Being a relative novice about categorial ontologies, but with some interest in, and experience with, lambda calculus, I was very interested in a paper about a categorial ontology presented by John Bigelow and Martin Leckey at the 2018 NZAP conference (at which conference I also enjoyed presenting).
Categorial ontologies that accompany formal category languages are ontologically prolific. They are ontologically inflationary compared to the ontologies of particulars familiar from first order formal logic. Normally associated with categorial logical languages, categorial ontologies admit to the ontology as real universals, properties and relations, in addition to the extensional particulars or individuals admitted by first order formal logic. The categorial ontology proposed by John Bigelow and Martin Leckey both countenances, and admits to the furniture of the world, properties and relations. It also admits properties of properties, properties of relations, relations between relations, relations between properties, properties of properties of properties, and so on, recursively - in what John Bigelow recently called 'worlds without end'.
Thus an important element of adherence to a categorial ontology like that of Bigelow and Leckey is a commitment to realism about more than just particulars or individuals, and universals:
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| Bigelow, J and Leckey, M. Categorial Ontology, Presented at the 2018 NZAP Conference at Wellington University of Victoria. |
In their 2018 NZAP presentation, Bigelow and Leckey identified as a motivation the quandary faced by W.V.O Quine with respect to universals. Quine was determinedly anti-realist about universals, but, in accordance with the Quine-Putnam indispensability argument, begrudgingly acceded to the inclusion of abstract mathematical particulars in the furniture of the world. This made Quine both nominalist about universals and - in a certain sense - realist: a very nearly contradictory position - if not a full-blown contradiction. According to The Quine-Putnam indispensability argument, abstract mathematical particulars get referred to in our best and most effective scientific theories with apparent realist commitments, and so the positivist and/or scientistic philosopher should bow to scientific intuition and practice and admit abstract mathematical entities into the ontology, realist-wise.
Syntax and Semantics
Due to these ontological commitments, formal categorial languages like that of Bigelow and Leckey have certain syntactic requirements. Bigelow and Leckey's categorial ontology comes with a very clean notation inspired by the work of M.J. Cresswell (Monash University associate of Bigelow) who adapts the work of Polish logician Stanislaw Lesniewski.
The first semantic challenge for the formal categorial language designer is deciding what to assign the category labels to. In work on the deep structure of sentences in natural language, which discipline influences categorial ontology, the most basic chosen labels/categories are names and sentences. Cresswell's syntax involves the use of angle brackets ⟨ ⟩ and category labels 0, 1. Slightly confusingly, according to Cresswell's approach it's the name that is '1' and the sentence that is '0':
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| (Cresswell, M. J., 'Categorial Languages', Studia Logica 36 (1977), 257-269.) |
The term 'functor' can be confusing in this context, since it is a term of art and technical term in linguistics denoting function words: words with little or no semantic content that are used to bind and combine semantically richer words to build meaningful phrases, clauses, and sentences. In the context of a formal categorial language, however, the term 'functor' denotes - in accordance with the definition in logic and algebra - a morphism between mathematical and/or logical categories.
According to Cresswell's deployment of Lesniewski's system, John is a name (1), and runs is an intransitive verb. Thus runs would be in category ⟨ 0,1 ⟩ because it takes/maps a name to a sentence. The two place transitive verb loves is in ⟨ 0,1,1 ⟩ because it takes two names (e.g. Chris, category 1, and Naomi, category 1) to a sentence (category 0) 'Chris loves Naomi'.
The same approach applies to categories where what is labelled is logical truth functional connectives and operators. Thus logical negation ~ is category ⟨ 0,0 ⟩ because it will take a logical sentence to another sentence:
'All frogs are amphibians'
∀x(Fx → Ax)
becomes another sentence
'All frogs are not amphibians'
~∀x(Fx → Ax)
And, as stated above, sentences are category label 0.
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| (Cresswell, M. J., 'Categorial Languages', Studia Logica 36 (1977), 257-269.) |
The treatment in the above quotations from reference #3 is, perhaps, not as clear as it could be. Cresswell's presentation in another paper (reference #4) is superior. In the following excerpt, the lambda calculus abstraction of Alonzo Church is introduced. This presentation makes it clearer how the complex categories work. The first term in the brackets is the 'output' category term or the result category:
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| (Cresswell, M. J. (1980). Quotational Theories of Propositional Attitudes. Journal of Philosophical Logic, 9(1), 17–40. https://doi.org/10.1007/BF00258075)
Note that there is an editorial/typographical error (missing angle bracket) at the category definition of the ∀. It should read:
F⟨ 0, ⟨ 0,1⟩⟩ = {∀}
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Cresswell continues to develop the formal language, introducing the lambda λ-abstraction symbol which enables working with functions as arguments of functions, and which is also motivated by the need to exclude set theoretic ontological considerations from the formalism:
Church's lambda λ here serves to denote that the application of the function expression α to the variable x.
The purpose of the lambda calculus and notation is to provide an alternative way of defining functions for λ-categorial languages: non-extensionally as a parameter or argument expression instead of extensionally as a set of ordered pairs corresponding to a morphism. Its use is straightforward once the category language apparatus as described above is in place. Instead of defining a function in set-theoretic terms, one specifies an expression α in the logical language, and then treats both the expression and the variable for the values x as arguments. The λ simply denotes this application of expression argument to variable argument.
It's called λ-abstraction because the expression is treated as an argument that can take any specified value and with no function name - only an expression. In functional computer languages this is also referred to as an anonymous function. No function definition is used and so no morphism between sets or set of ordered pairs is picked out as an extension. In this way, the function expression can be used as an argument passed to a function or ⟨λ, x, a ⟩ expression in λ-categorial languages I'll not pursue further investigation of the lambda calculus here.
What's of further interest in this post is Bigelow and Leckey's adaptation of the Dσ notation for domains of properties and relations of individuals:
Bigelow and Leckey denote a domain of individuals/particulars as
Dι
D⟨ ι ⟩
and then for the domain of properties of those properties:
and so on, recursively ad infinitum, while relations between any of the individuals and properties in these categories are denoted by:
D⟨ h, k, ...⟩
‘the category for relations among things in
categories h, k, …'
'For instance, if we might set the variables h = ι and k = ⟨ι⟩. Then
D⟨ h, k ⟩ = D⟨ ι, ⟨ ι ⟩⟩.And, in that case, ‘D⟨ ι, ⟨ ι ⟩⟩’ is to read as the category for‘ relations between individuals and properties of individuals’. (From Bigelow, J, and Leckey, M., 2018 Categorial Ontology URL https://aap.org.au/ABSTRACTS-2018)
Bigelow and Leckey's adaptation of the Cresswell-Lesniewski syntax for property and relation categories is syntactically elegant. It does render the semantics into a manageable and compact form of syntactic expression that allows for tidy representation. However, I suspect that any deployment of comprehensive logical proofs will involve what I will call a 'LISP' effect for longer proofs, according to which outcome (see next paragraph) the proof would become difficult to follow and read due to nesting of parentheses. It's a style and readability issue, potentially.
λ-calculus was the inspiration for a number of functional computer programming languages in the 1970s and 1980s (with a recent resurgence in interest with programming languages like Microsoft's F#.) One of the known problems with some of these languages was human readability for complex programs. The acronym LISP literally stands for 'Lots of Irritating Silly Parentheses'. It's not for nothing that there is a 'LISP showoff page' on Wiki:
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| LISP code excerpt from Wiki |
For those not familiar with procedural computer programming: much like logical proofs, any slip is death when it comes to syntax (and with computer code that can mean the aircraft traffic control systems!) Tracking the number of parentheses on the left and right is no longer a manual task in most programming IDEs (integrated [software] development environments), but when LISP was introduced to the world, things were less sophisticated and programmers had to count the parentheses and check their placement to ensure that the code would compile, and that if it did the logic was right. If a logician is not using a CA tool for proofs in logic, then the same situation prevails for readability and robustness in The point is that some formal logics have readability and clarity challenges, and Bigelow and Leckey's multiple nested ⟨ ⟩'s might become cumbersome. I should provide an experimental proof to test this speculation, but I will defer this until another day due to time and space constraints.
Potential Vagueness Problems
Leckey and Bigelow are aware of the challenges for their Platonic variety of realism about properties and relations. They bite the bullet on the Platonic commitments, and echo the Quine-Putnam indispensability argument for the realist commitments their categorial ontology, recursively replete as it is with properties of, and relations between, any and all of the possible elements in all available categories.
As with all formal language development, category languages involve problems and challenges. These are historically well known.
Bigelow and Leckey address issues of definitional in-exactitude for names in both natural and formal languages, referring to the work of semanticists in their treatment of the deep structure of natural languages, and to the closely related work of logicians interested in applying categorial languages to capture deep-structure natural language semantics and to provide an ontological basis and semantics for a formal logic.
In first order logic, only individuals, predicates, and names (for specific token individuals) are admitted: one can have a predicate P and apply it to a variable x or else to a name a. Importantly, in natural language:
there are syntactic categories approximately shadowing the names, variables, and predicates of the predicate calculus; but there are complications. The syntactic category for nouns, for instance, behaves in some respects as if these were names; but often we translate nouns into predicates when we regiment our assertions in the predicate calculus
(From Bigelow, J, and Leckey, M., 2018 Categorial Ontology URL https://aap.org.au/ABSTRACTS-2018)
In linguistic (natural language) semantics (as opposed to the logical-philosophical variety for formal languages and the study of meaning itself), theorists attempt to identify the deep structure of the meaning of sentences and how it relates to the surface structure. The classifications of nouns and natural language terms in formal categorial language hierarchy is often not straightforward.
Leckey and Bigelow, like Cresswell, also identify ambiguous syncategorematic terms in natural language as being problematic for a formal categorial language treatment because they do not seem to belong to one category only:
"Deviations of human language from ontological stratification:
In framing a description of a stratified categorial ontology, some of the words we have been employing do not invite interpretation by assigning to them a single thing, as a ‘semantic value’ or ‘meaning’. They seem, instead, to scatter their reference across more than one, sometimes perhaps infinitely many, distinct ontological categories.
Thus, for instance, our use of the word ‘property’ resists the assignment to any one unique ‘semantic value’, which could be located within any one domain of properties within the ontological hierarchy. Instead, the word ‘property’ seems to cast a scattered reference all the way up the hierarchy.
...Wittgenstein...suggested that there may be ‘family resemblance’ concepts that spread their reference by a recursive pattern that might work roughly like this.We might apply the word ‘game’ to something with salient characteristics A, B and C. Something new comes along with salient characteristics B, C and D. So we apply the same word to this new thing. And so on. Eventually we apply the same word to something with salient characteristics P, Q and R, which has no salient properties in common with the things that we initially called games because they have properties A, B and C.
A similar pattern can be elicited concerning our theoretical applications of the word ‘property’. We apply the word ‘property’ to things in category D⟨ ι ⟩ because they stand in a relation that we call ‘being instantiated by’ to the individuals that are found in the domain Dι
Then, at categorial levels just a few steps higher – say, categories D⟨ h ⟩ and Dh – we will find a relationship that resembles the ‘instantiation’ relationship between D⟨ ι ⟩ and Dι . Consequently, we transfer the reference of the words ‘being instantiated by’ up to that level as well. This recursive mechanism will smear the reference of the linguistic expressions ‘property’ and ‘is instantiated by’ all the way up the hierarchy, by a process that closely echoes Wittgenstein’s conception of a ‘family resemblance’. "
(From Bigelow, J, and Leckey, M., 2018 Categorial Ontology URL https://aap.org.au/ABSTRACTS-2018)Cresswell notes that some category language theorists have introduced features that allow multi-category expressions:
Again here there is some difficult typography.
"...for any function a in D⟨0,1⟩ , ζ(a) is the function such that for any x in D1 , (ζ(a) = ω(a(x))
Syncategorematic natural language represents a problem for category language. However, I think that one other possible related challenge for this kind of categorial language and ontology arises due to logical vagueness (common examples include identifying at exactly what point on a colour spectrum dark red becomes purple, or at what number of hairs a person attains the property of baldness). They are related to this kind of syncategorematic language ambiguity, but are not the same.
One example hinted at by Bigelow and Leckey themselves is that of the property of being approximately spherical, with a matching category for all of the individuals that are approximately spherical. Now, the natural language label for this category is easily stated. However, it would seem to be a requirement of categorial language and ontology of the kind Bigelow and Leckey desire that the categories be unambiguous and distinct,even if the semantic content of certain terms is spread across them. Yet, I suspect that the obvious sorites paradox associated with the concept of approximate sphericity (at what point does some geometric object that is not spherical become approximately spherical, and how is this decided and why?) troubles the base category of individuals, and by recursion all of the categories containing instantiations of property and relations that are based upon the category of individuals.
It's not so much that the property of being approximately spherical is cross-category in terms of expression that refer to it (it may well be so). It's not that it involves a progressive family resemblance problem such that the surface meaning of the term has slipped or migrated across deep meaning categories. It's that the property itself is ineliminably vague. Thus the category - against the requirements for category language categories - is also vague or perhaps undefinable in the right way. Why do I surmise that this might be a problem for Leckey and Bigelow? Because their proposed category ontology is supposed to be superior to, and more coherent than, those ontologies associated with formal languages which do not admit such a wide range of Platonic entities onto the menu of existing things.
There are pathological cases and problematic cases for all formal logics. Bertrand Russell's own paradox of FOL is perhaps the locus classicus example.
The domain for the property instantiated by the property of being approximately spherical would seem to be dead on arrival. Moreover, it is not clear what happens to the 'is instantiated by' relation and its domain in this case. Perhaps this pathological case is not something that should be admitted into consideration.
Perhaps the members of the set of approximately spherical things will have to be a Zadehian fuzzy set, given that adducing exactly when something crosses the threshold between approximately spherical, and not approximately spherical, might be prohibitive.
References
- Bigelow, John, (1988), The Reality of Numbers: A Physicalist’s Philosophy of Mathematics, Oxford: Clarendon.
- From Bigelow, J, and Leckey, M., 2018 Categorial Ontology URL https://aap.org.au/ABSTRACTS-2018)
- Cresswell, M. J., (1977) 'Categorial Languages', Studia Logica 36 , 257-269.
- Cresswell, M. J. (1980). Quotational Theories of Propositional Attitudes. Journal of Philosophical Logic, 9(1), 17–40. https://doi.org/10.1007/BF00258075
- Linnebo, Ø. (2018). Platonism in the Philosophy of Mathematics. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Spring 2018). Metaphysics Research Lab, Stanford University. Retrieved from https://plato.stanford.edu/archives/spr2018/entries/platonism-mathematics/
- Gallin, D. (1977). Review: M. J. Cresswell, Logics and Languages. J. Symbolic Logic, 42(iss. 3), 425–426.
- Rennie, M. K. (1974). CRESSWELL, M. J.: “Logics and Languages” (Book Review). Australasian Journal of Philosophy, 52(Generic), 277.