REASONING

The reasoning of traditional syllogistic logic is built upon the representation of truths in the form of propositions or premises (or sometimes called sentences).

In pattern logic, a premise is composed as an occasion of predication (SEH\\) occurring between two quantified terms.  

The terms of a premise are modeled as any indeterminate (conceptual) entity, but most simply as the primitive occasion SEH--, or  "Anything".

The four main quantifiers of pattern logic and their mapping to usage in first order logic, mereology and syllogistic logic have been introduced elsewhere and they are: universal, existential, non and partial quantification. 

In the rivulet diagram above we see a labeling of a universal premise which is also known as an A-premise.  The premise might commonly be rendered something like "all dogs are mammals" but a technical rendering in pattern logic would be more like "ALL dog IS-PRESENT-IN EXISTENT mammal".  

In this example, we see the universal quantification of "All dog" in purple and the existential quantification of "EXISTENT mammal" in orange.  The full premise is captured by the green predication at the lower center.  The additional two predication relations are requirements of the quantification of terms patterns.

The formalization of quantifiers in pattern suggest a method of expressing logical premises that expands the count of recognized premise forms within the system as shown below.

The red boxes in the diagrams above simply highlight the A, E, I and O premises from the classical syllogistic system but these are augmented with the black boxes representing the additional premises of the pattern logic syllogistic. One of the 4 types of major premise in the left-most box supplies a middle term that is universally- or non- quantified and then paired with (by following the arrows) a similarly-quantified middle term in the minor premise to produce (following the arrows a second time) one of the 16 conclusion premises. 

For example, we can start with an A-premise (second box down in the Major Premise grouping at the left of the Deduction diagram for universal middle term elimination) like “ALL mammal IS-PRESENT-IN EXISTENT animal” and follow the dashed line to a B-premise (top box in the Minor Premise Grouping at the center) like “All dog IS-PRESENT-IN ALL mammal” and then continue to follow the dashed line to arrive at a Necessary Conclusion of an A-premise “All dog IS-PRESENT-IN EXISTENT animal”.  This would be an example of how elimination is performed for a Barbara Syllogism (two universal A-premises lead to a universal A-premise conclusion).

The two deduction diagrams are exhaustive of the eliminations that result in necessary conclusions in the pattern logic syllogistic.

But if we consider a similar set of patterns which have an Existentially or Partially Quantified Middle term we can produce possible conclusions. These are not arbitrary conclusions: they are derived systematically from the same knowledge base as the deductive reasoning inferences. Therefore these represent a subset of all of the premises that could exists: they are the reasonable hypotheses of abductive inference.

As an example of abduction, if we alternatively started with a knowledge base containing a major P-premise (second box down in the Major Premise grouping at the left of the abduction diagram for existential middle term elimination) like “EXISTENT mammal IS-PRESENT-IN EXISTENT animal” and follow the dashed line to an A-premise (top box in the Minor Premise Grouping at the center) like “All dog IS-PRESENT-IN EXISTENT mammal” and then continue to follow the dashed line to arrive at a Possible Conclusion of an A-premise “All dog IS-PRESENT-IN EXISTENT animal”.

Now we have arrived at the same A-premise but our certainty about this premise is weaker with this abductive inference than it was with the deductive inference. 

This is a good hypothesis though, and it is useful in so far as it could inspire a search for evidence that would move it from a possible conclusion to a necessary conclusion. And we know exactly what the evidence we need is if we look back to the middle term. We know that there exists a mammal that is a dog. Can we assert that all dogs are mammals or are there dogs that are not mammals? We know that there exists a mammal that is an existent animal. Can we assert that all mammals are an existent animal? If so, then we could deduce, as above, that every dog must be an existent animal.

The example here is quite trivial to a human being familiar with the taxonomy of animals but the point is that even the more convoluted formulations of logical facts as premises are computable with appropriate certainty and speculation through these pattern logic methods of deductive and abductive reasoning. And a computer using these methods can evaluate knowledge bases in which a human’s ability to extract necessary truths, let alone reasonable hypotheses, becomes quite limited.  Think of medical decision making, drug discovery for treatment of diseases or sifting through military intelligence with varying degrees of certainty about the facts.  Automated reasoning via abduction helps to identify the right questions to ask to arrive at greater certainty.

Note that a “valid syllogism” in the classical syllogistic system must conclude with one of the four premises of the system: an A, E, I or O premise. The pattern logic syllogistic would consider any of the sixteen premises that can be concluded by elimination of a universal or non- quantified middle terms as being “valid” but these would not be recognizable in the classical system. 

In the proof path diagram above we see the results of applying transformation and elimination rules to the known classical syllogisms to prove their validity. Those with a dashed line are the “conditionally valid” syllogisms. Pattern logic explains this issue of “conditionally valid” as any proof path that involves a step that applies the Universal Subject Inclusion (USI) rule.

Interestingly, the Bocardo syllogism (OAO-3 at the upper right of the diagram) does not find proof of validity in pattern logic but there is an easily-confused syllogism that does. The classical O premise can be rendered in English as “Some S is not P”. In pattern logic it is rendered more precisely as “SOME S IS-PRESENT-IN NO/NOT P”. The O premise, as can be seen in the Transformation Grid already presented, cannot undergo any logical transformations: it is essentially a dead-end. However the N-premise which is rendered “EXISTENT S IS-PRESENT-IN NO/NOT P” reads very near the O-premise and without a clear articulation of the difference between existential and partial quantification, the N- and O-premises could easily be confused. The N-premise is able to participate in a possible elimination as a major premise paired with an I-premise and the result in an O-conclusion. Then, the converted A minor premise of the NAO-3 "Bocardo-like" syllogism can be transformed into an I premise through serial application of the rules of Universal Subject Inclusion (USI), Existential Subject Aversion (ESA) and Convertive Aversion & Reversion CAR). 

In summary, an understandable mis-reading of a Bocardo premise (N major premise instead of an O major premise) can be shown to be at least possibly truth preserving. 

Aside from this wrinkle with Bocardo, the validity of the other 23 classically valid syllogisms is affirmed by pattern logic. Inspection of the means of proof in pattern logic also demonstrates that there are no other starting points that would result in an A, E, I or O premise conclusion so the classical investigation seems to have correctly exhausted the search for validity within its defined parameters.

However, if we were to “lower the bar” of validity and use abductive reasoning to identify syllogisms that would conclude on possible A, E, I, or O premises, the Abductive Reasoning diagram tells use exactly what sort of syllogisms to look for:

PAA-1, IDA-1, NAE-1, ODE-1, PII-1, IJI-1, NIO-1, OJO-1

Setting terms to a PAA-1 possibly valid syllogism we might imagine an example syllogism like:

Major P-premise: EXISTENT ‘individual in a blue shirt’ IS-PRESENT-IN EXISTENT ‘individual I met after the game’

Minor A-premise: All ‘winners of the game’ IS-PRESENT-IN EXISTENT ‘individual in a blue shirt’

Concluding A-premise: All ‘winners of the game’ IS-PRESENT-IN EXISTENT ‘individual I met after the game'

This example syllogism imagines a scenario in which you have knowledge of some game: that the victor(s) of the contest wore (a) blue shirt(s). Then some time after the game you meet an individual wearing a blue shirt. Have you met a/the game winner? It’s possible. Depends on which blue-shirted individual we are talking about? Is it the same shade of “blue”? It also depends on how many winners of the game “all” refers to: is it one person or a team of blue-shirted players? Maybe you did meet one member of the winning team but not all of them. Regardless, this sort of knowledge might make you bold enough to inquire of anyone who you talk to that is wearing a blue shirt whether they are a winner. 

Abductively speaking, it is a reasonable hypothesis to assume that anyone you talk to after the game who is wearing a blue shirt could be a winner. These additional syllogisms provide us a formal basis upon which we can know the right sort of hypotheses to entertain given what we know already. And given this knowledge, you would similarly not ask this sort of question about "being a winner" of someone wearing a red shirt because abductive reasoning would not have suggested this possibility.