Wednesday, March 03, 2010

Inference Rules for Complex Predicates

By a complex predicate I mean a predicate such as one has in the following:

Every student in the class is clever or married.

Tom is both a mayor and a father.

Such complex predicates in effect break up into two types: conjunctive and disjunctive. In principle, it is possible to have conditional predicates as well; English is not very favorable to them, but they occasionally show up. They are a particular kind of disjunctive predicate.

I will use term-functor logic to characterize them. If I have two terms in my conjunctive predicate, +P and +Q, then I would represent the conjunctive predicate as +<+P+Q>, as in:

-S+<+P+Q>

which says, "All S is P and Q". Conjunctive predicates are commutative, as disjunctive predicates will also be; it makes no difference whether you say +<+P+Q> or +<+Q+P>. Disjunctive predicates are negations of conjunctive predicates; we rule out the one case they disallow. So if I have two terms in my disjunctive predicate, +P and +Q, I will rule out the one case where -P and -Q both occur; the predicate will be -<-P-Q>, as in:

-S-<-P-Q>

which says, "All S is P or Q". Note that -S-<-P-Q> is not equivalent to -S+<+P+Q>, but is a weaker claim. This is because of a key asymmetry between affirmative and negative quality. If there is only one predicate term, they are equally strong: the affirmative includes one thing and excludes one thing, and the negative does the same. You can go back and forth without much concern. But as the terms in the predicate increase, the affirmative excludes more and the negative excludes less. You can always move from a single conjunctive predicate to a single disjunctive predicate; you can never do the reverse. This gives us our first rule of immediate inference for complex predicates:

Conjunctive Predicate Weakening: From any proposition with a conjunctive predicate we can conclude a proposition with the disjunctive predicate that preserves the signs for the terms. For instance, from -S+<+P+Q>, "All S is P and Q," we can conclude -S-<-P-Q>, "All S is P or Q".


But conjunctive predicates and disjunctive predicates still are related by negation; the +<+P+Q> predicate, for instance, is equivalent to the --<+P+Q>. That is, we could have 'De Morgan rules'. There is however no reason to formulate them. De Morgan rules for complex predicates are just the ordinary logical rule of obversion; there is nothing special about it because it makes no logical difference to obversion whether the predicate is complex or not.

We can also move from a conjunctive predicate to a non-conjunctive predicate immediately:

Conjunctive Predicate Simplification: From any proposition with a conjunctive predicate we can conclude a proposition affirmatively predicating one of the conjunct terms of the subject. For instance, from +S+<+P-Q>, "Some S is P and non-Q," we can conclude +S+P, "Some S is P," or +S+(-Q), "Some S is non-Q".

More interesting things happen when we mix more than one premise where there are complex predicates. Here is an elimination rule for disjunctive predicates:

Disjunctive Predicate Simplification: From any universal proposition with a disjunctive predicate we can conclude a proposition affirming one of the disjunct terms of the subject if we combine it with a universal proposition in which the other disjunct term is denied of the same subject. For instance, from -S-<-P-Q>, "Every S is P or Q," and -S-P, "No S is P," we can conclude -S+Q, "Every S is Q".

Note that this is not available for particular propositions; we need to be sure that the predicates are said of the same subject, and particular propositions do not guarantee that. To put this in more technical terms: this rule is an application of the dictum de nullo. Here is an introduction rule for conjunctive predicates:

Conjunctive Predicate Strengthening: From two universal propositions, both of which affirm predicates of a subject, we can conclude a proposition affirming the conjunction of those predicates of the subject. For instance, from -S+P, "All S is P," and -S+Q, "All S is Q," we can conclude -S+<+P+Q>, "All S is P and Q".

As with the previous rule, the quantities must be universal; in technical terms, this rule is a particular application of the dictum de omni, and thus we must be sure we are talking about all of a subject. And finally we want to be able to throw something out if our complex predicate is inconsistent:

Complex Predicate Absurdity: From a universal proposition in which a contradictory predicate is affirmed of a subject we can conclude that the subject does not exist. For instance, from -S+<+P-P>, "Every S is both P and non-P," we can conclude -S, "There is no S".

So let's take an example. Suppose you have the following:

(1) All motion not from another would have to be either per se or per accidens.
(2) No motion not from another could be per se.
(3) All motion per accidens is either violent or natural.
(4) No motion not from another could be violent.
(5) All natural motion is either in some way self-moving or in no way self-moving.
(6) No motion not from another could be in no way self-moving.
(7) No motion that is in some way self-moving is motion not from another.

Putting this in term-functor form we get:

(1) -NA-<-PS-PA>
(2) -NA-PS
(3) -PA-<-V-N>
(4) -NA-V
(5) -N-<-SM-(-SM)>
(6) -NA-(-SM)
(7) -SM-NA

From (1) and (2) we get -NA+PA by Disjunctive Predicate Simplification. From this, (3), and (4) we can get -NA+N by the same rule plus the dictum de omni; another application of the dictum de omni gets us -NA-<-SM-(-SM)>. Combining this with (6) gives us -NA+SM; combining it with (7) gets us -NA+(-SM). Both of those are by Disjunctive Predicate Simplification. Conjunctive Predicate Strengthening lets us put these together to get -NA+<+SM-SM>. Using Complex Predicate Absurdity we draw the only possible conclusion: -NA, which says that there is no such thing as motion that is not from another.

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