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.    

On the Logical Priority of Logos

Theology's function is to interpret the kerygma into the context.   This much has always been clear to me.   But what are the limits of this interpretation?  What norms sort theological attempts between success and failure?   And what are the proper words to use here?   Ought we to speak of true theological statements over and against false ones?    Are theological claims made in this interpretation better thought to be felicitous or infelicitous?   Are some more fecund than others, and, if so, what are the marks of this fecundity?

Over three decades ago I decided that I wanted to do theology seriously.  But over the decades I have been paralyzed by the Herculean effort seemingly needed to make any true theological advance in our time.   I knew that I could not simply parrot putative truths of another time as if they were truths of our time, yet I did not want to say that the truth-values of theological statements were simply and facilely indexed to time.  I have watched contemporary theology (and theologians) come and go and I have marveled at how little their passage on the theological stage seemingly depends upon the strength of their arguments.  I have always assumed that the acceptance of theological positions ought not be like that of political ones.   Theology, the grand discipline of the west, could not be simply a matter of fad, whim, and immediate political, economic and social cash value.   It simply has to be something more, I have hoped.

The proclamation of the life, suffering, death and resurrection of Jesus the Christ has to be the starting point of theology.  The source of theology must be the Cross.  Of this, I have never had doubt.   An analysis of the cultural and intellectual horizon is necessary to the task of theology and, in some way, this horizon is itself a source of theological reflection.   However, this source is not of the same type as the other source.  While one has particular insight into the horizon, and while the horizon is something we "bump up against" in all experience, the horizon is not revealed.   The kerygma is revealed and the horizon is not.

Yet the two are given in a different way than our interpretative activity of unpacking the poles of kerygma and horizon, and carefully and patiently laying out, uncovering, or constructively articulating the relationships holding between those poles.  Our language, culture, philosophical assumptions, conceptual schemes, and own existences (including the socio-political) are the media by which the poles are refracted.  The hard task of locating the poles with respect to each other by specifying their connections is, of course, what the method of correlation is all about.   This creative, interpretive act of correlation is built upon previous acts of interpretation.   There is a hermeneutic of kerygma, a hermeneutic of horizon, and a hermeneutic correlating the deliverances of the first two hermeneutics.   Since the hermeneutical act is historically, culturally, conceptually influenced - - the product of the hermeneutic seems destined to be a here today, gone tomorrow, Johnny one-hit phenomenon.  Or so it seems on first reflection.

But perhaps we theologians spend too much creative energy wallowing in the quagmire of the seeming relativism based upon historical, cultural, and conceptual dynamism.  After all, it is not that the hermeneutical task - - and the hermeneutical circle and its effects - - infect what we do alone.   All intellectual activity proceeds by interpreting one thing, then interpreting another thing, and finally interpreting how those things fit, or don't fit, together.  It is what human beings do, and it is what we have always done.   Yet, there was once a time - - and there is in many other disciplines still a time - - when truth claims were/are vigorously asserted, supported, denied and repudiated on the basis of criteria that are abiding even within the flux of history, language, and culture.  It is not that everything is a Heraclitian flux only.  There is, after all, logos in the flux; there is order and reason.  We theologians have tended to concentrate so much upon the flux that we miss the order.   We tend to forget that the very categories we use in thinking and communicating the historical flux of thought are, in some sense stable categories.   In fact, the necessary condition for communicating flux is an ordered, coherent structure of thinking and being.  One cannot state change without perdurance.   This very old thought is either true or false, and I believe there are very good reasons to think it true - - Gorgias aside.  

What we theologians need again is a healthy dose of the reality of logos.  Our task is not dissimilar to Descartes'.   We must assume the worse-case scenario for theological knowledge, and try to uncover those stable structures presupposed by that worse case.  We must again learn to employ principle of contradiction:  If a theological position, or a hermeneutical interpretation of the hermeneutical situation ramifies a contradiction, then we must learn again to state clearly that the denial of that position is at least possible.  Moreover, we must learn again to think deeply enough theologically to spot the ways in which theological discourse is not generally a discourse of the contingent, and be able to conclude appropriately from this how the possible thus relates to the actual.  This is not easy work, but it is the work before us.

Just as flux presupposes logos, so does the historicity of the hermeneutical situation presuppose a metaphysics, that ontological correlate to the stable structural categories necessary even to state a non-completable hermeneutical dynamism.  It is precisely this metaphysics that theology has forgotten about, and it is precisely this that must be investigated again.   My hope is to begin this investigation soon.