Model-Theoretic Considerations for Theological Semantics

I

I have for many years been convinced that the theological enterprise cannot survive in our age without affording to its language robust truth conditions.  Contemporary men and women presuppose what Jaegwon Kim once called Alexander's Dictum, that is, "to be is to have causal powers."  We don't live in the 19th century where ideas themselves are thought to have a kind of reality; we don' t live in a time in which the conceptuality of God can remain important for vast numbers of people. 

In our days, people are not generally searching to find some overarching concept or principle that grounds our rational thinking about life and existence, a notion that might somehow explain why human experience is given as it is, and accordingly, somehow ground the preciousness and value of that experience.  While Madonna once sung of a "material girl," we generally acknowledge that, even in our churches, the cultural primacy of the physical reigns.  The new atheism talks breathlessly of its discovery of a worldview without divine agency and causality -- as if such a view of things is in any way new.  There is an assumption of the causal closure of the physical among many unwashed in the complexities of the actual relations holding among experience, theory and truth, among those who simply believe that the theories of the natural and social sciences simply state the way things are.  

The idea is easy enough to grasp.  Consider this structure <{x | x is a natural event}, C>.  This is a structure consisting of the set of all natural events and a causal operator C relating members of this set of events to each other.  This structure can satisfy these two assertions:  1) For all x, there is some x (or other) that causes x, and 2) For all x, if x is caused, then it is caused by some x (or other).  What is precluded by this structure is that there is an x that can be caused by some event or agency that is outside the set {x | x is a natural event}, or that x causes some event or state of affairs outside the set {x | x is a natural event}.  Simply put, there are no non-physical events causing physical events, nor no physical events causing non-physical events.  

In addition to the causal closure of the physical assumed by many impressed with the results and progress of the natural and social sciences, it is also supposed, though not always as clearly, by the heralds of late nineteenth century radical criticism, that human beings are somehow alienated when they fail to come to terms with the physicality of their fate.  Feuerbach, Freud, Marx, and Nietzsche, all, in their own way, argued that the illusion of the traditional God connects with the fundamental alienation of men and women.  Marx, for instance, argued that human value and ideology is determined by underlying economic processes, and that the concept of God simply operates to block human beings from understanding the basic materiality of their existence.  The God concept sanctions prevailing ideology and functions to keep in place value ideologies grounded in the unequal distribution of economic materiality.  

Most people that continue to practice the Christian life believe that there is a God and that God is active in the world, i.e., they assume that "to be is to have causal powers" and that God has causal powers.  They speak about the divine design of the universe, and about the power of prayer, particularly prayers of petition.  They assume that there are things that have come about that would not have come about were there no God, and that there are events and processed that have not come about that would have come about were there no God.   

The structure they assume is perhaps this: <{x | x is an event}, g, C, D> where there is a set of natural events and there is God, and that there is a binary natural causal operator C linking natural events to other natural events, and a binary divine causal operator D, linking natural events to divine agency, e.g., 'Dgp' means God divinely produces event p, with p being a member of the set of all natural events.  

Metaphysics remains crucial in theology because claims about that beyond the physical are by nature metaphysical, and assuming God to be with causal powers means that something beyond the physical is bringing about something physical.  This is clearly a metaphysical claim. To afford to theological language robust truth conditions in an age that assumes that to be is to have causal powers means that theology must be self-consciously and boldly metaphysical.  There must be intellectual honesty here.  Either theological language is broadly expressive of the self, its experiences and existential orientations and possibilities, or it is a rule-governed customary discourse by and through which human communities function and operate in the world, or it is a type of discourse that non-subjectively donates possible ways of being, or perhaps it is realist in its motivations; it states what its utterers believe is the ultimate constitution of things.  

One needs to think through these issues very clearly.  What are either the truth or assertibility conditions of theological language if one eschews realism?  Are sentences in the language rightly assertible simply because my tribe (the theological tradition) has traditionally asserted them?  But clearly the assertibility condition cannot simply be 'x is properly assertible' if and only if x has been asserted by normative theologians of tradition T over time t.  Why? In order even to begin to evaluate that claim we must know the identity conditions of 'normative theologian' and 'tradition' and 'time'.   

Are the assertions of theology then either descriptions of the self -- its experience and existential orientations -- or are they expressions of the self?  Clearly, embracing the latter is to give up on truth,  for it entails that assertibility must be understood broadly in terms of a "boo hurrah" theory of theological language.  But the former alternative is not much better, for on its assumption the truth-makers of all theological language are not theological.  On this view, models satisfying a set of theological assertions are not theological models at all because the sets, functions and relations of the models deal with the human.  Since human dispositions, experiences, and orientations are operated upon by relations and functions, these functions and relations ultimately concern the human. The fact that such models can satisfy a class of theological statements, should give us pause about what it is we are doing when we provide theological models. 

But there is another alternative, for we might hold that theological language somehow operates to disclose truth, that language, the Word, in its wording grants world and our place within it must itself be given a theological model.  But it is to me unclear exactly how this model can be constructed coherently.  Models or structures concern domains with functions and relations drawn upon those domains. But what can be the domain of the creative Word?  Remember that revelation is not insight.  Insight concerns an intellectual grasp of that which is already present.  Revelation, on the other hand, is a daring grasp of what is not present, but which shows itself eschatologically.  There is so much that can be said here, but I cannot in this brief essay say it.  We must move to the central issue of the influence that model-theoretic arguments might have for one who in her theological semantics, is broadly speaking realist. 

II

While I could only sketch briefly in the last section my prima facie reservations with non-realist construals of theological language, I will assume in this section that the reader is sufficiently persuaded by what I have said to give theological realism a try.  Theological realism, simply put, is the view that God, and divine states of affairs generally, exist and have the particular contour they have apart from human awareness, perception, conception and language.   Theological realism is thus a species of external realism, the view the world consists of entities, properties, events, relations and states of affairs which, broadly speaking, exists independently of our human perceptual and conceptual processing, or, more to the point, apart from our epistemic structures and capabilities.   We might call this the independence thesis with regard to external realism. 

I am convinced with many others that external realism makes two other important claims as well.  The first is the correspondence thesis which claims that statements about the world are true if and only if they correspond in appropriate ways with how the world actually is. (Clarifying what 'correspondence' and 'appropriate' might mean here is notoriously difficult.)  The other thesis of external realism one can be called the Cartesian thesis which states that although our theories about the world might meet all theoretical and operational constraints of an ideal theory, we could still be wholly wrong in our theory.  Since the theory is made true (or false) by how the world is apart from us, it is always logically possible to be wrong about everything that we might say about it.  Satisfying all theoretical and operational constraints does not a theory true make.  Only the way the world is can make the theory true or false.  The external realist thus seems committed to all of these: the independence thesis, the correspondence thesis, and the Cartesian thesis.  

Hilary Putnam in his famous "Models and Reality" distinguishes among three positions in the philosophy of mathematics.  These positions deal with both truth and reference in mathematics, and are thus, for him and for us, relevant to considerations of truth and reference with respect to external realism generally.  These positions are: 

• Platonism which, according to Putnam, "posits nonnatural mental powers of directly 'grasping' the forms" (Models and Reality, p. 24). This notion of grasping is primitive and cannot be further explicated.  Those familiar with Husserl's description of phenomenological intentionality will understand this quickly.  
• Verificationalism replaces the classical Tarskian notion of truth with verificational processes or proof.  Mathematic assertions are not true in any deep sense, but they are assertible on the basis of other mathematical procedures. Verificationist proposals within the philosophy of science of the last century are connected to this. 
• Moderate Realism, for Putnam, "seeks to preserve the centrality of the classical notions of truth and reference without postulating nonnatural powers" (Ibid.).  The idea here is that mathematical assertions are true, but that their truth does not involve one in a deep process of grasping or understanding the structure of some Platonic heaven.  

Putnam believes that arguments built upon the "Skolem-Paradox" are germane to a moderate realist perspective within mathematics and the external realist perspective in metaphysics generally.  These arguments are known in the literature as "model theoretic" arguments, and they basically exploit the difference in model theory between what might be intended and what might be said.  If one is a non-naturalist when it comes to semantics -- that is, if one thinks that semantic objects, properties, relations and functions are natural objects and does not involve non-natural magic -- then one has a problem with reference, because many models can make true the very same class of sentences.  This means, that one cannot naturally fix reference, that is, what the sentences say is logically independent from what one might mean to say in their saying.  

Putnam draws conclusions from this that are quite far reaching.  For instance, he claims that metaphysical realism (external realism generally) in incoherent, and that 'brain in vat' or 'evil demon" (Descartes) scenarios cannot even be coherently stated.  Putnam throughout tries to show that, because of the problem of reference, one cannot even state the conditions necessary to formulate the brain-in-vat/evil demon hypothesis. In other words, the necessary conditions for the possibility of posing the brain-in-vat scenario cannot obtain because a certain type of reference must be had by the language in stating the scenario, and since this type of reference cannot be had, the scenario cannot be coherently stated.  In other words, while it might appear that we could be a brain in a vat, we really can't be one, for to be one demands that we can refer to being a brain in a vat, and this we cannot do.  

Putnam employs a bit of a technical branch of logic known as model theory and there are considerable arguments in the literature about the effectiveness of his employment of these resources.  There are arguments as to the number and effectiveness of distinct model-theoretic arguments that Putnam uses, and their ultimate effectiveness in attacking metaphysical realism. All of this, I will lay out at another time.  What is important for us, however, is this question: Why is any of this important for theology? 

III

I believe that theological language must be given a realist construal if we are to retain it.  Long ago, I argued that the arguments for the elimination of theological language are strong, and that only a realist interpretation of theological language will likely stem the collapse of such language into reduction and ultimate elimination.  I can't rehearse that here, but know that I believe that theological realism best coheres with the principle that to be is to have causal powers. 

Notice now that if we afford to theological language realist truth conditions we seem to be interpreting it in ways that best connect to the classical Christian tradition.  Believers throughout the centuries assumed that there is a God, that one could refer to God, and that once could talk meaningfully about God's relationship with His universe, both in terms of creation and redemption.  It is extremely difficult, I think, to argue that the horizon of the Reformation is one in which one of the three following is not presupposed: theological realism, semantic realism, and theophysical causation.  The Reformers thought that God exists apart from human awareness, perception, conception and language, that our language about God is true or false apart from the ways in which we verify or come to hold it true or false, and that God is in principle capable of causal relations with nature and the historical realities of nature.  

So on the assumption of external realism when it comes to theology, what are the repercussions of model-theoretic arguments on theological semantics?  

At this point we must appreciate how important reference is for theological language.  We are using theological words and phrases, and if we must ultimately give a realist construal to theological language then reference turns out to be the key to theological semantics generally.  'God is in Christ reconciling the world unto Himself' is true only if 'God' refers, 'Christ' refers, 'world' refers, and the relation of 'reconciling' can be drawn between the world and Christ.  But now the question, if reference is so important, why cannot it be something intended?  Why can we not simply say that intentionality fixes reference and that we don't need to worry about model-theoretic considerations at all?   Remember, Putnam had said that model-theoretic arguments really apply to the moderate realist in mathematics and the metaphysical realist; they are not aimed at one who holds that intentionality can be fixed nonnaturally by something like Husserl's "ego rays."  If one wants to hold intentionality as a nonexplicatable primitive, then can't we simply say that our intentionality determines reference in the theological order, as well as the mathematical and metaphysical orders? 

Here is the problem with this response.  While one might hold that one can intend cherries or trees by nonnaturally fixing one's gaze upon them, one cannot seem easily to do that when it comes to God or the inner workings of the Trinity.  After all, "nobody has ever seen God."  How can one intend that which has no clear content?  The theological tradition knew the apophatic nature of God-talk.  We can never be given the proper content to think God, because the content of our thoughts pertain to the finite order and God is infinite. Our thoughts of God do not thus determine our reference to God; our intentionality cannot issue in reference, because we cannot be given that by virtue of which reference is determined. Instead of intentionality granting an intensionality that determines reference, our theological language -- the language of the tradition -- speaks about God and God's relationship to His creation.  The ways of talking about God are very important indeed!  God's name is that by virtue of which reference is established, and maybe for Christians -- or perhaps all the monotheistic religions of the west -- this happened at the burning bush.  (Recall here Kripke's "initial baptism" of the tretragrammaton at the burning bush in Exodus.) 

It is important here to grasp what is at stake. If intentionality cannot fix reference to the divine, and if we don't want to give up truth to some verificationist-inspired theological position -- that is to say, if we want to be realists in theology -- then we seem to find ourselves in theology with no other option than to have to take the model-theoretic arguments seriously with regard to theological realism.  This means that not only are model-theoretic arguments relevant to theology, they might be crucial to its very future.  If model-theoretic arguments yield a knock-out blow to external realism, of which theological realism is a species, and if realism is essential in providing a defendable semantics for theology, then model-theoretic arguments may pose a much deeper threat to theological discourse than we previously might have thought. 

So what is at stake with respect to model-theoretic consideration in theological semantics?  I think it likely that the future of theology itself might be at stake. But consideration of this must await another time.  It is upon that which I am toiling a new manuscript.  

Luther and Heidegger: Modeling the Destruction of Metaphysics

The International Luther Congress beckons this summer and I am thinking about doing something on Luther and Heidegger in the seminar on Luther and Philosophy. I am old enough now to remember Luther Congresses 35 years ago and more where this topic was not of deep interest. Having written a dissertation on Luther's theological semantics, I was from my first Luther Congress interested in these matters, and remember being introduced to the Finnish work in this area in Oslo in 1988. 

The following is the abstract for my paper on Luther and Heidegger this Sumer.  The seminar headed by Jennifer Hockenbery asks participants to relate Luther to the philosophical tradition through consideration of the notion of freedom. 

________

Much has been written about Heidegger’s indebtedness to Luther (along with Paul and Augustine) in the development of central themes of Being and Time e.g., death, fallenness, guilt, sin, freedom, etc. Heidegger breaks here with Husserl and western philosophy’s dream to frame a consistent and coherent theory adequate and applicable to all the facts, both physical and metaphysical. In the early 1920s Heidegger was interested in the phenomenology of Christian life, what it was to-be-unto-the-Parousia. He discerned in Luther a friend in uncovering the meaning of factical Christian existence, that primordial self-understanding from, and through which, any talk of theological “facts” can emerge.  

But the parallels between Luther’s critique of late medieval Scholasticism and Heidegger’s critique of Catholic theology in his time -- both are interested in the destructionof the abstract metaphysical in favor of the phenomenology of concrete lived existence – can occlude what profoundly differentiates the two approaches: Luther’s “Christian being” cannot be conceived apart from an encounter with the Other, an encounter that cannot be interpreted either as Zuhandensein or Vorhandensein. One must not confuse the experientia of Luther’s theologian with the experience of the peasant or particle physicist. The phenomenological ontological approach “laying bare” the being-in-the-world of both occludes the “stand on being” assumed in the approach itself, an approach that itself finally must stand before God. 

In this paper, I review the research into Luther and Heidegger with an eye toward towards an appropriation of the start differences between them, particularly with respect to the question of freedom. What is constructive here is my employment of model theory to show the truth-conditions of the sentences used in the analysis. Clarity on the semantics of sets of sentences about Luther’s experientia, Heidegger’s phenomenological ontology of Christian life, and the enterprise of their comparison provides greater precision and accuracy in evaluating the differences in their respective projects. 

_________

I have for some time thought that theologians should know the basics of model theory so that they might gain greater clarity into their own theological and ontological assertions. I will endeavor to provide a brief introduction to model theory in this summer's paper, and use it to clarify the difference between Luther and the early Heidegger's project of disclosing the primordial factic life of the Christian prior to the making and evaluation of abstract theological assertions.  

Theology and the Philosophy of Science: The Syntactic and Semantic Views

The Received View in the [hilosophy of science is the syntactic view.  Accordingly, scientific theory is construed as a set of sentences with the laws of the scientific theory being its axioms. By inputting initial conditions and conjoining these conditions to the laws (axioms) of the theory, one deduces future states of the system as theorems.  This is the theory's predictions. The syntactic conception of scientific theory is clearly in the tradition of Euclid, Aristotle, Newton, Carnap and the Logical Positivists. But as we pointed out in the last post, there are problems with the account. 

One problem is that the syntactic view presupposes the so-called analytic/synthetic distinction, that is, the distinction between what is true by definition versus what is true because of the way that the world is. The distinction is rooted in the work of Immanuel Kant (1724-1804). Kant famously claimed that an analytical statement or proposition is true because the meaning of the predicate is included in the meaning of the subject.  A synthetic statement, on the other hand is ampliative in that the meaning of the predicate is not included in the meaning of the subject.  The first effectively decomposes the meaning of the subject, finding that what makes the subject true also makes the predicate true. The second amplifies the meaning of the subject; it asserts of the subject that something is true that is not included within the very meaning of the subject. 

While this semantic distinction in Kant must be distinguished from the epistemological distinction between what is known "prior to" experience (the a priori) and what is known "after" or on the basis of experience (the a posteriori), we often today simply identify the a priori with analytical judgments and the a posteriori with synthetic judgments.  For instance, "a bachelor is unmarried" is a true analytic statement because one cannot think of married bachelors, but "a bachelor is happy," if it is true, would be a true synthetic statement.  We would know the second on the basis of experience, e.g., surveys, personal observations, controlled experiments, etc. 

W. V. O. Quine famously criticized the analytic-synthetic distinction about seven decades ago, calling it one of the "dogmas" of empiricism.  He claimed that the analytic-synthetic distinction is not a matter of meaning over and against experience, that it is not a matter of the necessary truth of the former over and against the contingent truth of the latter. The distinction is not absolute at all, he avers, but it is merely a matter of degree, of what statements we will give up last.  In our "webs of belief," we hold onto some statements longer than others.  We might say, "water is H20" and "water is odorless," and dutifully subject each statement to our "tribunal of experience."  It is clear that confronted with experience, we would hold onto the truth that water is H20 much longer than water is odorless.  In fact, I can imagine some experience which would compel us to claim that water is not in fact odorless.  Of course, the latter statement could be "saved" from repudiation by declaring that it is not water itself that is not being odorless, but something in the water that is smelling foul.  

Problems with the analytic/synthetic distinction were a profound challenge for the syntactic view of scientific theory because the "bridge rules" of the theory coordinating the theoretical and observational terms were supposed to be a matter of meaning alone.  This theoretical term just means this observational term. In fact, the higher level terms and propositions of the theory could be in principle reduced to phenomenal experience. The classic text of this approach is Carnap's The Logical Construction of the World.  Clearly, if analyticity does not hold by meaning alone, then the very notion of bridge rules is undermined. 

There were, of course, other difficulties with the syntactic approach. It turned out that rigorous axiomatic laws were too cumbersome to be used by actual scientists. Also, because scientific theory was construed in terms of sentences, endless debates in the philosophy of language ensued.  Finally, there were Goedel problems.  As it turns out, no axiom set and system of proof within a theory could prove all of the sentences regarded as true within the theory. The result was the overturning of the syntactic view of scientific theory.  The new approach was called the semantic view of scientific theory.

Emerging in the 1970s and 80s, the semantic view of scientific theory generally identified theories with classes of models or model-types along with hypotheses of how these models relate to nature. A theory thus could thus be cast as a "class of fully articulated mathematical structure-types" using set-theoretical predicates.  (See Demetris Portides, "Scientific Models and the Semantic View of Scientific Theories" in Philosophy of Science, December 2005, pp. 1287-98.)  

Models are thus included in the the theory structure, and are themselves constructed on the basis of data within a context of experimental design and auxiliary theories.  On the semantic view model A is equivalent to Model B if and only if there is a correspondence of the elements and relations of A and B.  (Some advocates claims there must be an isomorphism, some a partial isomorphism and some merely a similarity.) 

Advocates of the semantic view claim that a physical system is represented by a class of model types. Semantic theorists generally hold that data alone does not falsify a theory, but that  data, design and auxiliary theory are important in the construction of data structures. These data structures must be sharply distinguished from the theoretical model, in that the latter is a construction out of the data structure.  But the question arises: What exactly is a data structure? 

It seems that the models in question can be either more abstract, e.g., mathematical structures, or more concrete, e.g., visual models of molecules. Proponents of the semantic view often claims a superiority over the syntactic conception in that scientific theory now is understood as actually focussing on the actual things that scientists treat within their theories.  Moreover, they claim that the semantic view allows that scientific theories can be clearly seen as not simply related to actual chunks of the world, but rather to mathematical objects as idealizations that are connectable to the world. Such idealizations, they claim, are the true objects of science. Accordingly, abstract mathematical structures come to be understood as that which the theory is about. Thus, semantic theories privilege mathematics -- especially "set-theoretical" entities -- over first-order predicate logic.

Rasmus Groenfeldt Winther's article in the Stanford Encyclopedia of Philosophy distinguishes two general strategies within the semantic view generally.  The state-state approach focuses upon the mathematical models of actual science such that the scientific theory just is a class of mathematical models. Alternatively, the set-model theoretic approach emphasizes that the axioms, theorems and laws of a theory are satisfied, or made true by, certain mathematical structures or models of the theory.  The second approach is often deemed the more fruitful. 

I find Michael McEwan's 2006 article "The Semantic View of Theories: Models and Misconceptions," helpful in understanding what the semantic view is and is not.  McEwan points to the following slogan of the semantic view: A theory is a collection of models (1).  On what he calls the naive semantic view, the "is" here is the "is" of identity. Tarski famously connects models to semantic concepts through the notion of satisfaction.  He uses model-theoretic models in accomplishing this. A model-theoretic model is an interpretation which satisfies a class of statements by specifying a domain of individuals and defining the predicate symbols, relations and functions on this set of individuals.  Accordingly, a theory is a collection of model-theoretic models (2).  

On the model-theoretic model the theory is a set of sentences and the models are interpretations in which the set of sentences turn out to be true. A model-theoretic theory is true for a given model just in case the sentences are true on that model. The class of model-theoretic models make true the model-theoretic theory.  McEwan calls the identification of the model-theoretic theory with the class of its models a naive semantic view.  If, however, the class of models satisfies the sentences of the model-theoretic theory, McEwan no longer dubs this a simple naive semantic view.  He specifies the naive semantic view as having the following conditions (3).

•  It is identified with M, the class of model-theoretic models,
• The models in M are directly defined, 
• The naive-theory is true for model n just in case n is an element in M. 

One problem with the naive theory is that it is difficult to see how any of it touches the world.  As it turns out, no n need represent the world at all! Another problem is that since the theory itself is just the class of models, it is what it is only when each model is true. This means that no model really instances the theory, for the theory would not be that theory if it had other instances!  As McEwan points out, the question of whether the solar system instances Newtonian mechanics is not a non-trivial one, but on the naive theory, it would be true just in case we stipulate that it is so (5).  Simply put, if the naive theory were true, then one could not axiomatize in model-theoretic theory without knowing in advance which interpretations would satisfy the model-theoretic theory.  But we do not always know in advance which interpretations satisfy our theory; there are sometimes unintended models. (Consider the non-trivial question of whether a newly discovered solar system obeys Newtonian laws.) Thus, by modus tollens, naive theory is not true.  McEwan puts the matter bluntly: "There is nothing above and beyond the models themselves to decide whether a theory is applicable to some model or not" (7). 

Fortunately, the semantic view is not identified with the naive theory.  Indeed, the semantic view realizes that the models of M must represent the world in some way.  Clearly, realists and many empiricists would want this to be so. Why not then simply identify n with a physical model?  But how can a physical system be an interpretation of a formal language?  This seems to have the matter backward.  

As it turns out, semantic views are plagued by the representation problem. Consider the claim that one of the models of M (say n) is the faithful representation of the physical world. But on what basis is n the representation? If the theory is the class of models, one of which is the real world, then why identify the theory with the class of models in the first place (8)?

It seems that the semantic view must somehow deal with the representation problem.  However, Bas von Fraasen a theory's models is identified with a class of structures.  He writes: 

The syntactic picture of a theory identifies it with a body of theorems, stated in one particular language chosen for the expression of that theory.  This should be contrasted with the alternative of presenting a theory in the first instance by identifying a class of structures as its models.  In this second, semantic, approach the language used to express the theory is neither basic nor unique; the same class of structures could well be described in radically different ways, each with its own limitations.  The models occupy center stage.  

So what of these model that occupy center stage? What becomes of realism on the semantic view?  If the models are mathematical structures, then are the objects in these models "real enough" for one to claim that one's scientific theory is true of the real world?  Is the wave function a mathematical object and thus real in the sense that a scientific realist wants?  What would distinguish a real physical object from other pretenders?  What about unobservables -- are they real?  What would distinguish an unobservable mathematical object from an on observable "real" one?  The representation problem is clearly a problem for realism. 

While one might claim that the semantic view is the new "received view" in the philosophy of science, there are very strong voices that have emerged which have pointed to the "extra-scientific" or "extra-rational" factors at work in science, factors that seem as almost as deadly to the semantic view as they are to the syntactic view. We shall attend to these in the next post. 

On Theoretical Entities and Causality in Theology

In Chapter Seven of De prescriptione haereticorum, Tertullian declares, "What indeed has Athens to to with Jerusalem?  What concord is there between the Academy and the Church?  What between heretics and Christians?"

Tertullian is not saying that philosophy should be silent when it comes to things theological, or that philosophy and theology are about different subject areas, or that philosophy and theology somehow constitute incommensurate forms of discourse.  He is saying that we should reject attempts to produce what he calls, "a mottled Christianity of Platonic, Stoic and dialectic composition."

In the following reflection I take Tertullian's intent to heart.  I will not thereby produce a mottled Christianity.  It does not follow, however, that not producing a mottled Christianity entails that philosophy has nothing to do with theology.  In fact, philosophy has a great deal of relevance for theology, particularly as both disciplines were classically conceived and practiced.  Since the time of Plato, western philosophy has been profoundly concerned with questions of semantics, with the meaning and truth of its expressions.  Since the time of Aristotle, philosophy has been deeply concerned with logic, with entailments, compatibility and modality, that is, with what propositions follow from others, what propositions can be jointly true, and in what way these propositions are true.  From both men philosophy learned about metaphysics; it learned to reflect upon being and to distinguish the different ways that something can be said to be.  Clearly, talk of God presupposes positions in semantics, logic and metaphysics -- even if these views are not explicitly held or asserted.

Consider the following expressions comprising a primitive theological theory:

1. God is incorporeal
2. God is eternal 
3. God created the universe
4. God has three persons 
5. God through Christ redeems fallen creation 

For many Christians these expressions are prima facie quite simple and plainly true.  It seems, in fact, that there is no particular problem with their meaning, truth and entailments, or even the being of those entities and properties referred to.   But looks can be deceiving.   

Think of the term 'God' and compare it with other terms you might use, e.g., 'block', 'bird', 'slab', etc.  Notice that while 'block' and 'God' both are nouns and presumably name some entity, the way in which they do so is markedly different.  Presumably, 'block' picks out a member of a class of particular empirical objects, while 'God' does not.  (Specifying the necessary and sufficient conditions for a particular object to be a member of the class of blocks turns out to be a surprisingly difficult matter.  As Wittgenstein pointed out, there seems not to be definite criteria of application for the word 'block', but rather the members of the class seem to bear some not quite specifiable "family resemblance" to one another.)  The point is that 'block' does seem to refer to an observable object, while the term 'God' does not seem so to refer.  

Once upon a time in the philosophy of science people believed that there was a pretty clear distinction between observational terms and theoretical terms.  The referents of the first could be encountered through sense perception, while those of the second could not.   Unfortunately, the distinction between the two could not be easily maintained.  In what sense is an object observable to sense perception -- with the naked eye or through an electron telescope?  Are the bubbles in a bubble chamber an observation of a moving electron, or a phenomenal event that through suitable "bridge laws" biconditionally ties to a theoretical electron?  

Perhaps it is not the observational/theoretical distinction that separates 'block' and 'God', but a semantic difference having to do with whether or not the term in question has its meaning determined through the axioms of the theory, that is to say, the meaning of a theoretical term depends upon how that term is incorporated into an overall theory.  In a scientific theory, the laws of the theory are essential for determining the extension of the theory's terms.  This means that the meaning of individual terms in the theory are determined within the theory's overall context.   Holger Andreas writes: 

The contextual theory of meaning, therefore, makes intelligible how students in a scientific discipline and scientists grasp the meaning, or sense, of scientific terms.  On this account, understanding the meaning of a term is knowing how to determine its referent, or extension, at least in part.  (See "Theoretical Terms in Science," The Stanford Encyclopedia of Philosophy (Summer 2013), Edward N. Zalta, (ed.) URL = <http://plato.stanford.edu/archives/sum2013/entries/theoretical-terms-science/>.  

When thinking of theology, it is clear that it too is a theory of a particular kind with some terms that are quite theoretical and some less so.  For instance, the term 'human being' used in theology seems to make easy reference to the world, while the term 'creation' is more problematic.  The first seemingly has a common reference in theology and sociology.  The word 'creation', however, apparently refers to the universe as such within an overarching theological theory, but makes no reference at all within sociology -- unless it perhaps refers to the manuscript the sociologist is writing.

The term 'God' seems to have meaning within a particular theological theory.  In (1) above, 'God' is predicated by 'incorporeal'.  Is incorporeality "present in" God or "said of" God?   If the former, then the being which is God has the property of not having a body in the actual world, but could have a body in another possible world.  If the latter, then it is not possible that any being which is God could have a body.

From the standpoint of the philosophy of science, 'God' is a theoretical term naming a theoretical entity, a term that seemingly has incorporeality as part of its very meaning.  Just as a bachelor is an unmarried male, so too is God incorporeal.

The same might be said about God's eternity.  Perhaps it is essential for God to be eternal, that is, nothing that is God can fail to be eternal.  If both eternity and incorporeality refer to God, then we might speak of a "conceptual tie or law": For any x, if x is God then x is eternal and incorporeal.  But this is not a paradigmatic bridge law because it is not a biconditional; it does state in addition that for all x if x is eternal and incorporeal, then x is God.  In addition, it does not "bridge" from observation events to the exemplification of a property by a theoretical entity.

If we do not, however, think of theological theory as having any bridge laws in the classic sense, but rather as constituted by a group of propositions having terms, many of which appear in a number of the propositions, we can speak of a term's meaning being a function of the way in which it appears in the other propositions in the theory.  (What is predicated of the term and what the term is predicated of.)  This implicit definition of the term then determines its extension.

Within our primitive theory, (1) and (2) presumably has a distribution of predication that differs from (3), for while predication of 'eternal' and 'incorporeal' in the theory does not allow for an x that is God to be predicated by 'not eternal' or 'corporeal', the x that is God can be predicated by 'creates the universe' or 'does not create the universe' because while one can have as a statement in the theory, 'did not create the universe at time t',  one cannot have 'is not eternal at time t'.  That the truth value of 'creates the universe' differs as a function of its temporal index, while the truth value 'is eternal' does not so differ, clearly shows that 'is eternal' means something quite different than 'creates the universe'.

Now consider the predicate in (4), 'has three persons'.  To say that the x that is God has three persons is quite different than saying that the x that is a small company has three persons.  Why?  Because one rarely if ever would say that an x that is a causal agent -- like in (3) -- could ever have three persons.  While a company could be said to be a group of people exhibiting certain relationships among them, God cannot be said to be a group in any sense, for the three persons having relationships among themselves is the simplicity of the one God.

Proposition (5) asserts that the x that is God causes it to be the case that the domain that God creates is now redeemed.  This analysis of 'redeems the world' can be given a temporal characterization like 'creates the world', thus showing that these terms must have different meanings than terms like 'incorporeal' and 'eternal'.  The phrase 'through Christ' adds further complication because it raises the question of whether 'God redeems' if and only if 'God through Christ redeems', and, if so, what does 'through Christ' add in meaning to 'God'.  To show that 'through Christ' has a different meaning, one needs to show that 'God' and 'God through Christ' cannot be substituted with each other salve veritate throughout the entire theological theory.

What I am suggesting here is neither terribly original nor novel.  I am merely suggesting that it might be instructive to look at theological theory with its theoretical entities in ways similar to how we might look at a physical theory having such entities.  We might do this simply to get clear on the semantics of our theological language.   What exactly is meant by a term appearing within a theological theory of a particular kind over and against a term appearing within a theory of another kind?  Since we have fewer empirical moorings in theology than physics, it is useful perhaps to focus more deeply on what it is we might be meaning when employing language of the first kind.

Model-Theoretic Semantics and Theology

All too often we unthinkingly assume a "magical" view of language.   We naturally suppose that our language is anchored to the world correctly, as if our language intends to link to the world in a particular way.  For instance, we might believe that 'dog' uniquely refers to that class of which the canine at my heels is a member, and 'laptop' to that class of which this object upon which I type is an element.

However, reflection about the nature of such intentionality does not support these prima facie intuitions.    'Dog' cannot and does not intend the canine at my feet, though through appropriate human context and practice it may refer to that animal.   'Laptop' is conventionally linked to the object upon which I type these words, though it may not have been the case.

Hilary Putnam famously advanced the "model-theoretic argument against realism."  In it he purports to show that that an entire linguistic system considered as a totality cannot by itself determinately refer.   Representations, no matter how involved, are not agents and thus have no power to intend objects in the world.  Language, considered formally and syntacticly, does not in itself have meaning and cannot thus refer to the world.  Any attempt to give language such an intentionality through the use of model-theoretic semantics must fail.  In order to understand what Putnam is saying and its relevance for theology, we must understand what model-theoretic semantics is.

Model theory provides an interpretation to formal systems.  For the various symbols of a language, it assigns an extension, i.e., particular individuals, sets, functions and relations.  Model theory recognizes that since language does not magically intend objects in the world, the elements of language can only map to structures of objects.  Simply put,  given a particular function f, and any non-logical term p, f(p) graphs to a unique object in the world o.  In other words, there is a transformation from language to its extensional interpretation, a correspondence that is itself conventional.   Accordingly, while a particular function f1 maps 'dog' to the class of objects of which the canine at my feet is a member, another function f2 maps 'dog' to the last horse standing at Custer's last stand.  When we think language magically picks out the elements of the world, we simply forget that many other functional images of our language are possible.  Simply put, we forget that our language can sustain a large number of multivalent interpretations.

Model-theoretic semantics proceeds by constructing models which satisfy classes of statements, that jointly makes true those statements.   Take, for instance, this class C of statements:  'The cat is on the mat', 'John understands that an equivalence relation is reflexive', and 'All mats are owned by John'.   A model is an extensional interpretation I making all members of C true.  This might happen when 'cat' refers to the set of all domesticated felines, 'mat' to the set of all objects upon which one wipes one's feet, 'on' to a two place predicate Oxy specifying the set of all ordered pairs {x, y} such that x is adjacent and above y, 'John' to a particular person,  'understands' to a dyadic predicate Uxy forming the set of all ordered pairs {x, y} such that the first is an epistemic agent and y is that which is understood, 'equivalence relation is reflexive' to a member of the set of all concepts, and 'owned by' to a two place relation Wxy forming the set of all {x, y} such that x possesses y.  In addition, 'the cat' is a definite description uniquely picking out some member of the set of all domesticate felines, while 'the mat' uniquely refers to one member of the class of all objects upon which one wipes one's feet.  

The reader should reflect upon how difficult it is to provide an adequate intensional characterization of the set of mats or the set of things understood.   Fortunately, we don't have to pick all the properties that each and every member of the set has.  We can simply refer to the set whose members have these properties as well as others.  It is obvious that the three propositions above are true (or "satisfied") if there exists the sets in question and the members of these sets are related in the ways above specified. Let us call this interpretation I.  

Now notice that we can form I2 as follows:  Allow 'cat' to refer to the set of positive integers and 'mat' to refer to the set of negative integers, and "on to" (Oxy) to be the set of all ordered pairs {x, y} such that x is greater than y.   'The cat' now refers to a definite positive integer and 'the mat' to a particular negative integer.   Let 'John' refer to the positive integer 17 and 'understands' be the two place relation forming the set of all x such that x is the square root of y.   Assume that 'equivalence relation is reflexive' refers to 289, itself a member of the set of all odd numbers.  Finally, allow 'owned by' to refer to be the set of ordered pairs {x, y} in W, such that either x is greater than y v x=y v x is less than y.  While this interpretation may seem very artificial, it does in fact "satisfy" each member of C.  The point is that all sentences of C are true both on models I1 and I2.  

Model-Theoretic semantics provides abstract models satisfying classes of statements.  These models are sets obeying set-theoretic operations.  Clearly, we can think of the satisfaction of the classes of statements to be mappings from the constituents of those statements to unique set-theoretic structures; the relationship of the linguistic entities to their extensions are unique functions.  Each interpretation is a function from the linguistic to the set-theoretic because the following uniqueness condition holds where x is the linguistic and y the set-theoretic:  If and are members of f, then y = z. 

Putnam's argument purports to show that simply having a model that makes a class of statements true does not in and of itself determine reference.   There are an infinite number of models with different extensions that make the class of statements true!  Neither does representational similarity between the linguistic symbols and their extensions nor truth itself vouchsafe a unique reference for a language.

One way to grasp this is to consider Quine's gavagai example.   The anthropologist sees the native saying 'gavagai whenever presented with a rabbit.   But the anthropologist is sophisticated in his reflections and realizes that the native could mean 'undetached rabbit part' or 'rabbit event' or 'temporal rabbit stage'.   The model would seemingly be satisfied by any of these interpretations.   Language does not determine reference.

Putnam finds in the Lowenheim-Skolem theorem significant results which extend this insight.   The theorem holds that any satisfiable system -- that is, any system that has a model -- has a countable finite or infinite number of models.  Putnam generalizes the results of this theorem, showing that even in a system vast enough to incorporate all of our empirical knowledge, it would nonetheless be the case that there would be great numbers of models (and associated ontologies) satisfying all of the constraints of the system's theoretical and operational constraints.

While there is debate about whether Putnam's proof in "Model's and Reality" (see Realism and Reason, Cambridge: Cambridge University Press, 1983, pp. 1-25) commits a mathematical error, the general point is clear enough to anyone who has every taught an introductory logic course: Truth is always truth under an interpretation.   Agreeing on language does not an agreement make.   Agreement is only had if there exists agreement of language and a common interpretation or model.   Only if the same model is specified and there is agreement in truth-value among the relevant propositions can one speak of actual agreement.  

It should be obvious to anyone who reads theology that theological traditions have not always been clear about the interpretation of their language.   This becomes deeply clear in interfaith dialogues when two sides may use the same language, but mean something quite different with that language.   It happened, in my opinion, in the Evangelical Lutheran Church's adoption of three important documents between 1997-99:  Call to Common Agreement, the Formula of Agreement, and the Joint Declaration on the Doctrine of Justification.  The frustrating thing about those debates was that many of the participants either did not know that they needed to clarify the models they were using, or intentionally did not deeply reflect upon their interpretations for fear of losing the historic "agreement" between the parties that the ecumenical talks were supposed to engender.  

Maybe the proclivity of participants in ecumenical dialogues not to clarify the models they are assuming stems from a general historical practice among theologians to fail to specify the interpretations they employ in their own polemics and constructive work.

Take the following three propositions and assign them extensional interpretations I1 and 2.

• T1:   God creates the universe.
• T2:   All of creation has fallen into sin. 
• T3:   Through His Son, God redeems his fallen creation.  

Let I1 be the following interpretation: 

• 'God':    That being having all positive predicates to the infinite degree
• 'Creates':  A dyadic predicate whose extension is the relation {{x, y}: x causes there to be both the material and form comprising y}
• 'Universe':  All that exists outside of diving being
• 'Creation':   All that exists outside of divine being
• 'Falls':  A dyadic predicate whose extension is the relation {{x, y}: x is creation and y is the distortion of x under the conditions of present existence}
• 'Sin':  The distortion of creation under the conditions of present existence
• 'Son":  Hypostasis bearing the divine nature sustaining the following relationships of having been begotten by the hypostasis of the Father and spirating the hypostasis of the Holy Spirit
• 'Redeems':  A triadic predicate whose extension is the relation {{x, y, z}: x causes there to be reordering of y on account of z, such that x regards y as manifesting properties characteristic of the created universe 

Many readers may take issue with the extension I gave to T1-T3.   It would be an important exercise, I think, were all who employ theological language to attempt to provide a semantics like I just attempted.   It is by no means a simple task.   It is time, I believe, for theologians not simply to take responsibility for their theological language, but also for the interpretation they give that language.

Let I2  be the following interpretation:

• 'God':   To-beness in its totality.  That which is presupposed by the notions of being a particular being, and not-being a particular being
• 'Creates':  A dyadic predicate whose extension is the relation {{x, y}: x is conceptually presupposed by the class of all existing beings}
• 'Universe':  The set of all non-divine beings
• 'Creation':  The set of all non-divine beings
• 'Falls':  A dyadic predicate whose extension is the relation {{x, y}: x is creation and y is the set of attitudes, dispositions, and existential orientations of human beings phenomenologically present to human awareness as lacking the character of original creations
• 'Sin':  The existential of human existence towards the "what is" of the past rather than the "what might be" of the future 
• 'Son':  A symbol that points to and participates in the totality of being, and is capable of communicating the power of being itself phenomenologically to human beings
• 'Redeems':  A triadic predicates whose extension is the relations {{x, y, z}: x communicates the power of being itself to human beings (y) by means of the symbol of the Son (z)}  

The perceptive reader might find a trace of Tillich in interpretation I2.   The point to realize is that I1 and I2 both make T1-T3 true.   Both models satisfy a very small class of theological propositions.   Notice it is meaningless to ask if T1-T3 are true until a model has been specified upon which to evaluate their truth.  Here as everywhere in theology, truth is always truth under an interpretation.    

Towards a Lutheran Theological Semantics III

Imagine two theories T1 and T2 indiscernible with respect to their syntax. To give an interpretation to this syntax is to define an ordered pair <I, n> such that I specifies a domain D of entities named by the individual constants of the theory, some Fx specifies a subset of D, and Fx . . . k specifies a k-ary Cartesian product in D. Let n now designate a naming function from names in the language to D, monadic predicates to subsets of D, and k-ary predicates to k-tuples in D. The function thus assigns for each and every nonlogical symbol an extension in D. What we are doing here is assigning a semantics to our language. Obviously, if both T1 and Ts use the same >, they will mean the same thing. Two theories indiscernible with respect to syntax and having the same interpretation have the same model. We say that M models a theory T if and only if all the sentences of the theory are true given the projection of the language onto the model. Obviously, if M models T1, and T1 and T2 have the same interpretation and mapping function n, then M shall model T2 as well.

Within the practice of science, the syntax of theories change as a function of new empircal data and concomittant theory adjustment. In science generally, the method of projection of the syntax of the theory upon a model is for the most part invariant, and it is this invariancy that makes possible scientific progress generally. Words like 'electron', 'boson', and 'p orbital' retain their interpretation (reference) across different theories generally. (We might say that in a situation of revolutionary change in paradigm, new interpretations and naming functions might arise.)

However, within the practice of theology, things are far different. Scripture and theological tradition has worked to produce a rather loose 'theological theory' whose syntax does not in general change. But as times change, the syntax of this theological theory takes on a new interpretation. Imagine T1 being classical christological formutions at the Council of Chalcedon in 451, and T2 be the same classical christological formulations said by Paul Tillich in 1957. Here it is obvious that while the syntax of T1 and T2 remains the same, there is a change of the mode of projection of the syntax upon its model. Because T1 and T2 are both regarded as true, there are distinct models M1 and M2 that model the same syntax. The same syntax is modeled both by M1 and M2, or alternatively, M1 and M2 both model the same theory T1. (Remember that T1 = T2 syntactically.) The situation now is that we have two distinct models for the same syntax, two distinct ways that the world might be ordered that would make possible the truth of T1 = T2.

The question is this: What is the theological theory T1 and is it different then from T2 after all? The answer is, of course, that we do not regard scientific theory as mere syntax, but as syntax + an interpretation. Similarly, I aver, we ought not to regard theological theory as mere syntax but syntax + an interpretation. How can it be then that many today in theology, particularly in ecumenical theology, think that syntax alone does a theology make? How can it be that the Joint Declaration on the Doctrine of Justification can claim agreement on 'justification' because the syntax of the language is similar between Lutherans and Rome?

We should remember that Luther said he was not interested in agreement in words (verbis) but in things (in rebus). Although Luther was not using the language, he was, of course, interested in the disparate models the same theological syntax could sustain. When one thinks about it, this is how it has always been in theology. Was this not precisely what happened after Ephesus (431) that two sides used the same words while allowing different interpretations of the same language?

A theology that has lost interest in its interpretation and naming function, is a theology that has lost interest in truth, because only with the assignment of models is truth put in play. While sytax deals with form and structure, semantics deals with truth and meaning. Theology has always been about the latter. It is a mark of the recent theological poverty of our time that we could have been so bewitched for so long, and have not even noticed.

Theological Doctrine as Grammar: The Meaning as Use Ruse

Wouldn’t it be great to be able to assert the great doctrines of the Church without having actually to violate one’s ontological scruples about there being states of affairs referred to by these doctrines? What if one were to claim that the Lutherans and the Catholics actually pretty much agree on justification, as the Joint Declaration on the Doctrine of Justification declared, without either party actually having to change its very different understanding of the notion, an understanding that has separated the two groups for almost five centuries? What if one were to claim that there is an identity in difference, an identity sufficient for ecumenical agreement even when the groups have understood themselves in the difference? What if one could claim a unity without a change in the interpretation either side gives to its language? Would this not be too good to be true for proponents of contemporary ecumenicity?

Take the following sentence: ‘We are justified by grace alone through faith.” As it stands, it has an everyday meaning due to inchoate interpretations used both by Catholics and Lutherans. For Lutherans, the sentence has been regarded as true; for Catholics, it is false. It is true for Lutherans because God is the agent by which grace is given to the believer in faith. The Christian who is justified in faith automatically acts out of that faith, for a good tree must bear good fruit. In the Catholic tradition, faith must be formed by love; one is justified by grace through faith issuing in works of love. Catholics and Lutherans could disagree about how ‘faith’ connects to love. Lutherans tend to regard love as analytically entailed by faith, while Catholics deny this. (I am being very general here.)

One could, I suppose, count agreement between Lutherans and Catholics if the same sentence were uttered by each in relevant contexts with suitable linguistic cues. In other words, one could understand the sentence behaviorally. The sentence has the same meaning for two users if and only if the proclivity to utter it is similar given suitably similar linguistic cues. One might even claim that because the sentence is used the same way by the two different communities, thus the two theological communities assign to it the same meaning.

But this is, of course, a deeply unsatisfactory way for two different linguistic communities to affirm the same statement. After all, it is not the use to which it is put that gives the sentence the same meaning, but it is rather that both have a common meaning, and thus the sentences are used in similar ways. Clearly, in order for Catholics and Lutherans to agree on the statement, something more than merely uttering the statement in similar linguistic contexts is necessary.

It is standard in logic and semantics that an interpretation be assigned individuals and predicates of the language (non-logical terms) so that it can be determined what models of a sentence or groups of sentences satisfy them. See how this clarifies statements like ‘S is justified by grace through faith’ and ‘S is justified by grace through faith issuing in acts of love’. The first could be rendered as follows ‘(some x)(some y)[(Gx & Fy) & Jsxy]’ read as ‘there is something which is grace, and something which is faith, such that s is justified by that which is grace through that which is faith’. However, given that love issues from faith, we might add, ‘(some x)(some y){(Gx & Fy) &amp; (Ly & Jsxy)]’ read as ‘there is something that is grace, and something that is faith issuing in love, such that s is justified by that which is grace through that which is faith’. Now notice that the same string can be used to capture the Catholic view, that we are justified by grace, through faith issuing in love. Clearly, both have the same model, {(some x) is a member of {x : x has G}, (some y) is a member of [{y : y has faith} intersects {y : y issues in love}], (s, some x, some y) is a member of {(x, y, z) : x is justified by y through z}}. The two sentences are not only compatible by having a common model, they are equivalent because they are satisfied by exactly the same set of models. They have a common model-structure.

Ecumenical conversations would be greatly improved, in my opinion, if the dialogue partners were to pay profound attention to what is meant by the phrases they use. If both sides were disciplined in providing formal interpretations for their statements, it would become quite clear what, if anything, are the significant differences of meaning between the two.

Theological Semantics and the Problem of Interpretation

The sentence 'the cat is on the mat' is meaningless until it has been given an interpretation. We define a function from the sentence to objects within a domain. Standardly, we should say that 'cat' refers to {x: x is a cat}, 'mat' refers to the {x: x is a mat} and 'is on' refers to { (x, y) : x is on y} . Thus, we say that there is some member of the first set a, some member of the second set b, such that is a member of { (x, y) : x is on y}. To give an interpretation is to define a function from relevant linguistic units in the language to things in the world, such that the objects in the world form a functional image f* of the language. Thus, 'the cat is on the mat' is given by f*(cat), f*(mat), and f*(cat, mat) is a member of {(x, y) : x is on y}.

Now imagine providing such an interpretation for Trinitarian discourse. 'God is the Father', 'God is the Son', and 'God is the Holy Spirit', 'the Father generates the Son', and the Holy Spirit proceeds from the Father and the Son'. One could say that f*(Father) is a member of f*(God), f*(Son) is a member of f*(God), f*(Holy Spirit) is a member of f*(God), and that {x : x is God} has one member g. Thus f*(Father) = f*(Son) = f*(Holy Spirit) = g. 'The Father generates the Son' is thus f*(Father, Son) is a member of f*{(x, y) : x generates y}. Accordingly, 'The Holy Spirit proceeds from the Father and Son' is given by f*(Father, Holy Spirit) and f*(Son, Holy Spirit) is a member of {(x, y) : x proceeds y}. What follows, of course, is that it is a member of {(x, y) : x generates y}.

Now, taking 'G' to be "generates", we have that Ggg. Lombard and the Fourth Lateran Council reject Ggg because ascribing the reflexivity of generation to the individual g seems to deny simplicity, for there seems to be no possible world in which something can generate itself without dividing itself. (Notice how one can know oneself or think oneself without dividing oneself - - if one has intuitive, nondiscursive knowledge as has traditionally been thought to be true of God.)

Martin Luther, however, had no problem affirming the propriety of "the divine essence generates the divine essence'. When he said this, he meant that the Father generates the Son. If the Father is the divine essence, and the Son is the divine essence, and the Father generates the Son, then the divine essence generates itself, Ggg. He seems to have no problems with this because if Plato is a man, and Aristotle is a man, and Plato is a teacher of Aristotle, then it is proper to say that man is a teacher of man. Of course, the set M = {x : x is a man} is not a singleton set as is D = {x : x is God}. D has one member g, but M has billions of members.

When thinking the divine essence, one must not only subscribe to it a as a general essence, but one must claim a single instantiation, for if there was more than one instantiation, there would be a compromise of monotheism.

In order to make progress on the various claims in the late medieval period, we must be able to state clearly the ontological situation of the Trinity in the most perspicuous language we possess: first-order predicate logic with identity.

On the Theological Equivalency of East/West Trinitarin Models

Theologians are oftentimes too lax in getting clear on the types of claims they make. Take, for instance, the claim that the East and West made theologically equivalent claims about the Trinity, although they divided philosophically in their conceptuality about universals: The East assuming that a universal could be numerically one and multiply instantiatable, and the West denying this. Is it indeed true that the East and West make equivalent trinitarian claims?

Theoretical equivalency is understood differently on syntactical and semantic approaches. According to syntactical approaches, theories T1 and T2 are equivalent if they have, as their extension, the same or equivalent set of models. This syntactical approach to understanding theories and their equivalency was dominant through much of the twentieth century, and is oftentimes referred to as "the classical view" or "the received view." On this approach, a theory is a set of uninterpreted axioms in a specified formal language using a set of correspondence rules that provide a partial empirical interpretation to the theory by linking observable entities and processes to particular non-logical terms. A theory is true just in case its interpreted axioms are all true.

The semantic approach to understanding theories and their equivalency has developed over the last four decades. While syntactic approaches are interested in deducibility from axioms, semantic approaches focus upon the notion of "satisfaction." Objects which satisfy the axioms are models of those axioms. According to the semantic approach, the axioms comprise part of a theoretical definition. Whether or not such a definition is true of the world depends upon theoretical hypotheses. A theory is true just in case all of its associated hypotheses are true. In the words of F. Suppe, the semantic view "construes theories as what their formulations refer to when the formulations are given a formal (semantic) interpretation" (Suppe, The Semantic Conception of Theories and Scientific Realism, Chicago: University of Illinois Press, p. 4). This model-theoretic view identifies theories with the set of models that satisfies the theoretical laws of the theories. The models can be understood as being a pure structure: abstract entities and relations. Theories T1 and T2 are equivalent if and only if they are satisfied by sets if models M1 and M2 respectively, such that M1 and M2 are isomorphic to each other.

Applying this to the Trinitarian question, we might say that sentence "God is both one and three" is satisfied by two different sets of models, one employing a multiply instantiable universal, the other a property particular overlapping the compresent bundles of properties constituting the persons. If two separate sets of model structures satisfy the same set of Trinitarian propositions, these structures are equivalent and they are isomorphic with respect to each other.

It is ironic to find that East and West Trinitarian controversies may have been argued by people holding equivalent or almost equivalent Trinitarian views. Perhaps one is not making a different theological claim at all when "starting with the persons" or "starting with the essence." Maybe, in fact, the new social Trinitarianism is finally equivalent to older, more traditional Trinitarian positions.