Abstract
According to Suszko’s Thesis, there are but two logical values, true and false. In this chapter, we consider and critically discuss Roman Suszko’s, Grzegorz Malinowski’s, and Marcelo Tsuji’s analyses of logical two-valuedness. Another analysis is presented which favors a notion of a logical system as encompassing possibly more than one consequence relation. Moreover, in light of these considerations, we will point out that the relation between the notion of a truth value and the notion of entailment is even more intimate than the connection emerging from the interaction between properties of entailment relations and truth values. In some cases it is possible to draw a strong analogy between them, namely to interpret entailment relations as a kind of truth value, and such an interpretation seems to be both natural and promising.
[A] fundamental problem concerning many-valuedness is to know what it really is [125, p. 281].
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Notes
- 1.
This chapter brings together [233, 276].
- 2.
Sometimes Suszko’s Thesis is stated in more dramatic terms. Tsuji [253, p. 299], for example, explains that “Suszko’s Thesis maintains that many-valued logics do not exist at all”.
- 3.
Bivalent interpretations of many-valued logics have also been presented by Urquhart [256], Scott [220], and da Costa and later Béziau, see [30, 57], and references therein. In [12], the Suszko Reduction provides the background for discussing the merits of many-valued non-deterministic matrices.
- 4.
Wójcicki proved his theorem for consequence operations, but the presentations in terms of consequence relations and operations are trivially interchangeable. Moreover, it should be pointed out that the entailment relation defined by a matrix in this case takes into account only valuations which are homomorphisms, cf. Sect. 9.2.
- 5.
In [275] and Chap. 8 it is explained why it is not at all unreasonable to admit \({{\fancyscript{D}}}^+ \cap {{\fancyscript{D}}}^- \not = \varnothing\).
- 6.
Also Malinowski considers only valuations which are homomorphisms.
- 7.
It is assumed that an entailment relation \(\models _X\) with respect to a set \(X\) of bivaluations from the set of \({{\fancyscript{L}}}\)-formulas into \(\{0, 1\}\) is defined by
$$ \Updelta \models _X A \,\hbox{iff}\, \hbox{for every} \, \nu \in X, \nu(A) =1, \,\hbox{whenever for every} \, B \in \Updelta, \nu(B) = 1.$$Completeness thus means that for every set of \({{\fancyscript{L}}}\)-formulas \(\Updelta \cup \{A\}\), \(\Updelta \models _X A\) implies \(\Updelta \vdash A\).
- 8.
The problem of lifting the Suszko Reduction method from a reduction to a trivalent semantics to an \(n > 3\) is raised in [169], and Malinowski there presents some partial solutions to this technically as well as conceptually intricate problem.
- 9.
Often, consequence relations are defined for arbitrary sets. In this chapter we restrict our interest, however, to consequence relations defined on propositional languages.
- 10.
As Caleiro et al. [44] point out, “there seems to be no paper where Suszko explicitly formulates (SR)[the Suszko Reduction] in full generality!”.
- 11.
Basically, a generalized \(q\)-matrix is what has been introduced in the previous chapter as a “symmetric \(n\)-valued valuation system”, see Definition 8.1.
- 12.
In [29, p. 120] it is stated that “Malinowski constructs (using an extended concept of a matrix) a consequence relation which has no two-valued logical semantics because it fails to obey the “identity” axiom of Tarski. However it has been shown (cf. 148) that we can adapt in some way two-valued logical semantics even in the case of such kind of consequence relation”. Note that this “adaptation” is sketched in terms of \( two\) functions \( mod\) \(_1\) and \(mod\) \(_2\) each assigning to every formula and every set of formulas a class of models.
- 13.
Note the difference between the notion of \(t_m\)-entailment and the one of \(t\)-entailment introduced in Chaps. 4 and 8 by Definitions 4.1 and 8.12. Whereas \(t\)-entailment is defined in the context of truth-value lattices and is conceived as reflecting the corresponding lattice order (namely, the truth order), \(t_m\)-entailment is defined on a logical matrix, having the main characteristic to preserve designatedness of truth values. An analogous observation holds also for the notion of \(f_m\)-entailment introduced below. In fact, \(t_m\)-entailment and \(f_m\)-entailment are nothing else but the relations \(\models ^+\) and \(\models ^-\) defined in the previous chapter by (8.2) and (8.3).
- 14.
The idea of a logical system comprising two consequence relations may be detected already in [58, Chap. 6], where H.B. Curry distinguishes between deducibility and refutability. Curry assumes a theory \({{\mathfrak{T}}}\) generated by axioms and a theory \({{\mathfrak{F}}}\) generated by \(counteraxioms\). Moreover, it is assumed that a formula \(A\) is refutable, if a refutable formula is deducible from \(A\). Curry then introduces a notion of negation as refutation by requiring (i) that the negation of every counteraxiom is provable and (ii) that the negation \(\neg A\) of a formula \(A\) is provable, if the negation of a formula \(B\) is provable and \(B\) is deducible from \(A\).
- 15.
This is actually \(not\) Malinowski’s understanding of inferential many-valuedness. In [168] he explains that the chief feature of a \(q\)-consequence operation is that the repetition rule:
$${\text{rep}} = (\{A\}, A \mid A \in {{\fancyscript{L}}}) $$in general is not a rule of the operation. Moreover, \(q\)-consequence is the central notion of a purely \( inferential\) approach in the theory of propositional logics in the sense that “[t]he principal motivation behind the quasi-consequence \(\ldots\) stems from the mathematical practice which treats some auxiliary assumptions as mere hypotheses rather than axioms”. The fact that the repetition rule is not unrestrictedly valid allows Malinowski [168, Sect. 2] to define two congruence relations on \({{\fancyscript{L}}}\), inferential extensionality and inferential intensionality, which in general are independent of each other.
- 16.
Interpretations of distinguished sets of algebraic values need not appeal to truth or falsity. In a series of papers, Jennings, Schotch, and Brown have argued that paraconsistent logic can be developed as a logic that preserves a degree of incoherence from the premises to the conclusion of a valid inference, see [39, 136] and the references given there.
- 17.
That is, we disregard for a moment the \(t_m\)-entailment and \(f_m\)-entailment taken separately and concentrate solely on their intersection. Note that it is most important here to have a value which is neither designated nor anti-designated; otherwise \({tf}\)-entailment would trivially collapse into \(t_m\)-entailment (and \(f_m\)-entailment).
- 18.
A generalization of this definition to obtain a \(tf\)-entailment relation for \(any\) \(q\)-matrix \({{\mathfrak{M}}}\) based on a three-element set of algebraic values using Malinowski’s three-valued valuation \(t_\nu\) (as defined on p. 196) is straightforward.
- 19.
The notion of a \(k\)-matrix is not entirely new. Every \(k\)-matrix is a ramified matrix in the sense of Wójcicki [287, p. 189]. Ramified matrices are also called generalized matrices, see [55, p. 410 ff.]. Note, however, that Wójcicki associates with a ramified matrix a single entailment relation, namely \(\bigcap \{\models _{{{\mathfrak{M}}}} \mid {{{\mathfrak{M}}}}\,=\,\langle {{{\fancyscript{V}}}}, {{\fancyscript{D}}}_i, \{f_c: c \in {{\fancyscript{C}}}\}\rangle, 1 \leq i \leq k \}\).
- 20.
Note that since we defined the entailment relations from the \(k\)-\(models\), the semantics of \(k\)-\( matrices\) leaves room for other definitions of entailment.
- 21.
For Suszko, the relation between the algebraic and the logical values is established via characteristic functions, and the original intuitive interpretation of the algebraic values is uncoupled from the understanding of the logical values as \(truth\) and \(falsity\). Similarly, the understanding of the sets of designated values \({{\fancyscript{D}}}_i\) is detached from the intuitive understanding of the values in \({{\fancyscript{V}}}\). It may, of course, happen that there exists a bijection between \({{\fancyscript{V}}}\) and \(\{{{\fancyscript{D}}}_i , \ldots , {{\fancyscript{D}}}_k\}\). In the context of \(q\)-consequence relations, Malinowski [166, p. 83] explains that for some inferentially three-valued logics based on three-element algebras, referential assignments \(\ldots\) and logical valuations \(\ldots\) do coincide. This phenomenon delineates a special class of logics, which satisfy a “generalized” version of the Fregean Axiom identifying in a one–one way three logical values and three semantical correlates (or referents).
- 22.
See Chap. 3.
- 23.
For the sake of simplicity, we consider entailment as a relation between (single) formulas while keeping in mind its possible generalization to sets of formulas.
- 24.
We say “primitive” and have in mind the possibility of considering further definitions obtained by combining the original conditions.
- 25.
Incidentally, this observation shows that in the context of \({{\mathfrak{B}}}^*_4\), \(\models^p\) and \(\models^q\) are not Tarskian even if these two relations coincide. (Recall that an entailment relation is Tarskian iff it satisfies (Reflexivity), (Monotonicity) and (Cut). In this respect generalized \(q\)-matrices differ from non-generalized \(q\)-matrices since, as Frankowski has shown in [99, p. 198], if a relation defined on a non-generalized \(q\)-matrix is both a \(p\)- and a \(q\)-entailment relation, then it \(is\) Tarskian.
- 26.
We call a \(q\)-matrix \(proper\) iff \({{\fancyscript{D}}}^+ \cup {{\fancyscript{D}}}^- \not = {{\fancyscript{V}}}\) holds for it.
- 27.
We leave as an open question whether in \({{{\mathfrak{K}}}{{\mathfrak{P}}}}^*_3\, \models^p \, \subseteq \, \models^+ \cup\models^- \). An anonymous referee for [234] turned our attention to the fact that it is not difficult to define a particular \(q\)-matrix in which, e.g., \(\models _p\,=\,\models _t\,=\,\models _f\). But it is also possible to define a \(q\)-matrix such that \( \models^+ \cup \models^- \, \subset \, \models^p\). Indeed, add to \({{{\mathfrak{K}}}{{\mathfrak{P}}}}^*_3\) a unary truth function \(f_{\#}\) such that \(f_{\#}\)(T) = I and \(f_{\#}\)(I) = F. Then obviously \(A \models _p \# A\), but \(A \not \models _t \# A\) and \(A \not \models _f \# A\).
- 28.
Again, a generalized \(q\)-matrix is \(proper\) iff it satisfies the conditions: \({{\fancyscript{D}}}^+ \cup {{\fancyscript{D}}}^- \not = {{{\fancyscript{V}}}}\) and \({{\fancyscript{D}}}^+ \cap {{\fancyscript{D}}}^- \not = \varnothing\).
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Shramko, Y., Wansing, H. (2011). Generalized Truth Values and Many-Valued Logics: Suszko’s Thesis. In: Truth and Falsehood. Trends in Logic, vol 36. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-0907-2_9
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