The quantity may be either universal All or particular Some and the quality may be either affirmative or negative. These categorical propositions are denoted by a label, either A universal affirmative , E universal negative , I particular affirmative , or O particular negative. A categorical syllogism , then, is a sequence of three categorical propositions ordered in such a way that two propositions are premises and the last one is a conclusion. Within the premises there is a term that appears in both premises but not in the conclusion.
This particular term works as a link between the remaining terms and is known as the middle term, which we will denote with the term-schema M. According to this term we can set up four figures that encode and abbreviate all the valid and only the valid syllogisms Table 2. VENN Venn, is a sound and complete diagrammatic logical system Shin, that represents syllogistic perspicuously. VENN can be defined as a diagrammatic logical system with a well-defined vocabulary, syntax, and semantics Shin, Briefly, the vocabulary is determined by the next elements: the closed curve, the rectangle, the shading, the X, and the line Figure 8.
With this vocabulary a diagram in VENN is defined as any finite combination of diagrammatic elements Shin, A region is any enclosed area in a diagram. A basic region is a region enclosed by a rectangle or a closed curve. A minimal region is a region within which no other region is enclosed. Regions represent sets and the rectangle represents the domain. A shaded region represents an empty region and a region with an X represents a non-empty region. And with this syntax the categorical propositions can be represented as follows Figure 9 :.
Further, the semantics of VENN depends on a natural homomorphism with sets that help define the rules for developing diagrammatic proofs in this diagrammatic system Shin, 81 :. The rule of spreading of an X-sequence tells us that is obtained from if extra X-sequences have been added to some X-sequence of D. The rule of introduction of a basic region indicates that a basic region may be introduced by drawing either a rectangle or a closed curve. The rule of conflicting information says that if a diagram has a region with both a shading and an X-sequence, then we may transform the given diagram into any diagram.
If any region of D is shaded or has an X-sequence, then it has a counterpart in either D 1 or D 2 which is also shaded or has an X-sequence and conversely. The previous rules may be summarized into the rules of erasure , addition , and unification Nakatsu, As an example, consider a diagrammatic proof of a syllogism of the form Figure According to the previous rules we begin with an introduction of areas step 1 and then a unification is applied step 2.
After that, we apply an erasure of an X-sequence step 3 and then a spreading of an X-sequence step 4.
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Finally, by the erasure of a closed curve rule, we obtain a final diagram step 5. Since the conclusion got drawn by drawing down the premises, the inference is valid and corresponds to a Darii syllogism.
My first goal is to present LEIB by reconstructing its vocabulary, syntax, and semantics. From these diagrams we can infer that the vocabulary of LEIB has the following basic diagrammatic elements: the solid horizontal line and the dotted vertical line Figure With this vocabulary we can define the syntax of the wfds for LEIB. The solid horizontal lines stand for terms and the vertical lines stand for a relation between terms. Given two horizontal lines representing terms, one could be completely included in another; they could be completely disjoint; they could partially intersect each other; or one could be partially not included in another Figure The semantics of these wfds is straightforward: a diagram of a proposition A shows that all that is S is indexed in P , but not inversely.
Proposition E shows that no S is indexed in P and vice versa. Proposition I represents the fact that some S is indexed in some P , and vice versa. Proposition O states that some S is not indexed in all O Figure To exemplify how LEIB works let us represent a syllogism in a diagrammatic fashion. Figure 15 shows what a Barbara syllogism looks like:.
The construction of the previous syllogism is relatively simple. In order to do that we draw the proposition MAP Figure Hence, following the reconstruction, a Barbara syllogism would look like the following diagram:. I shall show what canonical syllogisms look like in LEIB , but before I do that I would like to focus on the most interesting aspect about LEIB : a linear time diagram-based algorithm of decision, call it A, that takes any syllogism as an input and provides a decision about the in validity of such syllogism by checking whether the wfd of the conclusion is automatically represented by representing the wfds of the premises otherwise, the syllogism is invalid Table 3.
In Figure 19 it is easy to see how the conclusion was automatically obtained by representing the premises, i. Finally, to provide a more comprehensive explanation of how LEIB works I would like to show a couple of examples of invalid syllogistic forms. LEIB shows that such syllogism is actually invalid because the conclusion is not a free ride, but an overdetermined alternative because the diagram of the conclusion is not even a wfd Figure After this exploration of LEIB as a logical system I would like to pursue my second goal by suggesting a proof for a set of propositions that cover attributes of soundness, completeness, decidability, autarchy, and some sort of equivalence with VENN as well as a subset of BOOL Boole, In Prior Analytics A.
The idea in this proposition is that all valid syllogisms can be reduced, by following some precise instructions like the ones depicted in Table 4 , to other valid syllogisms. We can see that this idea holds in LEIB. Proposition 1.
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In Figure 25 we can see Cesare and Celarent preserve the same diagram by reflection w. Festino and Ferio preserve the same diagram by reflection w. Baroco is reduced to Barbara by applying a contradiction of the conclusion using it as a premise. Bocardo is reduced to Barbara by contradiction. Fresison is reduced to Ferio by reflection w. Proposition 2.
We prove this proposition by cases. Since there are four figures, we need to cover each valid syllogism from each figure, which is precisely what we have done in the previous section Figures 20 , 21 , 22 y We prove this by contradiction. Indeed, algorithm A is a mechanical and diagrammatic procedure that is sound and complete; but perhaps the importance of these results lies in their relation to two important properties: autarchy and some sort of equivalence with VENN using three sets or regions, call it VENN 3.
The autarchy of a diagrammatic system corresponds to a trade-off between free rides and overdetermined alternatives, that is, a compromise between valid and invalid diagrams i. Since LEIB is sound, complete, and decidable, it follows that:. Proposition 6. From right to left the proof is similar.
What this brief metalogical exploration shows is that LEIB is not only a logical system, but an actual bona fide diagrambased logic because it produces valid soundness and only valid inferences completeness by providing a time efficient O n mechanical method of decision decidability that helps the automation of perceptual inference autarchy , while preserving equivalence with VENN 3 w.
This fact is important because it means that LEIB does not allow us to infer particular propositions from universal propositions due to issues with empty classes: in fact if, for instance, we try to get a Barbari syllogism in LEIB we will find that a such task is impossible and any attempt to build a Barbari necessarily yields a Barbara syllogism, due to autarchy, and the same happens with other syllogisms that require existential import. In the opinion of Kneale and Kneale , Leibniz was by no means an Aristotelian purist, but he was committed to the assumption of existential import p.
Indeed, Leibniz thought that his diagrammatic system was capable of modeling and representing syllogisms with existential import such as Barbari , Cesaro , Fesapmo , Calemos , and so forth, which implies the possibility of deriving particular propositions from universal ones Phil. The previous situation leads to an interesting puzzle between consistency and eligibility: if Leibniz proposed his system only as a representation system, then it would be a trivial system because it would represent any syllogism, both valid and invalid; but it is quite clear that Leibniz himself argued against such a thesis.
Therefore, his system was proposed rather as a non-trivial reasoning system, but then, by Propositions , by being equivalent to VENN 3 and BOOL , it has a model in a modern interpretation of syllogistic that assumes empty terms. The puzzle is, thus: why would Leibniz argue that his system is capable of modeling such imperfect syllogisms implying the acceptance of empty terms when it actually does not implying the rejection of empty terms?
I believe the answer to this puzzle is beyond the logical scope, and so, we leave the question open. Finally, as part of our current and future work, I would like to add that I am developing similar metalogical reconstructions for other diagrammatic logical systems old and new, traditional and original in order to promote the study of mechanical and diagram-based reasoning as a research program with applications, mainly, in philosophy and Artificial Intelligence. And rewrite those insightful last words in order to restate: let us draw diagrams , without further ado, to see who is right!
I would like to thank the anonymous reviewers for their precise corrections and useful comments. I also thank Dr. Allwein, G. Logical reasoning with diagrams. New York: Oxford University Press. Allwein J. Barwise J.
Bellucci, F. Diagrammatic Autarchy: Linear diagrams in the 17th and 18th centuries.
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