Conjunctive and Disjunctive Predicates
Something we have not directly dealt with before is how to handle something like this:
Every dog is both a mammal and a vertebrate.
In this we are predicating a conjunction of attributes (mammal and vertebrate) to every dog. It turns out that conjunctive predicates are easily handled in Sommers notation. We could translate the above proposition along the following lines:
Recognizing this allows us to identify some obvious inference rules for conjunctive predicates. For instance, we can obviously have a Rule of Conjunctive Predicate Simplification (CPSimp):
±X+(+Y+Z) ∴±X+Y or ±X+Z
And associative shift works here, as well; from
-D+(+M+V) = Every dog is a mammal and a vertebrate (or, equally, Every dog is a mammal that is a vertebrate)
we can conclude
-(+D+M)+V = Every dog that is a mammal is a vertebrate
We could have more if we choose to elaborate them. But I want to ask the next major question, which is how to handle disjunctive predicates, as in the following:
Every dog is either a mammal or a vertebrate.
We can think this through and use what we know of how to handle conjunctive predicates to handle this new kind of predicate. If every dog is either a mammal or vertebrate, what is definitely ruled out is the case where all the dogs are both non-mammals and non-vertebrates. Thus we can take disjunctive predicates to deny this conjunctive predicate:
We could develop rules for handling disjunctive predicates, as well; but we will not do so here.
In propositional logic every use of a variable technically requires a domain of discourse (often called a universe of discourse). In Sommers notation, as with any term logic, domain of discourse is much less important, but it still can be defined for every sentence, and usefully so, because every statement is true if and only if it denotes its domain of discourse. In Sommers notation every term has a domain (or, if you prefer, category), and the domain of discourse for any sentence is the intersection of the domains (or, if you prefer, categories) of all its terms.
For convenience, I will symbolize the domain itself (or, if you prefer, category) with a bolded term, like D, and indicate what falls within that domain with /D/. This latter is what I am calling 'category nominalization' or 'domain nominalization'. /D/ consists of everything that is D or nonD, where D and nonD are both taken to belong to the domain or universe indicated by D. Thus if we take the ordinary term 'red', then red is the domain that consists of red and nonred things (we could just as easily call it nonred or red/nonred), i.e., which consists of all and only those things which could intelligibly (even if wrongly) be characterized by ‘red’ and ‘nonred’; thus blue things might fall under this category, but not the number two or the pain in my left hand, because these are a different category (we can meaningfully say of them that they are not the sort of thing that could be either red or nonred). Similarly, /red/ indicates the actual members of this domain. We would use red to talk about the domain of red and nonred things itself; we would use /red/ to talk about what is included or excluded from it. Now, we can use this to handle a particular type of proposition:
Everything is P
Something is P.
In these the subject is the nominalized domain, so they are respectively translated as:
Or in other words, every member of the category or domain P is P; some member of the category or domain P is P.
It is by combining domains with conjunctive and disjunctive predicates that we are able to handle noncategorical logic: propositional logic derives from an analysis of predication. To see this we need to learn how to nominalize propositions and therefore use whole propositions as terms.
In order to get to noncategorical propositions, the question we want to ask ourselves is: How can we handle propositions about propositions? Consider the sentence we noted above:
Some girls who think that all love is easy are unhappy.
If we use [p] to indicate the proposition, "All love is easy" (or perhaps more accurately, to indicate what we mean when we say, "(such) that all love is easy"), we get the following rendering:
But because [p] is a complex term, we can treat it as one, keeping it in square brackets to indicate that it is a nominalized proposition.
In nominalizing, we have embedded one sentence in another by treating it as a term. This allows us to do a number of things.