Monday, February 15, 2010

Sommers Notation, Part V

(Part IV)

Basics of Propositional Logic

Here I will diverge from both Sommers and Englebretsen a bit. In their scheme, we can represent conditionals (if p then q) as:

-[p]+[q]

Conjunctions are represented as:

+[p]+[q]

And disjunctions as:

--[p]--[q]

This works, and it ties in with the history of logic; for instance, the similarities between conditionals and A propositions have been recognized for centuries now.

However, I would suggest that it is a mistake to apply quantity directly to nominalized propositions; the minuses and pluses associated with them work much more like quality than quantity. We should perhaps see nominalized propositions as predicates that are predicated of the domain. Thus, “p and q” is predicated of some domain, e.g., T for the domain of truths:

±T+(+[p]+[q])

Then disjunction is related to conjunction as negative quality to affirmative quality.

±T-(-[p]-[q])

That is, “p or q” tells us that the one case that can’t occur is the one in which –p and –q are both true. And the material conditional, “if p then q,” is (as it is in other forms of logic) a kind of disjunction:

±T-(+[p]-[q])

All the standard rules of propositional logic, then, follow from rules for conjunctive and disjunctive predicates. For instance, conjunction simplification follows directly from CPSimp.

The subject term here does much the same work as an assertion sign here. We also see how judgment and quality are related, since they turn out to be the same kind of thing, just at different levels.
Likewise, we can handle metapropositions easily:

± T-[p] = [p] is false (not true) = It is not the case that [p]

And so forth.

Using propositional nominalization we can easily prove that modus tollens is valid:

1. ±T-(+[p]-[q]) premise
2. ±T-[q] premise
3. ±T-[p] MI from 1,2

And likewise with modus ponens:

1. ±T-(+[p]-[q]) premise
2. ±T+[p] premise
3. ±T+[q] MI from 1,2

And so with disjunctive syllogism:

1. ±T-(-[p]-[q]) premise
2. ±T-[p] premise
3. ±T+[q] MI from 1,2

We could continue indefinitely, but we won’t here. It would, however, be useful to have a rule linking conjunctive and disjunctive predicates to conjunctions and disjunctions of propositions. We can call it a Rule of Predicate Distribution; for conjunctive predicates it is (CPD):

-X+(+Y+Z) ∴±T+(+[-X+Y]+[-X+Z]

And for disjunctive predicates (DPD):

+X-(+Y+Z) ∴ ±T-(-[+X+Y]-[+X+Z]

We could derive these rules from what we already know about Sommers notation, plus some basic knowledge of what a domain is; but the point should be clear enough without the derivation.

Because the domain is always the subject, and because in general the domain has to be the same through an entire argument (otherwise one equivocates), we can usually just leave the domain implicit and focus on the predicates alone. It is interesting, however, to consider that by reasoning in terms of different domains one could probably also handle a number of paraconsistent and modal reasonings. We will not look into the question of paraconsistent reasoning, which is almost totally unexplored, but instead look at modal reasoning, which has been explored, but only partially.

(Part VI)

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