This is a reflection on John Bacon’s article “First Order Logic based on Inclusion and Abstraction”.
Aristotle’s syllogistic was a logic of general terms, and the same could still be said of traditional logic in the 19thcentury. Writing in 1900, Bertrand Russell explained the failures he saw in Leibniz’s philosophy by the simple point that Leibniz was held back by the form of Aristotle’s syllogistic, the only logic then in play.
Ironically, we still see the struggles and birth pangs of a new logic coming into being in Russell’s 1903 Principles of Mathematics. What sense could there be in the new ideas about sets or classes? A set is meant to be a collection. But by the very meaning of the word, a collection is a collection of things, “things” plural. Does it make sense to speak of a collection of just one thing? What is the difference between “I put an orange on the table” and “I put a collection of one orange on the table”? It does make sense to say “I put nothing on the table”, but what about “I put a collection of nothing on the table”?
One might say, not unfairly, that all these puzzles were put behind us when Frege (in effect) introduced set-abstraction. (Even if he did so a little injudiciously, even inconsistently, the idea was soon tamed and made practical.) Within limitations, x̂Fx denotes the set of all things F, the singleton or unit set appears as extension of x̂(x = y), and the empty set as x̂(x 
But then, can’t we imagine that traditional logic could have developed equally successfully, through that same innovation?
I read John Bacon’s paper as answering this question with a resounding Yes.
As Bacon shows, the new logic that we enjoy today can be seen as a straightforward development of the very ‘logic of terms’ that Boole and De Morgan (and Schroeder and Peirce and …) turned into algebra. And so I would like to explain in a general way just how he does that.
To begin, it is a recurrent complaint about syllogistic that it is really only a logic of general terms, and has no good place for singular propositions. “Socrates is mortal” is accommodated awkwardly and artificially .
But as soon as singletons are respectable denizens of the abstract world, there is no problem for a logic of terms:
“Socrates is mortal” is true if and only if {Socrates} ⊆ {mortals}.
The three primitive notions that Bacon uses to simultaneously construct a contemporary form of the logic of terms and reconstruct first-order logic in just that form are these: the relation of inclusion, an abstraction operator, and the empty set.
Vocabulary: a set of variables, a set of constants, x̂, ⊆, and 0. Grammar: variables and constants are terms, and andx̂A is a term if A is any sentence and x a variable; if X, Y are terms then X ⊆ Y is a sentence.
The constants are names of sets, they cannot include “Socrates” but only a name for the set {Socrates}. The variables stand for singletons. So “Socrates is mortal” can be symbolized as “x ⊆ Y”.
Reconstruction of propositional calculus
If A is a sentence then x̂A is the set of all things x such that A. In this part we’ll only look at the case where x does not occur in A or B, for example x̂[2 + 2 = 4]. All things are such that 2+2 = 4! So that set is the universe of discourse, it has everything in it. We have a neat way to introduce the truth values:
A is true if and only if x̂A = the universe
A is false if and only if x̂A = 0.
This already tells us how to denote the universe:
U = x̂(O ⊆ 0)
A quick reflection shows that, accordingly, (x̂A ⊆ 0) always has the opposite truth-value of A. So we define:
~A = (x̂A ⊆ 0)
What about material implication, the sentence A ⊃ B? That is false only if A is true and B is false. But that is so exactly and only if x̂A = U and x̂B = 0, so we define:
(A ⊃ B) = (x̂A ⊆ x̂B)
Since nothing more is needed to define the other truth-functional connectives, we are finished. It is at once useful to employ the definable conjunction & at once for a definition:
X = Y iff (X ⊆ Y) & (Y ⊆ X)
Reconstruction of quantificational logic
Let’s to begin think only about unary predicates. The variables here can occur in the sentences we are looking at.
“Something is F” we symbolize today as “(∃x)Fx”. But this is clearly true if and only if the set x̂[Fx] is not empty. We define:
(X ≠Y) = ~(X = Y)
(∃x)A = (x̂A ≠ 0)
Similarly, the universal quantification (∀x)Fx is true precisely if x̂A = U.
Symbolizing “All men are mortal”:
x̂[(x ⊆ X) ⊃ (x ⊆ Y)] = U
Relations
The famous puzzle for 19th century logic was to symbolize
If horses are animals then heads of horses are heads of animals.
Bacon had no difficulty explaining how the system can be extended to relational terms P2, Q3, and so on, with the superscript indicating the addicity. The semantics can be straightforward in our modern view: P2 is a set of couples, Q3 a set of triples, and so on, with the simple idea that (ignoring use/mention):
(*) Rab is true if and only if <a, b> is in R.
Then Bacon allows strings of variables as n-adic variables. So if x stands for {Tom} and y for {jenny} then xy stands for {<Tom, Jenny>}. The sentence (xy ⊆ P2) then means that {<Tom, Jenny>} ⊆ P2.
But Bacon does it in a particular way that has some limitations. For example, in his formulation, inclusion is significant only between relations of the same degree. That raises a difficulty for examples like:
If Tom is older than Jenny then Tom is in age between Jenny and Lucy.
But there is a simple way, Tarski-style, to do away with this limitation. When Tarski introduced his style of semantics he gave each sentence as its semantic value the set of infinite sequences that satisfy that sentence.
Above we took terms to stand for sets of things in the universe of discourse. Now we can take them to stand for sets of countably infinite sequences of things in that universe. For example, the predicate “is older than” we take to stand, not for a set of couples, but for the set OLDER of infinite sequences s whose first element is older than its second element.
We represent any finite sequence t by the set of infinite sequences of which t is the initial segment. The notation <a, b> now stands not for the ordered pair whose first and second members are a and b respectively, but for the set of all infinite sequences whose first and second member are a, b respectively.
The predicate “is in age between … and —” we take to stand for the set BETWEEN of infinite sequences whose first member is in age between its second and third member. So the intersection of OLDER and BETWEEN is the set of sequences u such that the first element is older than the second, and the third element is someone older than both of them. This intersection can be the extension of a predicate P2 and if Q3 has BETWEEN as its extension then P2 ⊆ Q3is true.
But this is a minor point. There are always different, equally good, ways of doing things. The important moral of Bacon’s article is that it is not right to say that our contemporary logic had to replace the deficient logic of the tradition. Rather, the innovations that made our logic possible would equally improve the traditional logic, to precisely the same degree of adequacy.
REFERENCES
Bacon, John (1982) “First-Order Logic Based on Inclusion and Abstraction”. The Journal of Symbolic Logic 47: 793-808.
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