← left arrow
→ right arrow
• bullet
Arrows must link terms or bullets; they cannot stand on their own. The basic categorical propositions are as follows:
All A is B
A → B
No A is B
A → • ← B
Some A is B
A ← • → B
Some A is not B
A ← • → • ← B
In addition, for purposes of manipulation, these each has a reversed form that is equivalent to it, but switches the positions of the terms:
All A is B
B ← A
No A is B
B → • ← A
Some A is B
B ← • → A
Some A is not B
B → • ← • → A
Because I and E are symmetrical diagrams, allowing reversals directly gives us two rules of immediate inference:
(1) I-Conversion: From A ← • → B, you can conclude B ← • → A, and vice versa.
(2) E-Conversion: From A → • ← B, you can conclude B → • ← A, and vice versa.
For obvious reasons, we can likewise recognize,
(3) Identity: You may at any time add A → A
which is equivalent to "All A is A"; and we can also have,
(4) Subalternation: You may at any time add A ← • → A
which is equivalent to "Some A is A". Combining reversal with Subalternation lets us have conversion per accidens for A propositions.
When two propositions share terms at the extremes, they can be concatenated. So, for instance, A → B and B → C can be superposed at the 'B' in order to get A → B → C. And we can delete any term (but not a bullet) if it occurs between two arrows going the same way. This is enough to start getting syllogisms.
Barbara:
All B is C: B → C
All A is B: A → B
concatenate to get A → B → C
delete to get A → C, All A is C.
Celarent:
No B is C: B → • ← C
All A is B: A → B
concatenate to get A → B → • ← C
delete to get A → • ← C, No A is C.
Darii:
All B is C: B → C.
Some A is B: A ← • → B
concatenate to get A ← • → B → C
delete to get A ← • → C, Some A is C
Ferio:
No B is C: B → • ← C
Some A is B: A ← • → B
concatentate to get A ← • → B → • ← C
delete to get A ← • → • ← C, Some A is not C.
Thus all the First Figure syllogisms are simple cases of concatenation and deletion. For other figures we will sometimes need to use the equivalent reverses (once for Second Figure and Third Figure, twice for Fourth Figure); for all weakened syllogisms, we will need also to use our Subalternation rule. Subalternation effectively functions as a bullet-introduction rule.
The notation tracks distribution of terms. If a term is at the tail of an arrow, it is distributed; if it is at the head of an arrow, it is not. Thus if we look at one of the standard distribution rules for syllogisms, The middle term must be distributed at least once, we see immediately that a middle term allows concatenation; but it needs to have an arrow proceeding away from it if it is to be deleted and not show up in the conclusion. The second distribution rule, Terms distributed in the conclusion must be distributed in the premises, forbids just flipping arrows on their own. Pagnan notes that it's thus also possible to look at the syllogisms within the system by considering the facts that bullets cannot be deleted and that any bullets in the conclusion have to come from the premises. Then:
(a) You can only get S → P, which has no bullets, if there are no bullets in the premises; only the universal affirmative categorical proposition has no bullets. Therefore a universal affirmative conclusion requires universal affirmative premises.
(b) A universal negative conclusion, S → • ← P, is only possible if our premises have one bullet total and both arrows directed toward it. So one premise has to be universal affirmative, and the other has to be universal negative.
(c) A particular affirmative conclusion, S ← • → P, is only possible if our premises have one bullet total and both arrows directed away from it. So one premise has to be universal affirmative, and the other has to be particular affirmative.
(d) A particular negative conclusion, S ← • → • ← P, is a little trickier. But it will require two bullets that get us the arrows pointing the right alternating way. If each bullet in the conclusion comes from a different premise, one has to be universal negative and the other has to be particular affirmative to get two bullets and alternating arrows. If the two bullets come from one premise, that premise has to be particular negative (the only categorical proposition with two bullets) and the other premise has to be universal affirmative (which has none).
From (a), (b), (c), and (d) together we can see that every possible combination has one affirmative premise; none of the possibilities has two negative premises. Likewise, every possible combination has one universal proposition; none of them has two particular propositions. Every possibility with a negative premise has a negative conclusion. This is enough to get us the standard 'rules for syllogisms' in any of the usual forms that you find.
Pagnan also notes that you can build the square of opposition with what we have, if you add one additional consideration, namely,
(5) Noncontradiction: A ← • → • ← A may never be either a premise or a conclusion.
This, of course, reads as "Some A is not A". Any propositions that put together would yield a proposition of this form cannot be combined. So let's take
A → B
A ← • → • ← B
Concatenating gives us the contradiction (with B instead of A). The same will happen with
A → • ← B
A ← • → B
That's enough to get us a Boolean square of opposition; adding our Subalternation rule gets us the rest of the classical square of opposition.
Given that, we could also prove the validity of the Second, Third, and Fourth Figures by Aristotelian reduction to First Figure syllogisms. Take Datisi:
M → P
M ← • → S
Therefore S ← • → P
It's a Third Figure in which we can reverse the minor, concatenate, and delete; it is valid on the grounds we've noted. But, of course, by reversing the minor, we have turned it into a Darii syllogism. They're usually not that easy, of course. Let's take a Baroco syllogism:
P → M
S ← • → • ← M
Therefore, S ← • → • ← P
Baroco is converted to Barbara by contradiction, as the nasty little 'c' in its mnemonic tells us. Assume the contradictory of the conclusion, which would thus be S → P, since putting that with the actual conclusion would violate Noncontradiction. If you concatenate this with the major premise, P → M, we get S → P → M, which is equivalent to S → M, which is the contradictory of the minor premise (S ← • → • ← M), since if you put those together you violate Noncontradiction. So when we assume the opposite of the conclusion and use it for a Barbara syllogism, we get a conclusion that is inconsistent with the other premise; from which we can know that the conclusion does follow from the premises and Baroco is valid.
We have not considered two immediate inference rules, obversion and contraposition. Obversion of E and of O are extremely easy. "No S is P" is:
S → • ← P
The obverse is "All S is non-P"; but you can get this if you see • ← P as a negative and then read it all as one term. Using parentheses to make it a bit easier to see:
S → (• ← P)
Moving from "Some S is not P" to "Some S is non-P" works exactly the same way -- the obverses of the negative are already built in. Obversion of A and I require a new rule:
(6) Double Negation: → A and → • ← • ← A are equivalent.
Then we can see that "All S is P", S → P, is equivalent to S → (• ← • ← P), and that "Some S is P", S ← • → P is just like S ← • → (• ← • ← P).
Double Negation also allows us to do contraposition for A, since contraposition is the conversion of the obverse. S → P becomes S → • ← • ← P this is then reversed to get P → • → • ← S, "No non-P is S". With contraposition for O, we don't need Double Negation, we just reverse: "Some S are not P", S ← • → • ← P becomes P → • ← • → S, "Some nonP is S".
On this basis we can give translations for propositions with complemented terms:
All nonA is B
A → • → B
All nonA is nonB
A → • → • ← B
Some nonA is B
A → • ← • → B
Some nonA is nonB
A → • ← • → • ← B
No nonA is B
A → • → • ← B
No nonA is nonB
A → • → • ← • ← B
Some nonA is not B
A → • ← • → • ← B
Some nonA is not nonB
A → • ← • → • ← • ← B
If we wanted to, we could add parentheses to make the negated terms easier to pick out, but this wouldn't affect anything.
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Since we can do complemented terms, it also follows that we could use Pagnan notation to do propositional logic that can be simulated by syllogism (although we have to drop Subalternation). For instance, "All A is B" is like "If p, q", so the latter can be p → q, and then a hypothetical syllogism would work exactly like a Barbara syllogism. For something like modus ponens or modus tollens, we would need to allow bullets to be terminal; that is p → • is "It is not true that p" and • → p is "It is true that p". Given this, we can always turn p into • → p; that is to say,
Assertion: p and • → p are equivalent.
Then modus ponens is:
p → q
• → p
by concatenation and deletion we get • → q.
Disjunction 'p v q' would be p → • → • ← • ← q. Disjunctive syllogism would be
p → • → • ← • ← q
p → •
Reversing the first premise and concatenating, we get q → • → • ← • ← p → •
But by Double Negation, → • ← • ← p is equivalent to → p, so
q → • → p → •
But then p can be deleted to get
q → • → •
But then by Assertion we have, • → q → • → •
but then by double negation that is equivalent to • → q.
And so it goes. In any case, this is all just a side effect of the fact that syllogisms can simulate propositional logic; we could do the same with any other logical fragment that can be simulated by syllogisms, like basic mereology (A → B for "A is part of B" and A ← • → B for "A overlaps B"), or binary modal logic (e.g., we could take A ← • → B to mean "A is compossible with B"), or (as Pagnan does in one of his articles) rudimentary linear logic. Syllogistic is an extraordinarily powerful thing, so a notation that can handle syllogistic fairly easily can do a lot with only minor modifications.
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* Ruggero Pagnan, "A Diagrammatic Calculus of Syllogisms", Journal of Logic, Language, and Information, Vol. 21, No. 3 (Summer 2012), pp. 347-364; "Syllogisms in Rudimentary Linear Logic, Diagrammatically", Journal of Logic, Language, and Information, Vol. 22, No. 1 (Winter 2013), pp. 71-113; see also "Ologisms", Logical Methods in Computer Science, Volume 14, Issue 3 (August 31, 2018), arXiv:1701.05408.
