Now we get to the good stuff: actual inferences.
Loosely following Aristotle, we will call a perfect inference any inference that does not require a proof of its validity (because it is clearly valid); one in need of a proof of validity is imperfect. All basic rules of inference are patterns for perfect inferences. For our purposes we will not count as perfect inferences anything with more than two premises, however obviously valid it may be. Thus there are three types of perfect inference: two-premise perfect inferences, one-premise perfect inferences, and zero-premise perfect inferences. It may seem odd to talk about a 'zero-premise' inference; but what is meant is that you don't need a premise to introduce it. Thus all tautologies are zero-premise inferences; they can be introduced whenever you please.
We know that there are two necessarily contradictory types of proposition:
All S aren't S.
Some S aren't S.
Denying these, we get:
All S are S.
Some S are S.
These are tautologies; as will be "All S that are M are S" and "Some S that are M are S". From this we can formulate our first rule of inference, the Rule of Identity (Id):
∴ ±(±S±M)+ (±S)
where M may be empty (i.e., discardable). (In this rule, as in all the others we will use, ± means 'either plus or minus', and the assumption is that you will be consistent in which you choose.) Basically, the rule is that we may introduce a universal or particular tautology whenever we wish, without regard for prior premises. You may add –S+S or +S+S whenever you require. This will be the only zero-premise perfect inference we require.
A single-premise perfect inference is also called an immediate inference. The most obvious rule for such inferences is the Rule of Self Inference (SI):
±S±P ∴ ±S±P
Another rule of immediate inference has to do with universal affirmations; it is the Rule of Contraposition (Contrap), which I will break into two parts purely for the sake of clarity:
-(+S)+(+P) ∴ -(-P)+(-S)
-(-S)+(-P) ∴ -(+P)+(+S)
Basically this says that you can infer "All nonP is nonS" from "All S is P", and vice versa.
The third rule of inference has to do with particular affirmations; it is the Rule of Conversion (Con):
+(±S)+(±P) ∴ +(±P)+(±S)
That is,if you have +S+P, you can always conclude +P+S, and if you have –S-P, you can always conclude –P-S So we have rules governing affirmations. But we need rules for doing things with denials. The easiest way to do this is to formulate a rule for converting denials into affirmations; then the rules we have already will apply for all our needs. The Aristotelian claim that any denial is equivalent to an affirmation is very plausible. So we can formulate a Rule of Internal Negation Distribution (IN), which again I will split into two parts for clarity:
±S-(+P) ∴ ±S+(-P)
±S-(-P) ∴ ±S+(+P)
Related to this, we have a Rule of Double Negation (DN) to simplify and expand expressions; any (+T) term can expand to (-(-T)) and any (-(-T)) can simplify to (+T).
A useful additional rule is the Rule of Commutation (Com), which basically says that for any terms we can rewrite (+P+Q) as (+Q+P). If we want to, we can throw in a Rule of Association (Assoc) to handle groupings:
+S+(+Q+P) ∴ +(+S+Q)+P
It's also possible to have a Rule of Relational Simplification (Simp), to handle inferences like "John smiles at Mary; therefore John smiles." Thus we could things like:
±S+(+Q+P) ∴ ±S+Q
One last, and very useful, rule of immediate inference is the Rule of Composition (Comp):
-S+P ∴ -(+S+Q)+(+P+Q)
where Q is an adjective term (whether monadic or relational doesn't matter). This allows you to handle inferences like "All circles are figures; therefore all drawers of circles are drawers of figures" and "All men are mortal; therefore all rational men are rational mortals."
With immediate inferences in hand we are ready to tackle syllogistic, which in term logic covers all mediate inferences (two premises or more). To do this we need to add just one basic rule, the extremely important dictum de omni et nullo (DDO), which can be put in the same form as our other rules to constitute a Rule of Mediate Inference (MI):
-M+P, ±S+M ∴ ±S+P
The basic point is this: whatever is affirmed of all of something is likewise affirmed of what that something is affirmed of.
Given this, we are doing swell. We can show that the syllogism Darapti is valid by the following proof:
1. -M+P premise
2. -M+S premise
3. +M+M Id
4. +M+P MI from 1,3
5. +P+M Conv, from 4
6. +P+S MI from 2,5
7. +S+P Conv from 6
Thus every Darapti syllogism, i.e., every mediate inference of the form "All M is P; All M is S; therefore some S is P" is valid.
It's easy enough to identify the necessary and sufficient conditions for validity in Sommers notation. They are:
(1) A universal conclusion is only validly drawn from universal premises.
(2) A particular conclusion is only validly drawn from a set of premises that contains one and only one particular premise.
(3) The premises in a valid inference sum up to the conclusion.
So this is invalid:
(2) is irrelevant; it meets (1), but the sum of the premises is -M+P+S. This is also invalid:
(1) is irrelevant; it meets (3), but it tries to draw a particular conclusion from two particular conclusions.
Consider the following inference:
Some atheist mocks every prayer.
Every Shiite recites a prayer.
∴ Every Shiite recites what some atheist mocks.
In Sommers notation the premises will be:
The middle term here is P2. Thus (pretty much ignoring the parentheses, which I only put in here to make it easier to translate without accidental shift of meaning) by MI we can conclude:
This is clearly a valid syllogism; the first condition for validity is irrelevant, and it meets both the second and the third. And taking this conclusion, by Com we get:
Relational Simplification Arguments
Consider the following argument: John smiles at Laura; therefore, John smiles. This is easily handled by Sommers:
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
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 Sommers notation 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.
Consider the argument, "Paris loves Helen; therefore Helen is loved by Paris." In Sommers notation:
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
So far we have considered only categorical propositions. Sommers notation can also handle noncategorical propositions. Before we get to them, however, it will be helpful to look at two other features of Sommers notation: its handling of conjunctive and disjunctive predicates and its handling of domains of discourse.