Because these congresses occur at intervals of five years, they make for retrospection. I find myself thinking back over a century of logic. A hundred years ago George Boole's algebra of classes was at hand. Like so many inventions, it had been needlessly clumsy when it first appeared ; but mean while, in 1864, W. S. J evons had taken the kinks out of it. It was only in that same year, 1864, that DeMorgan pub lished his crude algebra of relations. Then, around a century ago, C. S. Peirce publ ished three papers refining and extend ing these two algebras-Boole's of classes and DeMorgan's of relations. These papers of Peirce's appeared in 1867 and
1870. Even our conception of truth-function logic in terms of truth tables, which is so clear and obvious as to seem in evitable today, was not yet explicit in the writings of that time. As for the logic of quantification, it remained unknown until 1879, when Frege published his Begriffsschrift; and it was around three years later still that Peirce began to be come aware of this idea, through independent efforts. And even down to little more than a half century ago we were weak on decision procedures. It was only in 1915 that Lowen heim published a decision procedure for the Boolean algebra of classes, or, what is equivalent, monadic quantification the ory. It was a clumsy procedure, and obscure in the presenta tion-the way, again. with new inventions. And it was less
A shorter version of this paper appeared in the Akten des XIV. inter nationalen Kongresses fiir Philosophie, vol. 3, 1969.
On the Lirnits of Decision 157
than a third of a century ago that we were at last forced, by results of Godel, Turing, and Church, to despair of a decision procedure for the rest of quantification theory.
It is hard now to imagine not seeing truth-function logic as a trivial matter of truth tables, and it is becoming hard even to imagine the decidability of monadic quantification theory as other than obvious. For monadic quantification theory in a modern perspective is essentially just an elabora tion of truth-function logic. I want now to spend a few min utes developing this connection.
What makes truth-function logic decidable by truth tables is that the truth value of a truth function can be computed from the truth values of the arguments. But is a formula of quantification theory not a truth function of quantifica tions ? Its truth value can be computed from whatever truth values may be assigned to its component quantifications. Why does this not make quantification theory decidable by truth tables ? Why not test a formula of quantification theory for validity by assigning all combinations of truth values to its component quantifications and seeing whether the whole comes out true every time ?
The answer obviously is that this criterion of validity is too severe, because the component quantifications are not always independent of one another. A formula of quantifi cation theory might be valid in spite of failing this truth table test. It might fail the test by turning out false for some assignment of truth values to its component quantifications, but that assignment might be undeserving of notice because incompatible with certain interdependences of the com ponent quantifications.
If, on the other hand, we can put a formula of quantifica tion theory into the form of a truth function of quantifica tions which are independent of one another, then the truth table will indeed serve as a validity test. And this is just what we can do for monadic formulas of quantification the ory. Herbrand showed this in 1930.
The method exploits what Boole called constituent func tions ; when adapted to quantificational notation they might be called constituent quantifications. For a single predicate letter 'F' the constituent quantifications are two in number :
158 Theories and Things
' ( 3 x ) Fx' and ' ( 3 x ) - Fx'. For two letters 'F' and 'G' they are four in number ' O x ) (Fx . Gx ) ', ' O x ) (- Fx . Gx ) ', ' O x ) (Fx . - Gx) ', and ' ( 3 x) ( - Fx . - Gx) '. For n letters, simi larly, there are 2n constitutent quantifications. They corre spond to the cells or uncut regions of the Venn diagram. Each constituent quantification says of its cell that it is not empty. Now Herbrand showed that by distributing and confining quantifiers and expanding conjunctions in familiar ways we can transform any monadic formula of quantification theory, in n predicate letters, into an explicit truth function of the constituent quantifications in those n predicate letters. These constituent quantifications are mutually independent ; con sequently the truth-table test of validity can be brought to bear on any monadic formula of quantification theory once we put the formula into such a normal form. We simply con struct the formula's H erbrand truth table, as I shall call it, which assigns a truth value to the formula for each assign ment of truth values to all constituent quantifications ap pearing in the formula's normal form. If we care to exempt the empty universe, as is usual and convenient, we have merely to ignore the simultaneous assignment of falsity to all 2n constituent quantifications.
Example :
0 x) (Fx . Gx ) ::J (x) (Fx . Hx . ::J . Gx) .
Expressing it as a truth function of constituent quantifica tions, we have :
O x) (Fx . Gx . Hx ) v O x ) (Fx . Gx . - Hx ) . ::J
- O x ) (Fx . - Gx . Hx) .
It comes out true in five of the eight lines of its Herbrand truth table, namely these :
( 3 x) (Fx . Gx . Hx) 1 T 1 T 1 ( ::J x ) (Fx . Gx . - Hx) 1 T T 1 1 ( ::J x ) (Fx . -Gx . Hx ) T 1 1 l 1 L
On the Limits of Decision 159
We have now observed this much kinship between monadic formulas of quantification theory and pure truth functions based on sentence letters : the monadic formula has the con stituent quantifications as its mutually independent truth arguments, that is, arguments that it is a truth function of, j ust as the pure truth-functional formula has its sentence letters.
It is also possible, by dividing matters differently, to re veal another kinship. Besides seeing a monadic formula in n
letters as one truth function of up to 2n constituent quantifi cations, we can see it also as corresponding to a set of truth functions, or Boolean functions, of just its n predicate let ters. These Boolean functions are the models, as I shall call them, of the monadic formula. Diagrammatically speaking, a region of the Venn diagram is a model of a given monadic formula if the formula comes out true when all cells of the region are occupied and all else is empty. For example, the formula ' (x) (Fx = - Gx) has three models, 'FG', 'FG', and
'FG v FG'; also the null region, if we choose to recognize the empty universe. The previous example in three predicate letters has 160 models. Clearly any monadic formula is de termined uniquely, to within equivalence, by a list of its models. A valid formula is one whose models comprise all
22n Boolean functions of its n predicate letters (minus one for the empty universe) .
There is a simple mechanical method for eliciting all the models of a monadic formula. It is cumbersome in practice but worth noting in theory. It proceeds from what I shall call the exhaustive Herbrand truth table. The ordinary Her brand truth table assigns truth values to the constituent quantifications that occur in the formula. The exhaustive one, on the other hand, assigns truth values to all � con stituent quantifications in the n predicate letters, whether the quantifications occur in the formula or not. Where n is 2,
the table runs to four columns and fifteen rows (if we de duct one for the empty universe) ; where n is 3 it runs to eight columns and 255 rows. Now the models of a monadic formula can be formed from the favorable rows of the ex haustive Herbrand truth table, that is, the rows in which the
160 Theories and Things
formula comes out true. At each favorable row we simply form the alternation of the constituent quantifications that are marked true in that row, and then delete the quantifiers and bound variables ; what remains is a model.
Example : ' ( x) (Fx = - Gx) ', expressed as a truth func tion of constituent quantifications, becomes :
- ( 3 x ) (Fx . Gx) . - ( 3 x ) (-Fx . - Gx) .
One of the favorable rows of its exhaustive Herbrand truth table is : ( 3 x ) (Fx . Gx) 1 ( 3 x ) ( - Fx . Gx) T ( 3 x ) (Fx . - Gx) T ( 3 x ) (- Fx . - Gx) 1
The corresponding model is 'FG v FG'.
So the relation of truth functions to monadic formulas can be seen in two ways : the monadic formula is a truth func tion of up to 2n constituent quantifications, and also it is de termined by a set of up to 22n - 1 Boolean functions of its n predicate letters.
However viewed, the relation invites extrapolation. Truth functional formulas are to monadic formulas as monadic formulas are to what dyadic ones ? Let us try extrapolating. The truth arguments of the n-letter truth-functional for mulas are just the n letters. The truth arguments of the n-letter monadic formulas are the 2n constituent quantifica tions. Now the new formulas will be certain n-letter dyadic formulas, having 22n truth arguments which are built from the 2n constituent quantifications as constituent quantifica tions were built from the n predicate letters. Each of these
22n new truth arguments will be, in short, an n-letter super constituent quantification, and will have this form :
( 3 x) [± ( 3 y) ( ± FiXY . ± F2xy . . .. ± Fnxy) . ± ( 3 y) ( ± FiXY . ± F2xy ... . . ± FnXY) . . . ± ( 3 y) ( ± Fi xy . ± F 2XY . . . ± F nXY) ]
where each '±' may represent affirmation or negation. The new sort of dyadic formulas to which we are extrapolating will be the truth functions of such super-constituent quanti fications.
On the Limits of Decision 161
The n-letter monadic formulas, though they were truth functions of the 2" constituent quantifications in those let ters, were of course not ordinarily written explicitly as truth functions of these. To rewrite them thus was to put them in a certain normal form. Now the same is to be true of our new n-letter dyadic formulas ; we may take this new lot as broadly as we please, so long as all are convertible into a normal form which represents them explicitly as truth func tions of the super-constituent quantifications in those letters. A natural class of dyadic formulas meeting this requirement is the class of what I shall call the homogeneous dyadic for mulas, defined as follows : a homogeneous dyadic formula is any formula of quantification theory in which each occurrence of each predicate letter is followed by the specific pair of letters 'xy' in that order, and each of the (possibly numerous) quantifiers containing the letter 'y' stands in the scope of one of the quantifiers containing 'x'. By the same moves by which Herbrand was able to transform any monadic schema into an explicit truth function of constit uent quantifications, namely, by distributing and confining quantifiers and expanding conj unctions, we are able also to transform any homogeneous dyadic formula in n predi cate letters into an explicit truth function of the super constituent quantifications in those n letters.
Consider now the nature of a super-constituent quantifica tion, as displayed above. If at first we think of the x as fixed, in that formula, then each line of the formula affirms the non emptiness or emptiness of a constituent function, or cell of the Venn diagram ; and all these 2" lines together then give the whole story regarding all 2n cells of some one Venn dia gram. Each way of settling all the affirmation-negation signs determines one fully marked Venn diagram for the n predi cates. For each particular way of settling all the affirmation negation signs, therefore, what the whole super-constituent quantification tells us is that that particular Venn diagram is fulfilled by the classes obtained by projecting the relations
F1, F2, • • • , Fn on some one object x.
The 22n super-constituent quantifications are mutually in dependent, since a different object x can serve each time.
162 Theories and Things
There is, therefore, a decision procedure for a homogeneous dyadic formula : j ust put it into normal form and build a whacking truth table on these 22n truth arguments. Inciden tally we can again exempt the empty universe if we please by ignoring appropriate rows.
We saw how an n-letter monadic formula, besides being a truth function of constituent quantifications, corresponds to a set of Boolean functions of the n letters themselves. Now an n-letter homogeneous dyadic formula can be seen to cor respond similarly to a set of monadic formulas in those n letters, and hence to a set of sets of Boolean functions of those n letters.
Another step of extrapolation leads from the homogeneous dyadic formulas to homogeneous triadic formulas, which again are decidable. Their truth arguments are super-super constituent quantifications. And so on up. In general, the homogeneous k-adic formulas are k-adic formulas of quanti fication theory meeting restrictions like those noted in the dyadic case ; namely, the variables must stand in a fixed order after the predicate letters, and the quantifiers must be nested always in that order.
Our natural tendency to associate monadic quantification theory with general quantification theory is in a way mis leading. The monadic has stronger affiliations with truth function logic than with general quantification theory. All these homogeneous polyadic formulas likewise are, is es sential respects, of a piece still with truth-function logic.
What makes the difference between all this and the un decidable general quantification theory is not, we see, the presence of polyadic predicates. What evidently gives gen eral quantification theory its escape velocity is the chance to switch or fuse the variable attached to a predicate letter, so as to play 'Fyx' or 'Fxx' against 'Fxy'.
The mercurial quality of general quantification theory can subsist, we know, in seemingly modest fragments of general quantification theory. The general theory of a single sym metrical dyadic predicate is undecidable.1 So is the general theory of dyadic formulas in which there is no quantifier
On the Limits of Decision 163 beyond a single initial cluster ' ( 3 x ) (y) ( 3 z) '.2 Perhaps the time will come when what makes for undecidability in quan tification theory will seem as obvious as the decidability of the monadic case. That time is not yet.
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