Abstract
Logical inferentialism maintains that the formal rules of inference fix the meanings of the logical terms. The categoricity problem points out to the fact that the standard formalizations of classical logic do not uniquely determine the intended meanings of its logical terms, i.e., these formalizations are not categorical. This means that there are different interpretations of the logical terms that are consistent with the relation of logical derivability in a logical calculus. In the case of the quantificational logic, the categoricity problem is generated by the finite nature of the standard calculi and one direction in which it can be solved is to strengthen the deductive systems by adding infinitary rules (such as the ω-rule), i.e., to construct a full formalization. Another main direction is to provide a natural semantics for the standard rules of inference, i.e., a semantics for which these rules are categorical. My aim in this paper is to analyze some recent approaches for solving the categoricity problem and to argue that a logical inferentialist should accept the infinitary rules of inference for the first order quantifiers, since our use of the expressions “all” and “there is” leads us beyond the concrete and finite reasoning, and human beings do sometimes employ infinitary rules of inference in their reasoning.
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Notes
- 1.
Proof-theoretic logical inferentialism accepts the idea that the meanings of the logical terms are given by their proof theoretic roles, but maintains that these meanings should be characterized only by using proof-theoretic concepts.
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- 4.
When sentential classes are considered, (Carnap, 1942: 38–39, 1943: 107–108) introduced the following special cases: the universal class that comprises all sentences could be constructed either conjunctively (˹V&˺ is the universal conjunctive, which is true if all sentences are true) or disjunctively (˹Vv˺ is the universal disjunctive, which is true if at least one sentence is true), and the null class that comprises no sentence could also be taken either conjunctively (˹Λ&˺ is the null conjunctive, which by definition is true, since it contains no false statement) or disjunctively (˹Λv˺ is the null disjunctive, which by definition is false, since it contains no true statement).
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In addition, non-normal interpretations are possible even if we consider the objectual interpretation of the quantifiers (see Garson’s Theorem 14.3 below).
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- 7.
Since the existential quantifier can be defined as ˹~∀~˺, one rule of inference is sufficient. This remark applies to all the approaches for obtaining categoricity discussed below.
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Likewise, at that time (Tarski, 1933/1983: 295) believed that the use of infinitary rules like the ω-rule “cannot easily be brought into harmony with the current view of the deductive method, and finally that the possibility of its practical application in the construction of deductive systems seems to be problematic in the highest degree”. Later on, however, Tarski became more flexible on this matter and even investigated calculi with infinite long expressions; see (Scott & Tarski, 1958) and (Tarski, 1958).
- 9.
If negation is normal, then disjunction is also normal, otherwise the Disjunctive Syllogism Rule (AvB, ~ A ˫ B) would become unsound (i.e., if both “A” and “B” are false and negation is normal (thus, “~A” is true), then “AvB” cannot be true). However, since negation and disjunction form a functionally complete set of connectives, then all the other connectives will be normal.
- 10.
The problems with the local models that (Garson, 2013: 42–43) identifies are that of incompleteness in the case of classical propositional logic and that the local models do not generalize properly at the quantificational level. We shall discuss these problems in Sect. 16.4.4) below when the approach of obtaining categoricity by convention will be explored.
- 11.
Certainly, a rule of inference in the sequential format says that if a sequent is provable, then another sequent is also provable. Thus, if a proved sequent is taken to be valid, then the requirement of validity preserving seems reasonable. Some authors, as (Bonnay & Westerståhl, 2016: 724), consider it to be too strong since it involves a complex grasp of logical consequence, in particular that it requires an understanding of validity preserving mechanisms.
- 12.
More generally, Garson shows that: (1) if we consider the axiomatic formalizations of propositional logic, then they fail to uniquely determine the standard meanings of the propositional operators no matter what kind of models (deductive, local, global) we use; (2) the multiple conclusions sequent formalizations determine the classical meanings no matter what models of the three we use; (3) if we consider the natural deduction format and the global models, then the meanings of the operators is the intuitionistic one. If we use the local models however, then the meanings of the operators are the classical ones.
- 13.
If we consider the ∀-introduction rule and the set Г, which does not contain x free, the validity of this rule tells us that: if Г⊧[S∀] A, then Г ⊧[S∀] (∀xA). Since v*(Ay/x) is true for each variable y, and v*(∀xA) is false, for preserving the validity of the rule, v*(Г) has to be false. Hence, the global validity of the rules from S∀ is consistent with this valuation v*, but v* provides the universal quantifiers with a different meaning than those defined by the standard substitutional (‖s∀‖) or objectual (‖d∀‖) semantics. Therefore, these semantics cannot be the natural semantics for S∀. Moreover, for the same reason, this valuation v* seems to do the same thing even if we consider the local validity of S∀. The valuation v* preserves the sequent satisfaction of the (meta)rule of ∀-introduction (provided that v*(Г) is false), and thus the rule is locally valid, while the universal quantifier is false even though its instances are true.
- 14.
The latter two approaches are discussed in details in Brîncuș (2024).
- 15.
See (Brîncuș, 2021) for a discussion of this second assumption of McGee’s approach.
- 16.
(McGee, 2015: 179) suggests a very interesting new approach for obtaining categoricity at the quantificational level through an open-ended application of Hilbert’s rule for the ε-operator. (Carnap, 1937: 197) referred to Hilbert’s version of the omega rule as being sufficient for eliminating the non-normal interpretations, but made no remarks in this respect on the rules for the epsilon operator. For some brief remarks on the epsilon operator see Sect. 16.5 below.
- 17.
- 18.
For instance, (Warren, 2020: 85) wonders whether “a version of Carnap’s problem” appears for the standard quantifier rules. Still, as we discussed above, (Carnap, 1943) had a unitary treatment of the non-normal interpretations both for the propositional connectives and for the first-order quantifiers.
- 19.
(Warren, 2021) approaches the possibility of infinite reasoning from a naturalist view on human cognition and on this view inferences are seen as “causal processes realized by our brains”, although they are not exhausted by these processes. The possibility of accepting infinite many premises becomes plausible, he argues, once we accept a dispositionalist account on acceptance and believing. We can have some behavioural dispositions for accepting a sentence without considering it in advance. Hence, as long as the premises of the omega rule are recursively enumerable, we can have the behavioural dispositions to accept them without considering each of them individually in advance.
- 20.
A valuation v satisfiess ˹Г⊢ φ˺, where s is a variable assignment, iff, in v, either s fails to make true some δ∊Г or s makes true φ. A sequent ˹Г⊢ φ˺ will be thus satisfied by a valuation v iff v satisfiess ˹Г⊢φ˺ for every variable assignment s.
- 21.
Garson suggests that a valuation may satisfy ˹Г⊢Ft˺ without satisfying ˹Г⊢(∀x)Fx˺, but the inference from ˹Г⊢Ft˺ to ˹Г⊢(∀x)Fx˺ is not an instance of the ∀I-rule, as Murzi and Topey formulate it, since there is no free variable in the premise. An instance of it would be rather the inference from ˹Г⊢ Ft˺ to ˹Г⊢(∀x)Ft˺, in which the use of ∀ is vacuous. Murzi and Topey do not consider, however, Garson’s valuation v* discussed in Sect. 16.3 which, as I suggested in footnote 14 above, may also be used against the determinacy of the ∀-rules when local validity is used.
- 22.
It should be noted that (Murzi & Topey, 2021: 3407) prove this weakened thesis by using a restricted formulation of the ∀-introduction rule: if ⊢φ, then ⊢∀xφ. If we consider the rule in its general form ‘if Г⊢φ, then Г⊢∀xφ’, then Garson’s valuation v* could be used to show that the ∀-introduction rule is locally valid, but ∀xφ is not satisfied, although φ is satisfied (see also footnote 14 above). If Г is taken to be empty, then if φ is a theorem, its logical closure will also be a theorem. However, it seems to me that the problem which remains in this case is to describe the relation between φ and its instances, since the logical inferentialist needs individual constants in his language. If φ follows from the set of all its potentially infinite set of sentences, without following from a finite subset of it, then a transfinite rule of inference, as that of Carnap’s above, seems to be needed.
- 23.
(Murzi & Topey, 2021) embrace a naturalist standpoint and assume that we have some general dispositions to infer in accordance to logical rules. These dispositions are supposed to have a syntactic nature and they allow us to infer in an open-ended way, i.e. we can accept instances of the logical rules formed with expressions that are in some extensions of the original language that we use.
- 24.
The idea is that, as long as we do not have a generally accepted theory of dispositions, we cannot convincingly argue for or against the idea that human beings have dispositions for following infinitary rules. If we have the dispositions to accept a class of sentences without considering individually each of them in advance, as (Warren, 2021) argue, then it is plausible to accept that we can follow infinitary rules. Beside this, since we can prove that Peano Arithmetic closed under the ω-rule is deductively complete, it is clear that in a certain sense we can fruitfully use infinitary rules (see also footnote 30).
- 25.
Some logicians, as (Curry, 1968: 261), even took recursive effectiveness as a necessary condition for logical formalization.
- 26.
If S is a semantical system and there is a sentence S2 and a infinitely class of sentences C1 in S such that S2 is a logical consequence of C1 without being a logical consequence of any finite subclass of C1, then an L-exhaustive calculus C for S can be constructed only if transfinite rules are admitted. See (Carnap, 1939: 23).
- 27.
As (Tennant, 2008: 103) points out, if the finite serial structures are to be likened to geometrical points, then an infinite sequence of premises involved in an application of the ω-rule may be likened to a geometrical line, being an infinite sequence of such points. Thus, ‘simply being infinitary does not count against being purely formal’.
- 28.
(Lewis, 1918: 236), for instance, makes the assumption that “any law of the algebra which holds whatever finite number of elements be involved holds for any number of elements whatever.” This assumption is taken by him to be true and is grounded by the convention that the quantifiers are equivalent with (possibly infinite) conjunctions and, respectively, disjunctions.
- 29.
(Carnap, 1961) also acknowledged the indeterminate character of the ε-operator, but, nevertheless, he found it useful not only for logic and mathematics, but also for defining the theoretical concepts of scientific theories.
- 30.
For a very interesting and elaborate discussion on the ability of human beings to perform infinite inferences see (Warren, 2021). For a criticism, see (Marschall, 2021). Marschall assumes, however, that we do not have dispositions for following infinite rules and, thus, he concludes that only languages with recursive rules are permissible. His conclusion is a radical one, since he also applies this constrain to meta-languages. As we have seen in Sect. 16.2 above, even (Church, 1944) admitted the fruitfulness of the infinitary rules in the meta-theory. After all, we can prove new theorems precisely by using these rules (see also footnote 25).
- 31.
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Acknowledgements
I would like to thank my audiences in Salzburg (2019), Prague (2019), and Vienna (2022), where parts of my work on this paper have been presented. I also want to thank Mircea Dumitru, Julien Murzi, Gabriel Sandu, Iulian Toader and Brett Topey for helpful discussions. Special thanks to the reviewers for their very useful feedback. This work was supported by a grant of the Romanian Ministry of Education and Research, CNCS - UEFISCDI, project number PN-III-P1-1.1-PD-2019-0901, within PNCDI III.
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Brîncuş, C.C. (2024). Inferential Quantification and the ω-Rule. In: Piccolomini d'Aragona, A. (eds) Perspectives on Deduction: Contemporary Studies in the Philosophy, History and Formal Theories of Deduction. Synthese Library, vol 481. Springer, Cham. https://doi.org/10.1007/978-3-031-51406-7_16
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