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.    

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.