Sunday, February 22, 2015

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.    

Wednesday, February 18, 2015

Growth of the Institute of Lutheran Theology

I want to inform all of you about the growth and development of the Institute of Lutheran Theology.   I have not been updating my blog regularly because I am pretty busy with ILT these days.   If you have not been to our webpage for awhile, please take a look.   We are growing with celerity and probity.

The Institute of Lutheran Theology was blessed to take in more than $760,000 in cash donations last year, and had revenue of over $1,000,000.  It has now achieved these levels with respect to the most recent American Theological School (ATS) listings: 

  • ILT expended more than 28 other ATS accredited institutions in 2014.
  • ILT had a headcount greater than 29 other ATS accredited institutions in 2014.
  • ILT had a full-time equivalency of students (FTE's) higher than 23 accredited institutions in 2014.   
  • ILT is now larger than three of the four Canadian Lutheran seminaries.  
The Institute of Lutheran Theology has a permanent staff of 14 (11 full-time) with a faculty of 16 (7 permanent and 9 adjunct).  It has awarded four masters of divinity degrees, four masters of religion degrees, and 12 ministry certificates.   Its library has grown to 40,000 volumes available to faculty, students and friends.   

Our staff and faculty have eight members from the Lutheran Church-Missouri Synod (LCMS), six from the North American Lutheran Church (NALC), four from Lutheran Congregations in Mission for Christ (LCMS), three from the Evangelical Lutheran Church in America (ELCA), two from the Augsburg Lutheran Churches (ALC), and one each from the Evangelical Lutheran Church in Canada (ELCIC) and the Association of Free Lutheran Congregations (AFLC).  On our Board we have the senior justice of the Minnesota Supreme Court and the former President of the American Academy of Religion. It is comprised of five LCMC members, two from the ELCA, one from the NALC, one from the LCMS,  and one from the Canadian Association of Lutheran Churches (CALC).  It is remarkable to bring together Lutherans from many so many Lutheran church bodies to build one institution.   Soli Deo gloria et honor!!!

Because we believe that laypeople are quite capable of thinking theologically, we provide these resources for congregations and interested Christians of all kinds: 

  • Table Talk.  This devotional bulletin insert by Pastor Tim Swenson is used by scores of congregations around the country and ILT hopes that more of you know about this and use it.   The masters are distributed without charge or a small fee for mailing bulk copies.  You may click here to see examples of this product.
  • Word at Work Magazine.  A seasonal magazine with articles on subjects of interest for all Lutherans, it is distributed free of charge and is also available online here.   Register online to join the 4,000 households already on the mailing list.
  • Word at Work Classes.   No longer do we charge for these excellent video products.   Go to this page to see what they have to offer.  ILT is currently featuring Dr. Forell’s last lectures on Luther.   Other things of interest include Dr. Buddy Mendez’s class on communication and relationships, Pastor Scott Grorud’s confirmation class, and Pastor Moe Redding’s class on discipleship.
  • Word at Work Events.  If you go here you see that ILT is happy to come to congregations and do theological lectures on various topics.   For instance, the Academic Dean, Dr. Jon Sorum just did a series on “What You Don’t Know Can Hurt You: Facing the Crisis in the Church” in Mesa, Arizona, last week.  Dr. Paul Hinlicky is just one of many ILT professors willing to do these events.  
  • Weekly ILT Chapel (and daily sermons from the ILT community).  Go here to view past and present chapels.  
Grounded in Scripture and the Lutheran Confessions, ILT boldly asserts the reality of God and the causal linkage between God and His creation.  It believes that God really was in Christ reconciling the world to Himself (2 Cor. 5:19), and it believes that this truth can be effectively related to the contemporary intellectual and cultural horizon.  

Visit our webpage, our Facebook page, or our YouTube channel for more information.