What is an example of a monadic predicate calculus argument that cannot be represented by the 19 classical Aristotelian syllogisms alone?


While reading Wikipedia's description of the monadic predicate calculus, I read the following:

Inferences in term logic can all be represented in the monadic predicate calculus.


Conversely, monadic predicate calculus is not significantly more expressive than term logic.


Reasoning about such statements can, however, still be handled within the framework of term logic, although not by the 19 classical Aristotelian syllogisms alone.

I am looking for an example of such monadic predicate calculus reasoning that can be handled within the framework of term logic but not by the 19 classical Aristotelian syllogisms alone.


Wikipedia, "Monadic predicate calculus" https://en.wikipedia.org/wiki/Monadic_predicate_calculus

Frank Hubeny

Posted 2018-09-26T21:03:18.457

Reputation: 18 742

1Wikipedia itself gives an example:"Every mammal is either a herbivore or a carnivore (or both)". This sentence is not in syllogistic form, so adding "This mammal is not a carnivore" to conclude "This mammal is an herbivore" is not, strictly speaking, a syllogism. But a disjunctive sentence can be easily converted into an equivalent conditional, e.g. "If a mammal is not an herbivore then it is a carnivore", which brings this inference into the syllogistic "framework". – Conifold – 2018-09-27T00:29:22.667

@Conifold Putting monadic predicate calculus arguments in syllogistic form is supposed to be possible. What isn't possible is using only the 19 classical Aristotelian syllogisms alone. Does this example use other syllogistic forms than these 19? I don't even know which syllogistic forms these 19 are. – Frank Hubeny – 2018-09-27T00:50:11.317

Syllogistic doesn't have propositional connectives or allow algebra with them. So putting monadic predicate calculus arguments in syllogistic form is probably what Wikipedia means by "not by the 19 classical Aristotelian syllogisms alone". The conversion can only be external. 24 figures are counted, but only 19 are "traditional". The missing ones they are like “Some S is not P” when the stronger “No S is P” obtains.

– Conifold – 2018-09-27T01:05:42.087

@FrankHubeny Wikipedia does refer to the "19 classical Aristotelian syllogisms". But there is no other reference to 19. I see one list of 11 syllogisms and another of 24. What are the 19 syllogisms? – Mark Andrews – 2018-09-27T01:22:26.837

@MarkAndrews - see LogicBlog : "Aristotle identified 16 valid forms of categorical syllogisms (though he formally acknowledged only the first three figures). Some thirteenth-century logicians such as William of Sherwood and Peter of Spain recognized nineteen valid forms." Aristitle in Prior An has only three figures and total of 14 moods : 4+4+6.

– Mauro ALLEGRANZA – 2018-09-27T06:59:33.717

A fourth figure was discussed in ancient times (Theophrastus) as well as during the Middle Ages. Fourth figure : 5 moods. Total : 14+5=19.

– Mauro ALLEGRANZA – 2018-09-27T06:59:37.237

Finally, adding the 5 "subalternate" moods, we get the new total of 24. – Mauro ALLEGRANZA – 2018-09-27T07:04:19.673

How about, "I think, therefore I exist?" 'I exist' contains no predication, and so cannot be admitted into Aristotle's syllogism. – transitionsynthesis – 2019-07-31T18:33:50.213

@transitionsynthesis If it contains no predication I don't see how it can be handled within the framework of term logic. – Frank Hubeny – 2019-07-31T20:21:31.953

@FrankHubeny Sorry, missed the bit about term logic. – transitionsynthesis – 2019-08-05T06:41:09.630



The classical syllogisms consist of propositions which use only simple (ie, one "variable") terms in the subject and predicate, so any argument containing propositions which use complex terms isn't a valid (classical) form, although it can still be handled within term logic by extended inference rules.

A complex term can be any Boolean expression. Being a term, it doesn't have a truth value, but the usual rules of simplification, combination etc, apply to such terms.

Boole was the first to show how such non-Aristotelian arguments could be formulated and "solved", but although his system was primarily a term logic, it was expressed in the language of mathematics (high school algebra). Towards the end of the 19th century John Neville Keynes (father of the famous economist), building on the work of Boole, Jevons, and Venn, developed a "syllogistic like" logical system which used the conventional Aristotelian propositional forms of A, E, I, O in which terms could be arbitrarily complex.

The set of inference rules needed to handle such arguments, together with many worked examples, can be found in appendix C (A Generalisation of Logical Processes in Their Application to Complex Propositions) of the final edition of Keynes' book Studies and Exercises in Formal Logic.

See also A.J. Baker's 1966 paper Non-empty complex terms, which contains a more succinct and rigorous account, as well as addressing issues of existential import.

An example of an argument which requires some of the extended inference rules is the following (taken from David Kelley's The Art of Reasoning, 1st ed.) :

  1. Some juveniles (J) who commit minor offenses (C) are put into prison (P).
  2. Any juvenile who is put into prison is exposed to all sorts of hardened criminals (E).
  3. A juvenile who is exposed to all sorts of hardened criminals will become bitter (B) and learn more techniques for committing crimes (T).
  4. Any individual who learns more techniques for committing crimes is a menace to society (M), if he or she is bitter.

    Therefore, some juveniles who commit minor offenses will be menaces to society.

Here's a proof of the conclusion :

 1.  Some JC are P            premise
 2.  All JP are E             premise
 3.  All E are BT             premise
 4.  All BT are M             premise
 5.  All JP are BT            2, 3, syllogism
 6.  All JP are M             4, 5, syllogism
 7.  All P are M or ~J        6, disjunctive transfer
 8.  Some JC are M or ~J      1, 7, syllogism
 9.  Some JC are J(M or ~J)   8, add superfluous conjunct
 10. Some JC are JM           9, simplification of predicate
 11. Some JC are M            10, drop superfluous conjunct (conclusion)


Posted 2018-09-26T21:03:18.457

Reputation: 86

Excellent answer. – Frank Hubeny – 2019-07-31T13:19:40.957