01 Apr Generating and Qualifying Propositions
What are the limits of reasoning? Is it possible to reduce every cognitive activity (telling time, falling in love, inventing rockets…) to a set of premises and conclusions: propositions? LITTLE ANIMALS ARE FURRY is a very simple proposition. Can intelligence be defined by the complexity of the sequence of propositions we can balance in evaluating a situation? I have looked at dichotomous logic and multivalued or fuzzy logic as mechanisms of human reasoning, and concluded that the system that automated system that can understand the complexity of human intent would need some of each kind of logic. Let’s look a little more at the complexity question. Our cognitive mechanism is probably capable of generating all the complex propositions below relatively quickly (though possibly involving much cognitive effort on our part) to consider the justification for life imprisonment for first degree murderers. Here is a list of some of the underlying propositions:
|society involves harmonious coexistence||people enjoy society|
|offenders commit offenses||offenses disrupt society|
|punishment deters offenses||offenses are scaled|
|society imposes punishment||punishment is scaled to offenses|
|society determines scale of offenses||murder is an offense|
|first degree is the worst murder||life imprisonment is the most severe punishment|
John Sowa proposed conceptual graphs for documenting this kind of propositions. To try to graph the interrelations of these propositions may be a mammoth task, especially compared to the task of defining the interaction of simple attributes of different animals (RODENTS are FURRY). Though animals have many attributes to define and assign, they have much more regular and predictable forms of interaction than complex emotional questions such as the appropriate treatment of criminals.
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|reasoning cognition||Sowa 1984|
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|proposition constraint||Stanford Plato on Fuzzy Logic|
For some crimes/criminals, it is easy to get from the verdict to the best punishment. But justice, especially under mitigating circumstances, can involve very complex propositions, and an understanding of millennia of reasoning, jurisprudence and precedents. Balancing mercy with justice is one of the most complex types of reasoning humans are called on to perform.
On the complex end of the spectrum are propositions based on complex causal reasoning. The example presented earlier had to do with sentencing first degree murderers to life imprisonment. Some of the fundamental premises might take this form:
CRIME REQUIRES PUNISHMENT
That is a general premise. In this specific instance the premise may read:
1ST DEGREE MURDER REQUIRES LIFE IMPRISONMENT
These are complex premises, and may rise to the level of propositions when we see that there are many premises underlying these terse statements that may sound like facts or common sense to many. In our minds, we are able to aggregate many premises and logically leap forward to complex propositions. But the books of case law behind the legal arguments are full of details and premises that are used to build up these complex arguments.
Another important factor to consider is the necessity of accurately qualifying propositions. Some relations expressed as propositions can be constrained by narrowing the scope of the objects. This could be done with LITTLE ANIMALS by redefining the left object as RODENTS. The right object could also cleaned up slightly by changing FURRY to HAS FUR. Constraining the crime propositions, however, may require insertion of complex qualifiers. It is already constrained in the example above because murders that are not FIRST DEGREE have been eliminated. The left side could be further constrained by REPEAT OFFENSE to qualify that part of the proposition. The right side, LIFE IMPRISONMENT, could be further constrained by specifying when parole is possible. The whole causal chain shown above has thus become much more complex.
The laundry list of other realities that have some bearing on the proposition could also be interpreted as qualifiers. The critical question is one of divisibility. Can any of the premises be stated in a separate proposition and be linked meaningfully to the first? For MURDER, there is no clean dividing point. Tying REPEAT OFFENSE to a PAROLE ELIGIBILITY premise may better express the interaction of the two qualifiers, but it would not be of any use in the specific instance because the impact of repeat offenses on parole eligibility is different for each crime (PAROLE AT 26 YEARS).
Modeling Qualified Propositions
My model for graphing these propositions involves left and right objects connected by relations. We can qualify complex propositions such as criminal sentencing by adding constraints on the left and right side of propositions (the objects), or to the middle (the relation). By using a qualifier, exceptions and parameters can be added to propositions. This illustration depicts the impact of additional constraints on the shape of propositions. Where unconstrained propositions appear linear (X –> is Related to –> Y), constraints can radically disrupt the linearity. Here are some examples of how exceptions may be applied (read them as OBJECT X is a(n) RELATION of OBJECT Y that is characteristically QUALIFIER):
|Object X||Relation||Object Y||Qualifier|
This qualified structure of associations is more expressive of the information content than associations with unqualified relations. The word REQUIRES is a subjective and arbitrary term applied to a causal type of relationship. At least a dozen other names for the relation in this proposition could be justified. Causal relations are particularly ambiguous when there are multiple constraints implicit in the proposition. Some of the constraints implicitly affecting this proposition are society, justice, safety, people interacting, deterrence, past harm, and future potential for harm. In fact, many hundreds or thousands of propositions are related to this one proposition. Generating and qualifying propositions of this type is my idea of the core cognitive process – the very thing we want to model to make smarter computers.
1ST DEGREE MURDER (REPEAT OFFENSE) REQUIRES LIFE IMPRISONMENT
Object qualifiers, in many instances, can be eliminated by restricting the type or definition of the left (entity) or right (associate) objects. We did that with crime earlier. Instead of qualifying the relationship between the crime and the punishment by saying, in effect, “when it is a repeat offense,” we can state the relations between crime, repetition and punishment as separate propositions. This narrowing process at the object level is essential for expressing many hierarchical ties. For many other types of relations, the narrowing process can be expensive in terms of computational space and time and still fail to capture the intricacies of relations between data objects.
In the proposition above, there are constraints on both sides of the link. The constraints can be used in fuzzy reasoning processes in which there may be a scale or hierarchy of guilt and a corresponding scale of punishments. The fact of constraints going in both directions is important to our model and will be discussed further in future posts.
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