Abstract
PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts.
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Notes
See also Boccuni (2010) for PG with a mixed Boolos-Fregean semantics. There are two reasons of discontent with that theory: first, it may be quite disputed whether it embodies some form of logicism; secondly, the Julius Caesar problem arises. We shall see in what follows that the present theory with ACS solves both issues. See also Boccuni et al. (2012) on this.
To be read “a is among the Ys”.
For some similarities with PG, see Burgess (2005, 2.3d), where a second-order language with a full second-order comprehension axiom for concepts in general, a predicative second-order comprehension axiom, and an axiom stating that, to every predicative concept, there corresponds an extension is sketched. In this setting, not all definable concepts interact with extensions—some of them “float” over extensions. Analogously, in PG not all pluralities interact with extension-term formation.
For a proof of model-theoretic consistency for PG, see Boccuni (2011a). The consistency of PG is indeed a remarkable result. In fact, it has been argued that second-order systems with Basic Law V beyond \(\Updelta^1_1\)-comprehension are inconsistent. The consistency of PG is remarkable in that it makes \(\Upsigma^1_1\)- and \(\Uppi^1_1\)- plural formulæ safely interact with Axiom V. See Boccuni (2011a) also for some considerations on the mathematical strength of PG, which is likely equi-consistent with PA2.
The fundamental law of the ordered pair \((x,y)=(u,v)\leftrightarrow{x=u\wedge{y=v}}\) may be easily derived in PG, through several applications of the usual rules of inference, axiom V, and the definitions of the unordered and ordered pairs.
The formal proof of this theorem makes a crucial use also of Axiom V and of the definition of the singleton.
Proof
$$\begin{aligned}&1(1) \exists y (\{y\}=0) \, \, \, \,{\fancyscript{A}}\\& 2(2) \{a\}=0 \, \, \, \, \, \, \, \, \, \,{\fancyscript{A}}\\ &2(3) \{x: x=a\}=\{x:x \neq x\} \, 2, \hbox{Def. } \{a\} \hbox{and 0}\\ &(4) \{x: x=a\}=\{x:x \neq x\}\leftrightarrow \forall x (x=a \leftrightarrow x \neq x) \, \, \,\hbox{ Axiom V}\\ &(5) (\{x: x=a\}=\{x:x \neq x\} \rightarrow\forall x (x=a \leftrightarrow x \neq x)) \wedge (\forall x (x=a\leftrightarrow x \neq x)\rightarrow \{x: x=a\}=\{x:x \neq x\}) \,\, \, \, 4 \hbox{ Def} \leftrightarrow\\ &(6) \{x: x=a\}=\{x:x\neq x\} \rightarrow \forall x (x=a \leftrightarrow x \neq x) \,\, \, 5, \hbox{ E }\wedge\\ &2(7) \forall x (x=a \leftrightarrow x\neq x) \, \, \, \, 3,6 \hbox{ MP }\\& 2(8) (a=a) \leftrightarrow(a \neq a) \, \, \, \, 7, \hbox{ EU }\\ &1(9) (a=a)\leftrightarrow (a \neq a) \, \, \, \, 2,8 \hbox{ EE }\\& (10)\neg \exists y (\{y\}=0) \, \, \, \, \, \, \, 1,9 \hbox{ RAA }\\&(11) \forall y \neg (\{y\}=0) \, \, \, \, \hbox{ 10 by } \neg\exists x \phi x \equiv \forall x \neg \phi x\\ &(12) \forall y(\{y\}\neq 0) \, \, \, \, \, \, \, 11 \hbox{ by the usualdefinition of } `\neq\\\end{aligned}$$See, for instance, Pettigrew (2008).
A further argument to this aim, from the uniformity of substitution of predicate and individual letters in argument schemas, may be found in Boccuni (2010).
See Suppes (1999, p. 82) for this example.
I.e. arbitrary names.
Suppes (1999, p. 82). Of course, it is not always the case that using the same arbitrary name leads to invalidity, nor that different arbitrary names have to refer to different individuals. For instance, consider using “a” for eliminating the quantifiers both from ∀x Fx and ∀x Gx in the same derivation, where x varies over the natural numbers and both formulæ have a model in Peano arithmetic. Or consider using “a” and “b” for eliminating respectively the first quantifier and the second, where a and b may well be the same individual. In none of these examples, sameness of reference seems to lead to invalidity, but such an innocuousness does not by itself speak against the genuine referentiality of “a” or the importance of sameness of reference to derivations. It rather testifies that there are contexts in which the co-referentiality of all the occurrences of “a” (or of “a” and “b”, for that matter) is not problematic.
Russell (1967, pp. 156–157). “ABC” is a free variable.
Suppes (1999, p. 81).
Analogously as far as the rule of introduction for universal quantification is concerned. See Martino (2004, p. 110).
For further justifications and applications of arbitrary reference, see also Breckenridge and Magidor (2012).
Martino (2004, p. 119), En. transl. mine. Notice that VCP* follows from TIR also when non-denumerable domains are concerned. Even though a language may lack non-denumerably many names, TIR still holds, as the ideal possibility of directly referring to each and every individual in a non-denumerable domain may be performed via arbitrary reference, as in the case of, e.g. “let a be an arbitrary real number”.
See Martino (2004, pp. 112–113) on this.
Martino (2004, p. 112), En. trans. mine.
See Martino (2004, pp. 103–133), also for the act of choice clause for the formulæ of the form ∀YB.
Martino (2004, p. 131), En. trans. mine.
I am here referring respectively to J. Burgess “E Pluribus Unum. Plural Logic and Set Theory”, and Ø. Linnebo “Pluralities and Sets”.
Notice how arbitrary reference has been put to use in PG: through it, a whole system of terms, namely extension-terms, is interpreted, unlike Breckenridge and Magidor (2012) where arbitrary reference is considered only as far as particular terms are concerned.
Carrara and Martino (2010) suggests a nominalistic interpretation of the abstraction operator # in Hume’s Principle. To this extent, Hume’s Principle becomes a full-fledged definition of #. The same goes for {:}, as well as for any other abstraction operator. There is a sense in which this move is unsatisfactory. In order to make sense of the nominalistic interpretation of abstraction principles Carrara and Martino (2010) have to tell a story about what is meant by abstraction. The usual meaning is that abstraction introduces or individuates special abstract entities in the domain, e.g. numbers or extensions. Carrara and Martino (2010) reject this reading. According to Carrara and Martino (2010), through abstraction we abstract from the objects’ peculiarities, in order to consider them only under the respect of, e.g. numerosity. Where this may be appropriate for Hume’s Principle, it is not clear under which respect we would be considering objects in order to make sense of the extension-operator. In general, the equivalence relation on the right-hand side of any abstraction principle should provide a specific respect under which to perform abstraction. I doubt this is achieved: are identity of directions or equivalence of concepts enough fine-grained? Moreover, in Hume’s Principle or Basic Law V as definitions, the first-order variables on the right-hand side of the biconditional should not take numbers or extensions as values, because otherwise we would need a way to individuate them before we stipulate the definition of, respectively, # or {:}. Nevertheless, first-order impredicativity in abstraction principles is also their strength. I thus prefer to hold the usual reading of abstraction operators: they individuate objects. Still, these objects are not, in my view, intrinsically numbers or extensions.
It has to be stressed that PG’s first-order fragment is not ontologically innocent, since it proves the existence of infinitely many individuals. This may cast doubts upon the logicality of PG, especially if one holds the view that logic does not imply any ontological commitment at all. Consider, for instance, Boolos’ view: “In logic we ban the empty domain as a concession to technical convenience but draw the line there: We firmly believe that the existence of even two objects, let alone infinitely many, cannot be guaranteed by logic alone. After all, logical truth is just truth no matter what things we may be talking about and no matter what our (nonlogical) words mean. Since there might be fewer than two items that we happen to be talking about, we cannot even take ∃x∃y(x ≠ y) to be valid.” (G. Boolos, “The Consistency of Frege’s Foundations of Arithmetic”, in G. Boolos, Logic, Logic, and Logic, p. 199). Nevertheless, it may be the case the one holds a different view of logic, as Frege and the neo-Fregeans do, according to which logic is a theory about some special kind of truths with its own special objects, namely logical objects. So, after all, whether PG’s first-order fragment is logic or not depends upon the view of logic one subscribes to. Regarding Boolos’ previous quotation, in fact, Wehmeier (1999, p. 326) points out: “This argument is so anachronistic that it seems quite unsatisfactory to me: Evidently Frege wanted his theory to prove the existence of infinitely many objects and still conceived of it as logical. And what if there really are infinitely many logical objects—why should logic not prove their existence? Be that as it may, one might argue that the provability of the existence of infinitely many objects other than logical ones is a reductio ad absurdum of a logicist system.” If one holds on to a Fregean view of logic, then she may consider PG as an alternative form of logicism. Frege’s opponent simply will not, and that is fair enough. Of course, the issue posed by Wehmeier about the existence of non-logical objects requires a detailed argument on why we should not be bothered by it. This issue, though, can not be investigated in this paper, not at the length it deserves. So, I shall leave it open for further analysis.
See Breckenridge and Magidor (2012), where arbitrary reference is taken into account only as for instantial reasoning, namely for particular terms. In PG, on the other hand, ACS provides a way to apply arbitrary reference to a whole system of terms, namely extension-terms.
Possibly, using λ-notation would be more transparent. Though, this would unnecessarily complicate the notation, but it surely is a way to go if some confusion should arise.
See Boccuni (2011b) for some similar considerations.
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Acknowledgments
I wish to thank the British Academy, which generously funded a Visiting Post-Doctoral Fellowship for me to work on this paper at the Philosophy Department, University of Bristol. I also wish to thank both the Philosophy Department, UoB, and Prof Øystein Linnebo for providing the sponsorship needed to obtain the funding and, most of all, I sincerely thank Prof Linnebo for his valuable supervision. I am grateful to the participants of the Paris-Nancy workshop in Philosophy of Mathematics (Paris 2011). In particular, I wish to thank Andrei Rodin, Marco Panza, Simon Hewitt, Sean Walsh, Patricia Blanchette, Ignasi Jané, John Burgess, Brice Halimi, and Sebastien Gandon for their questions and remarks. I also wish to thank two anonymous reviewers whose very detailed and valuable comments helped me to improve this paper dramatically. Last but not least, I wish to thank Paola Cantù for an amusing and particularly useful conversation on the topic of this paper in front of a pint of beer in Nancy.
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Boccuni, F. Plural Logicism. Erkenn 78, 1051–1067 (2013). https://doi.org/10.1007/s10670-013-9482-z
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DOI: https://doi.org/10.1007/s10670-013-9482-z