## Why did the mid-19th century and earlier thinkers fixate on one-place predicates?

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A book I'm reading mentions the following:

A major barrier to the development of first-order logic had been the concentration on one-place predicates to the exclusion of many-place relational predicates. This fixation on one-place predicates had been nearly universal in logical systems from Aristotle up to and including Boole.

Why was that? Presumably, people weren't significantly dumber back then, and they could see that their everyday thinking and speaking involves relations.

I realize that "hindsight is 20/20", and "there is a first time for everything" (and for predicate logic, the first time happened to be in the 1860s). But still, "from Aristotle up to and including Boole" is a very long time keep missing an obvious blind spot.

3They were not "dumb" but they were "under the spell" of Aristotelian logic and metaphysics: it provides a comprehensive analysis of thought and language in terms of subject-predicate, i.e. objects and its properties, while "relation" emerged very very slowly: only Leibniz attempted (with no success) to analyze the world and the way we think at it in terms of relations. – Mauro ALLEGRANZA – 2020-09-16T09:30:36.180

"hindsight is 20/20" What do you mean with this? – Deschele Schilder – 2020-09-16T18:53:15.757

4@DescheleSchilder It's a common idiom; '20/20' refers to '20/20 vision' which is an American designation for high visual acuity. The meaning is that when we look at the past (hindsight) we can perceive things clearly, even though they might not have been obvious at the time. – dbmag9 – 2020-09-16T19:53:46.140

Now I know what the song "2020 Vision" means. I thought it had to do with the year we live in. But...the 20/20 in hindsight can be a false vision too. – Deschele Schilder – 2020-09-16T20:12:21.600

It's not just language and logic. For a realist, predicates represent properties. It's easy to understand properties as attributes of individual things, and this idea, too, is influenced Aristotelian ideas. Where is a property like shape or mass? It's in the thing that has the property. Relations, or relational properties, are more difficult to conceptualize. Where is the longer-than relation? It's not in either of the objects that are so related, and it's not floating in the air between them. See Russell's The Problems of Philosophy for a critique of earlier monadic-property views. – Mars – 2020-09-17T04:29:58.247

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Because there was a calculus for one-place predicates, Aristotle's syllogistic, roughly equivalent to monadic predicate calculus. Aristotle does discuss "relatives" in Categories, which refer to multi-place relations, or rather to objects entering them. What will later be called oblique syllogisms involving relatives is mentioned in passing in Topics. But the modern logic of relations (polyadic predicate calculus) is significantly more complicated than syllogistic, in particular, it is undecidable. A calculus for it was not worked out until de Morgan, Peirce and Frege in 1860-70s, and it required the transfer from Aristotle's term logic to propositional logic first, which was only made available by Boole two decades earlier. Ancient Stoic logic, which was propositional, did not deal with quantification and was largely lost during middle ages, although Leibniz showed interest in it. Traditional denying, after Aristotle, of ontological status to relations did not help developing a logic of them either.

It should also be noted that the translation of natural language into modern predicate calculus is generally considered artificial, see What are the advantages of Aristotle's term logic over predicate logic?, so it is disputable that "speaking involves relational predicates", at least if predicates are taken as functions on a domain of discourse as in the predicate calculus. So predicate calculus could not be read off of the natural reasoning like syllogistic could be, and alternative resources of natural language were, in fact, used for relational reasoning. Calculi for them have been developed more recently, see e.g. Englebretsen, Something to Reckon with, (and an intro on Siris), van Benthem, Natural Logic and Ben-Yami, Logic & Natural Language, ch. 6:

"In natural language, pluralities are introduced and specified by means of plural referring expressions; in the predicate calculus, a plurality, which is unspecified by the sentence, is introduced by presupposing a domain of discourse... In the predicate calculus, quantifiers specify how many particulars from a presupposed domain have a certain property; the quantifier in natural language, by contrast, specifies how many particulars of a plurality introduced by a general term have a certain property... The predicate calculus cannot even be seen as a simplified model of a fragment of natural language."

These devices have early traditional precursors, see Hodges, Traditional Logic, Modern Logic and Natural Language. E.g. Alexander of Aphrodisias and Ibn-Sina converted binary relational inferences into syllogisms by changing the domain of discourse to pairs. Other examples of "non-syllogistic inferences" were also discussed by Islamic scholars, scholastics, Leibniz and others, but only ad hoc, see Medieval Theories of Relations and Relational Syllogisms and the History of Arabic Logic. Ockham and Buridan present oblique syllogisms like "Every horse is black, you have a horse in the stable, so you have something black in the stable" as instances of applying dictum de omni et nullo. Wrote Leibniz in New Essays on Human Understanding:

"It should also be realized that there are valid non-syllogistic inferences which cannot be rigorously demonstrated in any syllogism unless the terms are changed a little, and this altering of the terms is the non-syllogistic inference. There are several of these, including arguments from the direct to the oblique — e.g. 'If Jesus Christ is God, then the mother of Jesus Christ is the mother of God'. And again, the argument-form which some good logicians have called relation-conversion, as illustrated by the inference: 'If David is the father of Solomon, then certainly Solomon is the son of David'".

Still, one would be surprised today by the unwavering faith of Ibn-Sina and many 16-17th century authors in the possibility of reducing Euclid's reasoning to syllogisms, De Risi, Leibniz on the Parallel Postulate and the Foundations of Geometry, 3.1 gives a nice review:

"Leibniz wanted to present all geometrical proofs (including those of axioms) as logical arguments in forma or chains of syllogisms (or other logical inferences) starting from the definitions. This kind of reduction of Euclid to syllogistic reasoning had already been attempted, it is true, in the past. The celebrated Analyseis Euclideae (1566) by the mathematician Christian Herlinus and his pupil Konrad Dasypodius had displayed in fact the first six books of the Elements as chains of syllogisms (or other propositional inference rules); and Clavius himself had quoted with praise the Analyseis in his commentary, reproducing Herlinus’ logical proof of Elements I, 1 and stating that a similar presentation of the subject could be attained for the whole of mathematics".

From the modern perspective, Euclidean geometry essentially involves multi-place relations like incidence, betweenness and congruence. In practice, "chains of syllogisms" had to be supplemented by inferences from diagrams to make up for the inevitable gaps. This was noticed by Kant, and led to his idea that geometric reasoning is "synthetic" rather than "analytic", i.e. not purely logical. Hindsight is indeed 20/20.

"Nevertheless, it is true that, even in his most daring geometrical constructions, Leibniz remained somehow entangled in the classical views, and he wavers between old and new concepts of geometry. Moreover, he saw the necessity of a logical treatment of relations, but lingered in an improved syllogistics that fell short of it. The most complete and historically accurate treatment of Leibniz’ theory of relations is Mugnai 1992."

2This is a good answer (and +1) but I disagree with "the translation of natural language into modern predicate calculus is generally considered artificial ... so it is disputable that "speaking involves relational predicates."" I think it's not at all disputable that amongst the things we do in natural language is talk about relational predicates, so I think the OP's claim "speaking involves relational predicates" is fine: it's a lower bound claim ("sometimes speaking involves relational predicates"), and the worse natural language is the stronger the distinction between it and monadic logic. – Noah Schweber – 2020-09-16T01:08:32.720

@NoahSchweber the worse natural language is the stronger the distinction between it and monadic logic Can you explain what you mean (perhaps as an answer)? Better speakers talk in unary predicates? – bobcat – 2020-09-16T02:55:53.640

@MaxB Sorry, I used "worse" in a somewhat sarcastic way: by "worse" I mean "more difficult to formalize" (so: arguably better, actually). My point was just that natural language certainly is at least as complicated as the basic logic of higher-arity relations, so the difficulty of formalization doesn't push back against your initial observation. – Noah Schweber – 2020-09-16T03:07:36.853

1@NoahSchweber I think it is fair to say that speaking involves relations, but the notion of predicate as a function on a domain of discourse is arguably artificial. It is not that formalizing natural language is more involved than just predicate calculus, but rather that it is a wrong kind of formalization. Ben-Yami (ch. 6) writes, for example:"if one wanted to develop an artificial language that could represent the semantics of natural language, one should depart from the predicate calculus to such an extent that the outcome could hardly be considered a modification of the latter". – Conifold – 2020-09-16T08:26:55.883

@Conifold OK, I changed that part back to the original "relations". – bobcat – 2020-09-16T15:39:35.490

1@MaxB My sense is that a more natural formalization of relational logic has a more complex calculus, so it makes sense that predicate calculus was worked out first. But it partly explains why people did not "see" it earlier as they did syllogistic and propositional calculus, even as they saw the use of relations. The hill was too steep to climb directly and required an abstract detour, for which mathematics developed only in 19th century. – Conifold – 2020-09-16T18:40:27.393

1Didn't Euclid live long before relational syllogism appeared in the spotlight? I think math has nothing to do with the question. With logic, yes. But nevertheless +1 – Deschele Schilder – 2020-09-16T18:57:46.033

3@DescheleSchilder That is right. What Euclid does instead, when he needs relational inferences, is appeal to the diagrams. This is almost explicit when he says that some point falls on a segment between the other two, for example, or lies on a particular line, or that two triangles are congruent. 16-17th century logicians were dissatisfied with such appeals and tried to eliminate them, but they did not have means at the time to do it fully. So, in a sense, diagrams were the first (specialized) "calculus" of relational logic – Conifold – 2020-09-16T19:10:03.150

1Incidence and betweenness went under the radar in Euclidean geometry until the Hilbert reaxiomatisation (I believe), and congruence is sort-of equality, but what about similarity (equality of ratios)? When Euclid in Book V starts writing A : B :: C : D (A is to B as C is to D) we have a four-place relation. How could that go unnoticed? Also, +1. – Lars H – 2020-09-17T10:11:59.293

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@LarsH Good point. Definition 3 of Book V explicitly says:"A ratio is a sort of relation in respect of size between two magnitudes of the same kind". Subsequent definitions lay out how to validate that one ratio is the same or greater than another, i.e. Euclid builds 4-place from 2-place relations. His reasoning about them can be split into syllogisms and applying common notions (like "if equals are added to equals, then the wholes are equal"). Kant will later say that algebraic reasoning too is "synthetic".

– Conifold – 2020-09-17T10:46:08.120

1@Conifold: Yes, but that is one rather soft definition, like Book I def. 2 "A line is breadthless length." The hard definition is def.5 "Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth, when, …" which does not refer to ratios as objects at all. (Also, fun fact: These definitions of Euclid do not require addition to be commutative — they work in any totally ordered semigroup. ) – Lars H – 2020-09-17T15:03:03.643

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An n-ary relation gives rise to parameterized unary predicates if one fixes n-1 arguments. Wilfrid Hodges argues that this is what logicians did before the nineteenth century. (There may be other works of his that better explain this.) More concretely, they would re-write the relevant statements by using natural language reasoning so that all relations are parametrized and that syllogistic calculus might be applicable, and then applied a step of syllogistic reasoning. One sometimes had to choose a different parameterization in the middle of a single proof. The paraphrasing process is not formal and may not be ideal from the modern point of view, of course.

Good point +1. Hodges in Traditional Logic, Modern Logic and Natural Language also mentions changing the domain of discourse between syllogistic steps as a way to handle relations. For example, Alexander of Aphrodisias and Ibn-Sina introduce a domain of pairs for binary relations, that way "A has the same parents as B; but B also has the same parents as C; therefore A has the same parents as C" can be reduced to Darapti. Ben-Yami and others name such "local formalizing" devices as a major difference between natural language and predicate calculus.

– Conifold – 2020-09-17T11:11:58.453

That's interesting. I wonder if alternating between fixing n-1 arguments and applying syllogisms (or any universally valid 1-place rules) is a complete proof method for FOL. – bobcat – 2020-09-17T15:45:20.360

@MaxB This is what Hodges calls "top-level processing", and it is too weak to capture all of relational reasoning. But look at van Benthem's reference I added, he discusses modern formalizations that extend dictum de omni et nullo used by scholastics to arbitrary depth. The result is not exactly FOL, but it might be as expressive. – Conifold – 2020-09-17T17:52:30.010