Ramon Llull

Ramon Llull

Ramon Llull, born in Majorca around 1232 or 1233 and conventionally dated to 1316 at his death, developed an evolving system he called the Art. His philosophical and theological purposes were central to it. His autobiographical tradition must be read as a later account of his life, not a transparent record of every reported event. The Stanford Encyclopedia discusses the evidence and project.

Llull used letters, diagrams, tables, and rotating figures to organize combinations of concepts and guide arguments. This is a useful subject for studying representation and search. It should not be described as a brass computer that automatically manufactured every truth or as an ancestor of every modern AI system.

What the figures actually organize

The Art changed over time. In its later ternary phase, nine letters, B,C,D,E,F,G,H,I,K, can denote different series of concepts. For example, the first series includes goodness, greatness, eternity or duration, power, wisdom, will, virtue, truth, and glory. Context determines which sense a letter carries. The same marks are not nine unambiguous Boolean variables. University of Barcelona explanation of the principles.

The Ars brevis, written in 1308, presents four figures. Its Fourth Figure uses a fixed outer circle and two smaller rotating circles. Calling this “Figure S” mixes different parts or phases of the Art. The University of Barcelona's specific account of the Ars brevis identifies the figures and explains their use.

Producing a combination is only one stage. Definitions, rules, questions, and interpretation supply its argumentative role. The system's theological premises are not neutral facts that all possible readers must accept. A modern comparison with search must preserve that distinction.

Worked mechanism: count the space before searching it

This is an original combinatorial exercise inspired by the nine-letter alphabet, not a complete simulation of Llull's Art. Suppose we want triples of distinct letters and regard order as irrelevant. There are 9*8*7 ordered selections, but each unordered triple appears in 3*2*1 orders. Thus there are:

9*8*7 / 6 = 84

Examples include {B,C,D} and {B,C,E}. A simple enumeration visits indices i<j<k. That condition prevents repetitions and ensures each unordered triple appears once. Given any three distinct indices, sorting them gives exactly one increasing triple, proving completeness and uniqueness.

Change the assumptions and the answer changes. If order matters but repeated letters remain forbidden, the count is 504. If order matters and repetition is permitted, it is 9³=729. A rotating display's raw alignments are not automatically identical to any one of these spaces; one must state which alignments are admissible and how they are interpreted.

The historical table discussed by the Barcelona center has 84 columns with 20 components each, giving 1680 components. That is not the same assertion as “three nine-position wheels have only 84 possible states.” Counting symbols and counting interpreted entries are different tasks.

Generation is not verification

To make the distinction precise, consider a separate modern propositional example. Let A, B, and C be Boolean statements. Suppose the premises are A implies B and B implies C. Does A imply C follow?

There are eight truth assignments for three Boolean variables. A verifier can enumerate them and reject assignments that violate either premise. For any remaining assignment, if A is true, the first premise forces B, and the second forces C. Thus A implies C holds in every model of the premises. This is a proof of entailment.

By contrast, C implies A does not follow. The assignment A=false, B=false, C=true satisfies both premises but falsifies that conclusion. One countermodel is enough. Generating the sentence C implies A would not make it true.

This example has explicitly defined semantics and a finite search space. Llull's interpretation of concepts and theological argument should not silently be replaced with this truth-table mechanism. The comparison is useful precisely because it exposes what a verification procedure must specify: syntax, meanings, premises, and a criterion for validity.

A careful connection to computer science

External symbols and mechanical aids can reduce the burden of enumerating possibilities. They do not eliminate the need to justify assumptions or interpret results. An exhaustive search is exhaustive only relative to a specified space. Searching all triples from nine symbols does not search every possible proposition, and validating a consequence of chosen premises does not establish the premises themselves.

Later thinkers engaged with Llull, but specific influence claims require specific documents. It is unnecessary to claim that Napier's rods descend from Llull's wheels or that he first asked whether machines think. His own project is substantial enough to study on its actual terms.

Exercises

1. Enumerate all unordered triples of distinct symbols from {B,C,D,E,F}. Prove your list is complete and contains no duplicates.
2. Count ordered triples from those five symbols when repetition is permitted, and when all entries must be distinct. Explain why a table with twenty interpretations per unordered triple is a third quantity.
3. Given premises A implies B and B implies C, assess both not C implies not A and C implies A. Provide a proof or countermodel for each, then explain why producing candidate statements is insufficient for verification.

Compare your reasoning with the worked solutions

1. Enumerate with an ordering rule

The triples are BCD, BCE, BCF, BDE, BDF, BEF, CDE, CDF, CEF, DEF. There are 5*4*3/6=10. Every three-element subset has exactly one alphabetical ordering, and the list visits all increasing choices. Therefore no subset is missing or counted twice.

2. Keep the sample spaces separate

Ordered triples with repetition number 5³=125. With distinct entries they number 5*4*3=60. The ten unordered triples with twenty interpretations each give 10*20=200 interpreted entries. An interpretation label is additional structure, so 200 does not contradict either raw triple count. The question being counted has changed.

3. Verify rather than merely generate

If C is false and A were true, the premises would force B and then C, a contradiction. Therefore not C implies not A follows. But A=false, B=false, C=true satisfies both premises and falsifies C implies A, so that conclusion fails. Enumeration can propose both sentences. Only applying the semantics and premises distinguishes a valid consequence from an invalid one.

Sources

• Ramon Llull, Stanford Encyclopedia of Philosophy: life evidence, doctrinal aims, phases, and the interpretive work required by the Art.
• The principles of the Art, Centre de Documentació Ramon Llull, Universitat de Barcelona: alphabet, figures, and the 84-by-20 table.
• Ars brevis, the same academic center: dated work description and correct Fourth Figure identification. These are scholarly explanations with illustrated historical material, not a complete primary-text edition. The combinatorial proofs and Boolean example above are original teaching constructions.