There have been several attempts to produce an ontology of biology and the life sciences in general. One of the more outstanding was Joseph Woodger's 1937 The Axiomatic Method in Biology, which was based on Russell's and Whitehead's Principia and the theory of types. In this, Woodger attempted to develop a logic system that would account for all the objects of the theories of biology, especially of embryology, physiology (including cell theory) and genetics. It was hard going even for logicians (Tarski himself wrote an appendix), and the theory thus elucidated seemed to be very post hoc - it was unclear how it helped theories in biology.
Nevertheless, the axiomatic method, or reducing theories to self-consistent logical models based on a few axioms, was very popular. Mary B. Williams tried to axiomatise evolutionary theory in 1970. Others have tried to axiomatise genetics. The hope is that by doing so, by reducing a theory to its axioms, one can uncover at least the basic commitments of the theory, that is, its ontology. Critics have held that biological theories are either not axiomatisable, or that they do not resolve the issues (is genetics a part of an axiomatised evolutionary theory? Williams thought not, while Rosenberg thinks it is). Most criticisms, starting with Haldane's critique of Woodger's book, have focused on the promise of axiomatisation to deliver new knowledge by deduction, which is neither here nor there for our purposes.
Axiomatisation must not be confused with a much older tradition of doing science by definition, which goes back to Aristotle. Here the notion was that if we define our terms, such as "Man", we will have uncovered the essential properties of being a man. Aristotle was not a pure rationalist, of course, but as far as metaphysics goes, where evidence is unavailable (can you measure a universal?), definition seems to be one way, if not the main way, to address the topics. Axiomatisation takes the prior empirically based theories as the foundation for the modelling of the conceptual commitments of biology (and any science that is axiomatisable).
More recently, a method known as "Ramsification" has become the standard way to define what the ontology of a scientific theory is. It was first proposed by David Lewis in 1970, after the work of Frank Ramsey, but it is also implicit in the work of Carnap. The approach is to set up, in a formal language, the theory as a sentence - a "Ramsey sentence" - and to treat the bounded variables of that sentence as the ontological commitments the theory requires. As Quine put it, to be is to be the value of a bounded variable. For instance, "x has charge" in first order logic comes out as [ppt of a talk by David Chalmers]
or, "there exists some phi such that phi has property P and x is phi"; where phi is the bounded variable for charge. Hence if that Ramsey sentence is true, there must be things that phi stands for. Any x that has phi is an instance of that class of charged things and charge is thus an element of the ontology of that theory. Of course, real theories are much more complex than that, and there are different logical languages into which they can be formalised, but you get the idea.
All of these approaches assume one thing - the ontology of the domain of a theory is just the ontology of the theory itself, assuming it is complete enough and empirically adequate. This is not, I think, a safe assumption. It puts the epistemic cart before the ontological horse again. For a start, why think there are domains at all in any sense that is clear? The world does not come in nice packages - molecular biology, developmental biology, and ecology all shade into each other in complex ways. If a theory explains some aspect of, say, ecology, it clearly is going to involve physiological processes, functional molecular biology, and evolution. The notion that there can be a self-standing ontology of evolutionary or ecological theory strikes me as unlikely at best.
In fact much of this arises in the context of reductionism in biology. Are there laws of evolution? Of ecology? Of cell theory? If so, then there are object classes that the laws require, and so an ontology of those laws and the theories in which they are embedded. But if not, are there in fact special ontologies of biology? If all biology resolves down to physics, some say that there is no ontology of biology (or at least, that all objects that biology has are just physical objects that are subject only to laws of physics).
If reductionism, or rather physicalism (which is the view that only physical things can be said to exist by science) is true, then the ontology of biology is just the ontology of physics, under a Ramsification approach, because all biological objects can be eliminated in the theory and replaced with physical objects. If we presume that an ontology is something one derives from theories, and biological theory reduces to physics (as I think it does) then it suggests that biology has no special ontology. Is there a way around this? I think there are two.
The first is to say that there can be objects and properties of a theory of biology which, while they do reduce down to physics, are nevertheless biological objects. The organism might be a candidate object of this type. Sure, organisms are physical objects, but they have properties that are specially biological, such as reproduction. The reduction lies in the explanation of those properties. Organisms do not disappear because we have a physical explanation for them. A cell exists even if its dynamics are all the effects of physics and chemistry.
The second, and this is one I particularly prefer, is to say that biological objects are not objects of a theory, necessarily. Where is it written that all objects of a domain must be derivable from a theory? [Oh, yes, in Quine. I forgot.] There are another class of objects in biology and other derivative sciences: phenomenal objects. That is, there can be objects of a theoretical domain which are merely observed rather than derived from the theory. I believe, and have previously argued, that species is a phenomenal object in biology, required by no theory but explained by several. We need to consider as we cover the various domains and periods of biology whether Quinean objects are all that the domain contains.
References
Haldane, J. B. S. 1938. The Axiomatic Method in Biology. Nature 141: 265 - 266.
Lewis, David. 1970. How to Define Theoretical Terms. The Journal of Philosophy 67 (13): 427-446.
Rosenberg, Alexander. 2005. Philosophy of science: a contemporary introduction. 2nd ed, Routledge contemporary introductions to philosophy. New York; London: Routledge.
Williams, M. B. 1970. Deducing the consequences of evolution: a mathematical model. Journal of Theoretical Biology 29: 343-385.
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I guess I just don't understand what you mean when you mean by a phenomenal object.
See Ontology 3! Due whenever...
But in practice we don't reduce biological theory to physics, so this doesn't seem like a sensible statement. I guess one can then ask why it doesn't, and look at where the reduction process breaks down, and what stops it. For ecology, I think you'll find that it stops at individuals (OK, there's some leakage into physiology, but it's still pretty much the individual that's the lowest object we focus on). I think this gets us back to the phenomenal object.
I suspect most areas of biology can be boiled down in this way. The exceptions, such as biophysics, are probably really physics, but done on biological objects. But although this base is in physics, most of biology doesn't get anywhere close to that. We don't want to be associated with denialist stamp collectors.
Lots there, more than I can comment on in one post!
Re: Woodger. Woodger made use of at least one logical tool that goes beyond W. and R.'s "PM": he made essential use of the ideas from mereology (theory of the part/whole relation, what Nelson Goodman calls "the calculus of individuals"). Tarski's appendix to Woodger's book is, i.i.r.c., largely about the rigorous axiomatic formulation of mereology: when David Lewis was writing his "Parts of Classes" (Blackwell, 1991), which also makes great use of mereological ideas, I borrowed Woodger's book from the University of Melbourne Zoology library for him so he could refer to T.'s appendix!
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What you can axiomatize in the theory of types plus mereology you CAN axiomatize in the theory of types alone, but doing without mereological ideas makes things a whole lot more artificial and complicated and unintuitive. I think the discussion, a decade or two back, of "species are individuals" might have been clearer if the discussants had been able to refer to the distinction between mereological and theory-of-types notions.
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Minor: the formula you display in explaining Ramsey sentences isn't exactly in "first-order" logic: phi is a predicate variable (it occurs as predicate in the "x is phi" conjunct), so this is SECOND-order. My slogan: Ramsey sentences are an application of Russell's theory of descriptions in second-order logic: they replace predicates with variables in the way that, e.g., Quine's elimination of names replaces names with variables.