$$
(Monotonicity)
For example, consider the following instance:
\(\text {If } (A \rightarrow B) \wedge A \models _{\tiny {{\textbf {K3}}}} B, \text { then } (A \rightarrow B) \wedge A, C \models _{\tiny {{\textbf {LP}}}} B\)
Now, consider the following interpretation: \(\sigma _{p}(A) = \{\ \}\), \(\sigma _{p}(B) = \{0\}\), \(\sigma _{c}(A) = \{1,0\}\), \(\sigma _{c}(B) = \{0\}\), and \(\sigma _{c}(C) = \{1\}\). Under this interpretation, we observe the following:
In the meta-premises: \(\sigma _{p}(A \rightarrow B) = \{\ \}\), so \(\sigma _{p}((A \rightarrow B) \wedge A) = \{\ \}\). Since \(\sigma _{p}(B) = \{0\}\), there is an interpretation that satisfies the meta-premises (by vacuity).
On the other hand, in the meta-conclusion: \(\sigma _{p}(A \rightarrow B) = \{1,0\}\), so \(\sigma _{p}((A \rightarrow B) \wedge A) = \{1,0\}\). If \(\sigma _{c}(C) = \{1\}\) (or \(\sigma _{c}(C) = \{1,0\}\)), that same interpretation does not satisfy the meta-conclusion.
Hence, I conclude that Monotonicity is locally invalid KLPL2. Therefore, since Monotonicity is a valid argument both in STL and in KLPL1, but is invalid in KLPL2, it follows that KLPL1 and KLPL2 are not identical logics.
4.3 Summary
I have evaluated a total of twelve logics. Below is the list of logics with their corresponding labels:
Bild vergrößern
The following list presents a set of summary observations in which I aim to identify which logics are identical to others already considered. These comparisons serve to highlight identities, redundancies, and distinctive features among the logics I have evaluated.
TSG is a logic without valid arguments, but all its meta-arguments are logically valid.
TSL is a logic without valid arguments, but not all its meta-arguments are invalid.
STG is identical to classical logic.
STL validates all arguments of classical logic, but it is non-transitive.
LK3G1 is identical to TSG.
KLPG1 is identical to classical logic and to STG.
LK3L1 is identical to TSL.
KLPL1 is identical to classical logic and to STL.
LK3G2 invalidates the meta-argument “If \(\models _{\tiny {{\textbf {LP}}}} A \rightarrow A\), then \(\models _{\tiny {{\textbf {K3}}}} A \rightarrow A\)”, which is valid in LK3G1.
KLPG2 is non-transitive, unlike KLPG1.
LK3L2 invalidates the meta-argument “If \(A\rightarrow A \models _{\tiny {{\textbf {LP}}}} B \) then \(B \models _{\tiny {{\textbf {K3}}}} A\rightarrow A\)”, unlike LK3G2.
KLPL2 invalidates the meta-argument “If \((A \rightarrow B) \wedge A \models _{\tiny {{\textbf {K3}}}} B\), then \((A \rightarrow B) \wedge A, C \models _{\tiny {{\textbf {LP}}}} B\)”, unlike KLPL1.
4.4 A Third and Promising Option
The question of how meta-arguments should be evaluated in heterogeneous logics remains open. While both Option 1 and Option 2 may be philosophically appealing, there is still a further alternative to consider. Recall that in a heterogeneous logic, the premises of an argument are evaluated using a different language than the one used for evaluating the conclusion. In the cases examined here, we refer specifically to interpreted languages.
Although the premises in Option 1 and Option 2 are evaluated within a specific interpreted language, they are not evaluated with respect to a logic per se. Here, the interaction between these two interpreted languages gives rise to what we consider a logic. A key contribution of each interpreted language is its set of admissible interpretations. For example, in the case of LK3, formulas in the premises can be either true or false, while formulas in the conclusion can be neither true nor false. In the case of KLP, formulas in the premises can be neither true nor false, while formulas in the conclusion can be both true and false.
At the level of formulas (that is, in arguments), heterogeneous logics allow for different patterns of evaluation. Therefore, at the level of arguments (that is, in meta-arguments), heterogeneous logics should also allow for different forms of evaluation. Traditionally, formulas are either true or false. However, when working with Dunn semantics, formulas may take on additional semantic statuses. By contrast, arguments are typically classified only as either valid or invalid. Yet, this does not necessarily have to be the case.
Imagine, for instance, a scenario in which certain arguments are both valid and invalid. As a precedent, we already have examples of contradictory logics such as those discussed in [29, 38]. A logic is contradictory if and only if both \(\models _{\tiny {{\textbf {L}}}} A\) and \(\models _{\tiny {{\textbf {L}}}} NA\) hold. What we propose here is to take the next natural step and accept that it may be the case that a logic both validates and does not validate the same argument; that is, both \(\models _{\tiny {{\textbf {L}}}} A\) and \(\not \models _{\tiny {{\textbf {L}}}} A\) hold. On the other hand, it is also easy to imagine a logic in which some arguments are neither valid nor invalid. We are not referring to arguments whose status is undetermined, but rather to arguments for which we can confidently assert that they are not valid and also not invalid (see [33] for a discussion of this type of approach).
In summary, the arguments of heterogeneous logics allow us to evaluate formulas in the following way:
In this option, we preserve the heterogeneity of meta-arguments by extending the semantic distinctions that apply to formulas directly to arguments themselves. This approach builds on Option 2 but represents a distinct way of evaluating meta-arguments. We set this option aside for now because, in logics such as K3 or LP, where arguments are exclusively either valid or invalid—with no arguments that are neither valid nor invalid—it effectively collapses into Option 2.
Since in both LP and K3 arguments form a collectively exhaustive and mutually exclusive partition (i.e., every argument is either valid or invalid), evaluation beyond Option 2 is neither novel nor necessary. Consequently, this approach does not yet constitute a genuinely new option given the logics considered here. However, as noted, there are logics where arguments can possess additional statuses. Only within such logics will this approach enable a more nuanced evaluation of meta-arguments, making it a promising subject for future research.
This perspective could be seen as an attempt to capture a broader range of logical phenomena. For example, it is often assumed that invalidity is simply the negation of validity. However, as we have argued, this need not be the case. One might consider that some arguments are both valid and not valid, thereby distinguishing between arguments that are invalid and those that are merely not valid. To clarify this distinction, consider the hypothetical scenario of a logic in which all arguments are invalid. A question that rarely arises is whether such a logic could be trivial in the sense that all its arguments are also valid. The standard answer would be no: if all arguments are invalid, then none are valid. Yet, another possibility is that all arguments are both invalid and valid, so none are merely not valid. Finally, a third possibility is that all arguments are simultaneously valid, invalid, and not valid. Expanding our conception of triviality in this manner may yield deeper insights into its nature and allow exploration at a more intricate level.
On one hand, this form of heterogeneity better reflects the true nature of meta-premises and meta-conclusions: the former are, after all, premises, and the latter conclusions. Treating them as such preserves their distinct roles within each argument and ensures that their evaluation respects the intended asymmetries of heterogeneous frameworks. On the other hand, continuing to rely solely on Option 1 for evaluating meta-arguments limits our formal frameworks to homogeneous assessments, thereby neglecting heterogeneous features of logics that may be philosophically and technically significant.
5 Conclusion
There is still a long way to go in the study of heterogeneous logics. In this paper, I have presented LK3 and KLP, two heterogeneous logics based on the interpreted languages of K3 and LP. Although these logics can validate the same arguments as logics like TS and ST, I have proposed three different options for evaluating meta-arguments. Some of these options allow me to understand these new heterogeneous logics, rather than mere variants of TS or ST.
One formal advantage of my proposals for evaluating meta-arguments is that they allow me to choose whether I want to apply global or local validity. This choice is especially relevant, since it remains an open issue in the field. Both alternatives allow for different sets of arguments to be classified as valid or invalid in heterogeneous logics, thereby enabling a more flexible exploration of the theoretical and philosophical consequences of each logic. Another formal advantage is that I have introduced, for the first time, more general definitions of global and local validity for meta-arguments, to cover a broader range of possible cases. Existing definitions tend to consider only those meta-arguments that preserve validity from valid meta-premises to a valid meta-conclusion. However, this view excludes an entire class of meta-arguments that I have presented in this work. While this new characterization is useful for my purposes, likely, many researchers will also find it philosophically valuable.
All the options I have presented preserve the heterogeneity in the evaluation of arguments and/or meta-arguments. Option 1 preserves heterogeneity at the level of arguments but loses it at the level of meta-arguments. This is the most conservative option, in the sense that it corresponds to the usual method for evaluating meta-arguments in classical logic. However, in logics with mixed consequence relations, such as TS and ST, we also find that meta-arguments are typically evaluated in the same way. In other words, although logics like TS and ST are usually presented model-theoretically through mixed consequence relations for arguments, this feature tends to go unnoticed when it comes to the evaluation of meta-arguments.
Option 2 preserves heterogeneity at the level of meta-arguments but loses it at the level of arguments. It is arguably the one with the greatest potential for generating philosophically interesting results. This is because, if I evaluate the meta-arguments of the heterogeneous logics I have introduced using this option, I can show that I am not merely offering new model-theoretic presentations of TS or ST, but rather proposing genuinely new logics in the strict sense. I have stated both explicitly and implicitly that this option seems to answer the central question of this paper: meta-arguments in heterogeneous logics should be evaluated using Option 2. Indeed, this option preserves the heterogeneity of meta-arguments, allowing for distinctions that are not possible under Option 1. Moreover, it also respects the asymmetric structure between meta-premises and meta-conclusions, which is central to the spirit of heterogeneous logics and is lost when meta-arguments are evaluated homogeneously.
Finally, with Option 3, I have opened an even more general door, allowing me to abandon the exclusivity and exhaustiveness of the distinction between valid and invalid arguments. Since more and more logics are emerging that include valid and invalid arguments, as well as arguments that are neither valid nor invalid, I will need sufficiently general frameworks that enable me to evaluate the different ways these arguments can be related. Option 3 provides me with a much broader framework for the evaluation of meta-arguments.
The following diagram summarizes the logics discussed throughout this paper. Solid arrows represent relations of greater logical strength. That is, the target logic validates all arguments (or meta-arguments) validated by the source logic, and possibly more. Dotted lines indicate that the connected logics are identical, even if they differ in presentation or formulation. This representation highlights both the hierarchical relationships among the logics and the subtle distinctions between them, especially regarding the validity of arguments and meta-arguments (Fig. 1).
With this paper, I hope to have taken a small step toward understanding what heterogeneous logics truly are and, above all, how we should evaluate their meta-arguments. However, I have not addressed here another type of heterogeneous logics that, while sharing the same interpretations, differ in the evaluation conditions used to define their connectives. This aspect has the potential to further enrich the approach and open new avenues for research. Furthermore, the potential benefits of heterogeneous logics in resolving paradoxes or providing a non-trivial framework for unusual connectives have not been thoroughly examined yet. For instance, the logic KLP might also serve to block various semantic paradoxes or give a formal account of unusual connectives, as has been evaluated with ST in [14, 31, 32]. The door is open for further research in this area.
Acknowledgements
I would like to thank the evaluating committee of the Raúl Orayen Unilog Logic Prize for México 2025 for their support, and the Mexican Academy of Logic (AML) for organizing and promoting this initiative. I am also grateful to Luis Estrada-González for raising the question addressed in this paper and for his valuable comments and feedback on the typescript. I owe special thanks to my student Valeria Ramírez-Licona, whose undergraduate thesis ultimately developed these ideas in full detail. Our years of collaboration and discussion played a crucial role in shaping both her work and the present paper. I also thank Lógica M\(\exists {x}\forall \), and particularly Manuel Tapia-Navarro, for insightful discussions and helpful suggestions. Finally, I extend my gratitude to the attendees to Trends in Logic 2025, especially Elia Zardini, for their comments.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Pailos [25] argues, quite correctly, that invalidity and anti-validity should also play a role in identifying a logic. The validity of arguments and meta-arguments as a criterion for identifying a logic is enough for my present purposes, though.
I distinguish Mexa-Explosion from Meta-Explosion, understood as: If \(\models _{\tiny {{\textbf {L}}}} A, \models _{\tiny {{\textbf {L}}}} NA,\) then \(\models _{\tiny {{\textbf {L}}}} B\), where N is a negation, as it has been developed in [10].
In fact, my definition now encompasses not only invalid arguments, but also any other type of argument. If desired, I could further explore a notational option that also accounts for antivalid or antisound arguments (for more on antivalid arguments, see [1]; for more on antisound arguments, see [16]).
This assumes a SET-FMLA framework, since, under a SET-SET evaluation, differences do arise. These differences have been explored by Blasio, Marcos, and Wansing [5]; in particular, see Theorem 12 (pp. 254-255) and Table 1 (p. 245) for reference.
It is indeed possible to distinguish ST from classical logic without appealing to meta-arguments, specifically by considering antivalid arguments. This was shown by Scambler in [36]. However, when the goal is to assess whether two logics are identical based on their meta-arguments, local validity seems to offer some insight, unlike global validity.
One solution explored in [33] involves interpreting that conditional as a Transplication. When interpreted this way, not all arguments are logically valid.
A standard (see more in [6, 7, 37]) is a condition that requires a logic to have a determinate interpretation within the premises or the conclusion in order to validate an argument. For example, when I deal with classical logic or FDE, there is a fixed interpretation that needs to be preserved from premises to conclusion, namely \(1 \in \sigma \). In contrast, a standard in mixed consequence relations, such as ST and TS, is a pair of interpretations that differ between the premises and the conclusion but need to be fulfilled to also validate an argument in the logic. The standards are usually represented by the letters s-strict and t-tolerant, but there are many more in the literature.
For those interested in exploring the differences between TS and LK3, as well as ST and KLP, Valeria Ramírez-Licona’s undergraduate thesis ([30]), provides a detailed study. Over the past three years, Valeria worked under my supervision investigating how to distinguish mixed logics from heterogeneous ones, particularly as their argument collections initially appeared coextensional. While her thesis offers a broad analysis of various distinguishing criteria, the present text focuses specifically on meta-arguments, offering a complementary perspective on the subject.
1.
Barrio, E., Pailos, F.: Validities, antivalidities and contingencies: A multi-standard approach. J. Philos. Log. 51(1), 75–98 (2022)MathSciNetCrossRefMATH
2.
Barrio, E., Pailos, F., Szmuc, D.: Substructural logics, pluralism and collapse. Synthese 198(20), 4991–5007 (2021)MathSciNetCrossRefMATH
3.
Barrio, E., Rosenblatt, L., Tajer, D.: The logics of strict-tolerant logic. J. Philos. Log. 44(5), 551–571 (2015)MathSciNetCrossRefMATH
4.
Barrio, E.A., Pailos, F.: Why a logic is not only its set of valid inferences. Análisis Filosófico 41(2), 261–272 (2021)CrossRef
5.
Blasio, C., Marcos, J., Wansing, H.: An inferentially many-valued two-dimensional notion of entailment. Bulletin of the Section of Logic 46(3/4), 233–262 (2017)MathSciNetCrossRefMATH
6.
Chemla, E., Egré, P.: From many-valued consequence to many-valued connectives. Synthese 198(S22), 5315–5352 (2018)MathSciNetCrossRef
7.
Chemla, E., Égré, P., Spector, B.: Characterizing logical consequence in many-valued logic. J. Log. Comput. 27(7), 2193–2226 (2017)MathSciNetMATH
8.
Cobreros, P., Égré, P., Ripley, E., van Rooij, R.: Tolerant, classical, strict. J. Philos. Log. 41(2), 347–385 (2012)MathSciNetCrossRefMATH
9.
Cobreros, P., Égré, P., Ripley, E., van Rooij, R.: Tolerant reasoning: nontransitive or nonmonotonic? Synthese 199(3), 681–705 (2021)MathSciNetCrossRef
10.
Da Ré, B., Rubin, M., Teijeiro, P.: Metainferential paraconsistency. Logic Log. Philos. 31(2), 235–260 (2022)MathSciNetCrossRefMATH
11.
Da Ré, B., Szmuc, D., Teijeiro, P.: Derivability and metainferential validity. Journal of Philosophical Logic, pages 1–27, (2022)
Estrada-González, L., Romero-Rodríguez, C.: Empty validity all the way up: An easy road. In Thirteenth Smirnov Readings in Logic: Proceedings of the International Scientific Conference, pages 62–65, Moscow, (2023). Lomonosov Moscow State University
14.
Estrada-González, L., Romero-Rodríguez, C.: How we learned to stop worrying and love tonk. THEORIA. An International Journal for Theory, History and Foundations of Science, (2025). https://doi.org/10.1387/theoria.26626,
15.
Ferguson, T.M., Ramírez-Cámara, E.: Deep st. J. Philos. Log. 51(6), 1261–1293 (2022)MathSciNetCrossRef
16.
Fiore, C.: Classical logic is connexive. The Australasian J. Log. 21(2), 91–99 (2024)MathSciNetCrossRefMATH
17.
Fitting, M.: A family of strict/tolerant logics. J. Philos. Log. 50(2), 363–394 (2020)MathSciNetCrossRef
18.
Fjellstad, A.: Logical nihilism and the logic of ‘prem’. Logic Log. Philos. 30(2), 311–325 (2021)MathSciNetCrossRefMATH
19.
Frankowski, S.: Formalization of a plausible inference. Bulletin of the Section of Logic 33(1), 41–52 (2004)MathSciNetMATH
20.
Frankowski, S.: \(p\)-consequence versus \(q\)-consequence operations. Bulletin of the Section of Logic 33, 197–207 (2004)MathSciNet
21.
French, R.: Structural reflexivity and the paradoxes of self-reference. Ergo, an Open Access Journal of Philosophy 3(5), 113–131 (2016)
Kaplan, D.: Demonstratives: An essay on the semantics, logic, metaphysics and epistemology of demonstratives and other indexicals. In Joseph Almog, John Perry, and Howard Wettstein, editors, Themes From Kaplan, pages 481–563. Oxford University Press, New York, (1989)
24.
Kleene, S.: Introduction to Metamathematics. North-Holland Pub, New York (1952)MATH
Romero-Rodríguez, C.: A Philosophical Study of Empty Logics. Doctoral dissertation, Universidad Nacional Autónoma de México, (2024)
34.
Russell, G.: An introduction to logical nihilism. In Logic, Methodology and Philosophy of Science—Proceedings of the 15th International Congress. Wiley-Blackwell, (2017)
35.
Russell, G.: Logical nihilism: Could there be no logic? Philosophical Issues 28(1), 308–324 (2018)CrossRef
36.
Scambler, C.: Classical logic and the strict tolerant hierarchy. J. Philos. Log. 49(2), 351–370 (2020)MathSciNetCrossRefMATH
