In Part I, I noted the basics of SETL, in a rough way.
In Part II and Part III, I discussed briefly some special cases and how SETL handles them.
In Part IV, I discussed some basics of argument using SETL.
Up to this point we have largely seen how SETL and modern predicate logic are equivalent (or, if Purdy is right, we have seen how SETL and modern predicate logic are equivalent except for cases of identity between variables; see Part II for my discussion of this). What I'd like to do here, following some of the work by Sommers, is to look at a few very simple sorts of argument that shed light on the differences between SETL and modern predicate logic. It will be seen that SETL has a few advantages, at least minor advantages, that modern predicate logic does not. (As noted before, sources and supplementary readings can be found in a post to come.)
General Points
First, it might be useful to note the obvious differences between the way we handle propositions in predicate logic and the way we do so in SETL. In predicate logic, the following are true:
(1) The subject is always singular.
(2) The predicate is always general.
(3) Denial of the predicate is treated as being the same as propositional negation.
(4) Affirmations are treated as logically simple, while denials are treated as logically complex.
(5) Every term is treated as if it were positive.
None of these are constraints on SETL. In SETL, both subjects and predicates can be singular or general; denial of the predicate is distinguished from propositional negation; affirmations and denials can be treated as being at the same logical level; terms may be positive (red) or negative (unred) and this can be portrayed in the logical formulation itself. What this, taken altogether, means is that SETL can take as simple things that predicate logic has to take as secretly complex. For instance:
All dogs are canines.
In SETL this is the simple proposition, -D+C. In predicate logic, we have to translate it as (x)(Dx → Cx). Or to give it in a rough, regimented English version, "For anything: if it is a dog, it is a canine." This is not simple at all; we have two predicates (where SETL has only one), an implication (which SETL doesn't need), and a variable (which SETL also doesn't need). The standard response is that the sentence, "All dogs are canines" has a 'logical form' that is more complex than we might think from the natural language version. And, indeed, this is true if we are putting things into predicate logic. But we don't have to do so; and when we use SETL it doesn't have a secretly complex logical form at all.
Simplification Arguments
Consider the following argument: John smiles at Laura; therefore, John smiles. This is easily handled by SETL (if we formulate a rule of association and a rule of simplification):
1. ±J+(S±L) premise
2. (±J+S)±L Assoc from 1
3. ±J+S Simp from 2
Similarly, for "John smiles at some person; therefore, John smiles":
1. ±J+(S+P) premise
2. (±J+S)+P Assoc from 1
3. ±J+S Simp from 2
Sommers notes, with regard to arguments of this last type, that they baffle attempts to formulate them in modern predicate logic. You can represent (1) as (∃x)(Sjx); but there seems to be no way to represent (3) in a different form. Of course, this is not a problem for modern predicate logic, since one could just say that this shows why the inference is a good one -- the 'logical form' of (1) and (3) are the same. But when we have a singular direct object, as with the first, it isn't quite clear -- although probably still could be argued -- that modern predicate logic handles it properly. The natural way to handle (1) would be to translate it as (Sjl); we can then get the conclusion, (3) by existential generalization, as (∃x)(Sjx). But this means that the first argument above is a different argument entirely from the second, and one can argue (as Sommers does) that it would make sense if the two arguments above were treated exactly the same way. The reason modern predicate logic fails to do this is that it, unlike SETL, lacks a predicative functor that joins the subject and predicate; it assumes predication as a given.
Associative Shift
Consider the argument:
Someone Thomas studied was Aristotle.
Therefore, Thomas studied Aristotle.
Compare it to:
Someone Thomas studied was the disciple of Aristotle.
Therefore, Thomas studied the disciple of Aristotle.
These are easily handled in SETL by association. The first argument becomes:
1. (±T1+S12)±A2 premise
2. ±T1+(S12±A2) Assoc from 1
And the second:
1. (±T1+S12)+(D±A2) premise
2. ±T1+(S12+(D±A2)) Assoc from 1
The form of argument is very simple and straightforward in both cases. To handle either of these arguments, however, modern predicate logic has to appeal to identity and Leibniz's Law. So, for instance, it would handle the first argument in the following way:
1. (∃x)(Rwx & x=a) premise
2. Rwa Leibniz's Law from 1
With a slightly more complicated premise, you can do the same with the second argument. There is no doubt that it can handle this form of argument; but it has to do so in a roundabout way, whereas SETL handles it very straightforwardly.
Passive Transformation
Consider the argument, "Paris loves Helen; therefore Helen is loved by Paris." In SETL:
1. ±P1+(L12±H2) premise
2. (±P1+L12)±H2 Assoc from 1
3. ±H2+(±P1+L12) Comm from 2
4. ±H2+(L12±P1) Comm from 3
In predicate logic, however, we can't represent the argument at all unless we treat the premise has having the same logical form as the conclusion. Thus we are in the same situation for passive transformation as we were for simplification. This is not surprising; Frege, for instance, explicitly considers passive transformation in order to conclude that it is not significant for logical matters because the active and passive forms have the same truth conditions. Sommers replies to this that then we should regard the change from "Some A is B and C" to "Some C is B and A" as insignificant for logical purposes, because they have the same truth conditions; and that the more powerful concept-script is the one that brings the most transformations within the field of what is significant for logical purposes. And as we've done with active and passive voice, so can we do with dative movement (e.g., "Dave sold the truck to June; therefore Dave sold June the truck").
None of this is particularly fearsome for the die-hard fan of modern predicate logic. Indeed, the reason that SETL handles these arguments so easily is that it is much closer to natural language than predicate logic is, and, of course, they are quite ordinary arguments in natural language. But it should give one some pause, at least to this extent: there are common arguments that can be expressed easily in SETL (if we decide that they are important enough to be expressed in it) that are not obviously handled properly in predicate logic.
So that's a brief survey of some of the differences. The next post on this subject will look at how one might extend basic SETL into modality.
