1. The concept of time has been been a main topic of philosophic reflection from Aristotle onwards. It is also an object of theoretical inquiry in several sciences — physics, physiology, and psychology. Many of the scientific theories of time are, moreover, of great interest to philosophy.
In spite of the existence of a large and varied philosophic literature dealing with the problems of time, there are but few attempts to treat the concept with the tools of formal logic. This fact may have something to do with the traditional character of logical theory. Logic, as we know it, is predominantly concerned with the conceptual scaffolding of a static world. Propositions, we are usually told, are either true or false, but not both true and false. In one and the same thing one and the same property cannot be both present and lacking. All this is true enough, granting that the world does not change. As such it is quite possible that one and the same thing both has and has not one and the same property — e.g., it first has it and then lacks it. And there is nothing implausible about the notion of a proposition whose truth-value varies with time. Such a proposition as, e.g., that it is raining, may now be false but later true.
A logic which is concerned with the conceptual scaffolding of a dynamic world is still largely a desideratum. In this logic the notions of change, process, and time will hold a prominent place.
Among the little that has been done to meet the desideratum in question the investigations of A. N. Prior foremostly deserve mention.1 They can be summarily characterized as attempts to create a tense-logic. The present study has a more restricted aim than that of Professor Prior's investigations. In its approach it is somewhat similar to Miss Anscombe's recent paper “Before and After”.2
In Professor Prior's tense-logic the ingredients new relative to
traditional systems of logic are two tense-operators, roughly of the type of the quantifiers and the modal operators. In the system which is studied here the new ingredient is a logical constant more reminiscent of the truth-connectives. It is a kind of asymmetrical conjunction. It can be interpreted, it seems, in several ways. On one interpretation it corresponds, roughly, to the meaning of the phrase “and next” of ordinary discourse. Other possible interpretations will not be considered in the present study. (See below Section 9.)
A first attempt to study this logical constant was made in my book Norm and Action.3 The present essay is a generalization and systematization of ideas which were first introduced there. I have had an opportunity to lecture on the topic to several audiences. In connection with one lecture a decisive contribution was made by Dr Lennart Åqvist of Uppsala University.4
2. The formal system which we are going to study will be called the T-calculus. It has the following vocabulary:
a. Variables p, q, r…
b. Truth-connectives ~, &, v, →, and ↔.
c. A connective T.
The variables are schematic representations of sentences (of a certain sort). The question may be raised, what the sentences express or what entities their meanings are. We shall call the entities generic propositions.5 Such propositions are not true or false “in themselves”. They have a truth-value only relative to a (point in) time. They may be true of one time, false of another, and they may be repeatedly true and false. Let the general proposition be, e.g., that it is raining. It may be true of to-day, false of to-morrow, but true again of the day after to-morrow. (The relativity of generic propositions to a location in space will not be considered here.)
The well-formed formulae of the T-calculus we call T-expressions. They are defined recursively as follows:
a. A variable is an atomic T-expression.
b. An expression formed of the connective T with a T-expression to the left and a T-expression to the right is an atomic T-expression.
c. Atomic T-expressions and their truth-functional compounds are T-expressions.
By a truth-functional compound (of atomic T-expressions) we understand a molecular compound which is formed of its constituents solely by means of truth-connectives. The atomic T-expressions of which a T-expression is a truth-functional compound we call its truth-functional components.
It is convenient to have a rule for the omission of brackets. We shall give it in the form of a convention concerning the “binding force” of the connectives. Beginning with the weakest and ending with the strongest, the order of the connectives shall be: ~, &, v, →, ↔, T. Thus we can, e.g., for ((((~p&q)vr) → s) ↔ u)Tv write simply ~p&qvr → s ↔ uTv.
Consider an atomic T-expression, other than a variable. That letter T, of which the expression is formed by putting a T-expression to its left and another to its right will be called its principal T. The expressions to the left and to the right of the principal T may themselves contain one or several occurrences of the symbol T.
The expression pTq may be read “p and next q”. For “p” put, e.g., the sentence “it is raining” and for “q” the sentence “the sun is shining”.
3. The T-calculus is a formalized axiomatic structure. Its axioms are the following:
A0. A set of axioms of (“classical”, two-valued) Propositional Logic (PL).
Al. (pvqTrvs) ↔ (pTr) v (pTs) v (qTr) v (qTs). Distributivity.
A2. (pTq)&(rTs) ↔ (p&rTq&s). Co-ordination.
A3. p ↔ (pTqv~q). Redundancy.
A4. ~(pTq&~q). Impossibility.
The rules of inference of our calculus are the following three:
R1. Substitution (of T-expressions for variables).
R2. Modus ponens.
R3. Extensionality. This means that provably equivalent T-expressions are intersubstitutable.
4. We shall next prove, or sketch the proof of, some theorems of the T-calculus (T-theorems).
(1) p ↔ (pTqv~q). A3.
(2) ~p ↔ (~pTqv~q). From A3 by R1.
(3) (pTqv~q) ↔ (pTq)v(pT~q). By virtue of Al.
(4) (~pTqv~q) ↔ (~pTq)v(~pT~q). Al.
(5) pv~p → (pTq)v(pT~q)v(~pTq)v(~pT~q). From (1)—(4) by PL.
(6) pv~p. PL.
(7) (pTq)v(pT~q)v(~pTq)v(~pT~q). From (5) and (6) by R2. Q.E.D.
By substituting p for q in T1 we immediately obtain as a corollary
I shall say of the four disjuncts in T2 that they stand for the four types of elementary changes or state-transformations.6
T3. ~(p&~pTq). The second Principle of Impossibility. (Cf. A4.)
(1) p&~p ↔ (p&~pTqv~q). From A3 by R1.
(2) (p&~pTqv~q) ↔ (p&~pTq)v(p&~pT~q). Al.
(3) p&~p ↔ (p&~pTq)v(p&~pT~q). From (1) and (2) by PL.
(4) ~(p&~p) ↔ ~((p&~pTq)v(p&~pT~q)). From (3) by PL.
(5) pv~p ↔ ~(p&~PTq)&~(p&~pT~q). From (4) by PL.
(6) pv~p → ~(p&~pTq). From (5) by PL.
(7) pv~p. PL.
(8) ~(p&pTq). From (6) and (7) by R2. Q.E.D.
Consider the conjunction of any two of the four disjuncts in Tl. For example, take (pTq)&(pT~q). By reason of A2, this is equivalent to (pTq&~q) which, by reason of A4, is provably false. Or, take (pTq)&(~pTq). By reason of A2, this is equivalent to (p&~pTq) which, by reason of T3, is provably false, too. This shows that the four disjuncts of T1 are, not only jointly exhaustive, but also mutually exclusive. The result immediately applies to the four disjuncts of T2. The four types of elementary changes, in other words, are mutually exclusive and form an exhaustive disjunction.
Because of the mutually exclusive and jointly exhaustive nature of the four disjuncts in T1 (and T2), the negation of any one of these disjuncts is, by reason of PL, equivalent to the disjunction of the three remaining disjuncts. Thus, for example, ~(pTq ↔ (pT~q)v(~pTq)v(~pT~q) is a theorem of our calculus.
T4. (pTq) → p.
(1) p ↔ (pTq)v(pT~q). From A3 by A1 and PL.
(2) (pTq) → p. From (1) by PL. Q.E.D.
T5. p&(qTr) ↔ (p&qTr).
(1) (pTqv~q)&(qTr) ↔ (p&qT(qv~q)&r). From A2 by R1.
(2) (pTqv~q) → p. A3.
(3) (qv~q)&r ↔ r. PL.
(4) p&(qTr) ↔ (p&qTr). From (1) — (3) by PL and R3. Q.E.D.
T6. ((pTq)Tr) ↔ (pTq&r).
(1) (pTq)&(pTr) ↔ ((pTq)&pTr). From T5 by R1.
(2) (pTq)&(pTr) ↔ (pTq&r). From A2 by R1 and R3.
(3) (pTq)&p ↔ (pTq). From T4 by PL, or from T5 by R1, PL, and R3.
(4) ((pTq)Tr) ↔ (pTq&r). From (1) — (3) by PL and R3. Q.E.D.
Consider an atomic T-expression which is so constituted that the expression to the left of the principal T is itself an atomic T-expression. Then, by reason of T6, we can sever from this second atomic T-expression, “chop off”, the part which stands to the right of its principal T and join it to the expression which stands to the right of the principal T of the first atomic T-expression, forming a conjunction. This process may become reiterated until the expression to the left of the principal T contains no occurrences of the symbol T. The principal T thus becomes the left-most T of the expression. There is, however, no method of moving the principal T to the right. pT(qTr) is not equivalent to p&qTr.
The connective T is not associative.7 (pTq)Tr is not equivalent to pT(qTr). The first expression refers, in fact, to two successive points in time only, the second refers to three.
5. Consider an arbitrary T-expression. We make a list of all the variables which occur in it. The variables of the list, we shall say, determine the Universe of Discourse, or simply the universe, of the T-expression.
Consider a universe of n elements (variables), p1, …, pn. The n elements determine 2n so-called state-descriptions. We call them s1, … , s2n.
Consider a T-expression of the form -T(-T(-T(- … T-))) … . Each occurrence of - indicates the place of a state-description in the expression. The first sequence of dots indicates an arbitrary number of repetitions of the pattern T(-. The second sequence of dots indicates as many closing brackets as the first sequence of dots indicates repetitions of the pattern T(-.
An expression of this form we shall call a history of the universe of n elements. By the length (or degree) of the history we understand the number of successive state-descriptions in it.
Examples. siT(sjT(skTsl)) is a history of length 4. It tells us which are the four successive (total) states of a certain universe (of some n elements). The same state may repeat itself, one or several times, in the course of a history. siT(siT(siTsi)) tells us that the universe remains in the same total state si when, metaphorically speaking, time passes from a first point to a fourth.
siTsj is a history of length 2. An individual state-description is a history of length 1. It is called a history “by courtesy” only. It tells us how the world is at a certain moment in time, — and nothing about how it changes or remains unchanged from one moment to the next.
It is easy to see that, in a universe of n elements, the total number of possible histories of a given length m is 2mn. For m = 1, this reduces to 2n or to the number of state-descriptions.
For the sake of convenience we shall introduce a convention to the effect that all pairs of brackets may be omitted from histories. Thus, for example, we may write siTsjTsk instead of siT(sjTsk). This must not be confused with the expression (siTsj)Tsk which has an entirely different meaning.
By a Tm, n-tautology we shall understand a disjunction of all the possible histories of exactly the length m in a universe of n given elements. By a Tm-tautology we understand a disjunction of all the possible histories of precisely the length m in a universe of some n elements. By a T-tautology, finally, we understand a disjunction of all the possible histories of some length m in a universe of some n elements. Every Tm, n -tautology is also a Tm-tautology, and every Tm-tautology is a T-tautology.
Examples. (pTp)v(pT~p)v(~pTp)v(~pT~p) is a T-tautology. The histories in it are of length 2 and the universe has only one element, p. The tautology is thus a T2,1-tautology. “By courtesy”, pv~p too counts as a T-tautology, viz. a T1,1-tautology.
6. By reason of A3, we can tautologically increase the length of a given history. Let this history be h1. Let the state-descriptions in the universe of the formula be s1, … , s2n. Then, by reason of A3, hi is equivalent to hiT(slv … vs2n). This is not, as such, an expression of the form which we call a “history”. But, by reason of A1, it may become distributed into a disjunction of histories (hiTs1) v ... v(hiTs2n). Since any two state-descriptions are mutually exclusive, it follows from the First Principle of Impossibility (A4) that the disjuncts, too, are mutually exclusive.
Histories may be tautologically increased, not only in length, but also in “width”. This happens through the introduction of new elements (variables) into the universe of the formula.
Consider the expression pTp. It is one of the four possible histories of length 2 in a universe containing one single element, p. We replace (R3) p by its perfect disjunctive normal form in terms of p and q, i.e. by p&q v p&~q. Then the initial expression becomes p&q v p&~qTp&q v p&~q. We distribute this expression according to A1 and get (p&qTp&q)v(p&qTp&~q)v(p&~qTp&q)v(p&~qTp&~q). This is a disjunction of 4 of the 16 possible histories of length 2 in a universe of two elements, p and q. By virtue of the First and Second Principles of Impossibility, we immediately prove that the four disjuncts are mutually exclusive.
Generally speaking, consider a history hi of some length m in a universe of some n elements p1, … ,pn. We introduce a new element pn+1 into the universe. Let sj be an arbitrary state-description which occurs in hi. We replace sj by sj&(pn+1v~pn+1) and similarly for all the other state-descriptions which occur in hi. Thereupon we use A1 for successively distributing the new expression into a disjunction of 2m histories. Take any two of the disjuncts and “co-ordinate” them using A2. Then, by the First or Second Principle of Impossibility, we may conclude that the disjuncts are mutually exclusive.
The histories of length 1, in a universe of 1 element, form an exhaustive disjunction. (pv~p is a T-theorem). From the possibility of tautologically increasing the histories in length and in width, it
follows easily that the histories of an arbitrary length m in a universe of an arbitrary number n of elements form an exhaustive disjunction, too.
By the complement of a given history we shall mean the disjunction of all other histories, but this one, of the same length in the same universe. Since the histories are mutually exclusive and jointly exhaustive, it follows that the negation of any given history is equivalent to the complement of that history. (Cf. above Section 4.)
7. Every T-expression is (in the T-calculus) provably equivalent with a disjunction of histories (of equal length) in the universe of the expression.
The notion of a disjunction (of histories) includes, as limiting cases, disjunctions of one member and of no member.
A disjunction of histories, which is equivalent to a given T-expression, will be called a T-normal form of the T-expression. Our meta-theorem above thus says that every T-expression has (may become transformed into) a T-normal form.
In order to prove the meta-theorem, we first consider the following process of successive decomposition of a given T-expression:
The T-expression, if it is not atomic, is first decomposed into its truth-functional components. These components are atomic T-expressions. Every one of them is thereupon decomposed into the two T-expressions which stand to the left and to the right of the principal T. These T-expressions are either atomic or molecular. The process of decomposition is continued for them. The process comes to an end when we reach expressions of PL, i.e. variables or truth-functional compounds of variables.
Example. The below table illustrates, how an initially given T-expression is successively decomposed until we finally reach expressions of PL only:
The process is now reversed and we embark upon a successive building up of the original expression from the parts, into which it has been decomposed.
First step. We start from the ultimate results of the decomposition, the expressions of PL. (Bottom row of the above table.) We replace each of these expressions by their perfect disjunctive normal forms in terms of all the variables in the initially given T-expression. (The variables make up the universe of the T-expression.) If the normal form is a 0-termed disjunction we write for it the symbol 0.
Second step. Thereupon we replace the atomic T-expressions, which are the penultimate results of the process of decomposition (see row 4 of table), by new atomic T-expressions in which the expressions to the left and to the right of T are in the perfect disjunctive normal form. If in some of these new atomic T-expressions the symbol 0 occurs to the left or to the right of the T, we replace the atomic T-expression itself by the symbol 0. (By reason of A4 or T3.) The other T-expressions we distribute, according to A1, into disjunctions of histories. All these histories are of length 2, i.e. they consist of two state-descriptions in the universe of the initially given T-expression joined by the symbol T.
The next or third step in the process of recomposition consists in the formation of truth-functional compounds of histories of length 2 and/or expressions of PL in perfect disjunctive normal form and/or the symbol 0. Since state-descriptions count as histories of length 1, we can also characterize this stage in the process of recomposition as the formation of truth-functional compounds of histories of at most length 2 and/or the symbol O. These compounds are equivalent to the antepenultimate steps in the decomposition process. (Cf. row
3 of table.) The compounds are then transformed into disjunctions of histories according to the following device:
We replace the histories of degree 1 by 2n-termed disjunctions of histories of degree 2 according to the method described in Section 6 of tautologically increasing the length of histories. Thereupon we transform the entire compound into its perfect disjunctive normal form in terms of the histories and/or the symbol 0. This normal form is a disjunction of conjunctions of histories and/or negations of histories of length 2 and, possibly, of symbols 0 or ~0. We replace negations of histories by their complements (see Section 6). We replace 0-signs with a negation sign in front (~0) by disjunctions of all the possible histories of length 2. We omit conjunctions in
which the 0-sign occurs unnegated. After these replacements and omissions we distribute the entire expression according to A1. We get a disjunction of conjunctions of histories of length 2. We apply A2 to these conjunctions. The conjunctions now become atomic T-expressions. If the expression to the left or to the right of the T is a contradiction of PL, we replace the atomic T-expression by 0. If the expression to the left and to the right of the T is not a contradiction we replace it by the state-description to which it is equivalent. The disjunction of conjunctions of histories of length 2 has now become a disjunction of histories of length 2 and, possibly, the symbol 0. We omit 0, if it occurs, from the disjunction. Then we are finally left with a disjunction of histories of length 2. (Should this disjunction happen to be 0-termed we write for it the symbol 0.)
The next or fourth stage in the process of recomposing the original T-expression consists in the formation of atomic T-expressions, in which the expressions to the left and to the right of T are disjunctions of histories of at most length 2 and/or the sign 0. (Cf. row 2 of table.) If in an atomic T-expression, thus formed, the sign 0 occurs, we replace the whole expression by 0. If the sign 0 does not occur, we distribute the expression according to A1 whenever possible. The expression then becomes a disjunction of atomic T-expressions, in which to the right and to the left of the principal T there is a history of at most length 2.
We focus attention on these atomic T-expressions. If the history to the left of the principal T is of length 2, we apply T6. In the expression, thus transformed, there is a state-description to the left of the principal T. To the right there is either a conjunction of state-descriptions or a conjunction of a state-description and a history of length 2. A conjunction of the first kind is equivalent to a state-description (or 0). A conjunction of the second kind is equivalent to a history of length 2 (or to 0). Replacing the conjunctions by their equivalents, the initial atomic T-expressions become histories of length 3 or 2, or they become 0.
The fifth stage in the process of recomposition is analogous to (the first and) the third. It consists in the formation of truth-functional compounds of histories of at most length 3 and/or the symbol 0. (Cf. top line of table.) These compounds are then transformed, first into their perfect disjunctive normal forms (in terms of the histories), and ultimately into disjunctions of histories of length 3.
The sixth stage is analogous to (the second and) the fourth. It
consists in the formation of atomic T-expressions in which stand, to the left and to the right of the principal T, disjunctions of histories of at most length 3, or the symbol 0. These atomic T-expressions are then transformed into disjunctions of histories of at most length 4, or they become 0.
The nature of the last stage in the process depends upon whether the initially given T-expression is atomic or molecular. If it is atomic, the stage is of the type of the stages two, four, and six. It consists in the formation of one atomic T-expression in which the expressions to the left and to the right of T are disjunctions of histories of at most a certain length m-1, or the symbol 0. This atomic T-expression is thereupon transformed into a disjunction of histories of exactly the length m, or it becomes O.
If the original T-expression is molecular, the last stage is of the type of the stages one, three, and five. It then consists in the formation of truth-functional compounds of histories of at most a certain length m, and/or of the symbol 0. This compound is then transformed into a disjunction of histories of exactly length m, or it becomes 0.
8. A T-expression is a T-theorem, if and only if, it has a T-normal form which is a T-tautology.
If the technique of decomposing and recomposing T-expressions., which we described in Section 7, is applied to the axioms A1 — A4, we obtain disjunctions of histories of length 2. The disjunctions answering to A1 and A2 contain 22*4 or 256 disjuncts; they are T2,4-tautologies. The disjunctions answering to A3 and A4 contain 22*2 or 16 disjuncts, they are T2,2-tautologies.
Any formula which is derived from a T-tautology by means of R1 or the usual rule of variable substitution must be itself a T-tautology. — A T-tautology presents us with all possible concatenations of some 2n state-descriptions to form histories of a given length m. If a variable is replaced by one and the same formula in all these concatenations of state-descriptions, the resulting increase in length and width of the original T-tautology, if any, must be of the “tautologous” kind which we described in Section 6. If no increase in length or width takes place as a result of the substitution, we are left either with identically the same T-tautology or with a “shrunken” one with a smaller number of variables.
Any formula which is derived from a T-tautology by means of R2 or the rule of modus ponens must be itself a T-tautology. — If
f is a T-tautology, ~f is 0. If f → g is a T-tautology, then ~f v g is a T-tautology, too. But if ~f is 0, g alone must be a T-tautology
Finally, any formula, which is derived from a T-tautology by means of R3 or the rule of extensionality, must be itself a T-tautology. — Assume that f ↔ g has been derived from the axioms without the use of R3. Then f ↔ g is a T-tautology. Assume that t is another T- tautology, and that f occurs in it. We replace some occurrence of f in t by g. The resultant formula t' must be a T-tautology. For, by reason of the tautological nature of f ↔ g, the formula g is equivalent to exactly the same disjunctions of histories as f in the universe of t.
9. The connective T, we have said, may be read “and next”. This suggests an analogy with conjunction (&, “and”).
If in the axioms A1 — A4 we replace T throughout by &, we get tautologies of PL. The T-calculus thus has an interpretation in PL.
T may be characterized as an asymmetrical (and non-associative) conjunction.
“p and q” means the same as “q and p”, but “p and next q” clearly has a different meaning from “q and next p”. This is reflected on a formal level in the facts that p&q ↔ q&p is, but that (pTq) ↔ (qTp) is not a T-tautology. If we transform the second expression into a T-normal form in terms of the variables in its own universe, we obtain a disjunction of 10 histories of length 2. Of the 16 possible histories of this length in that universe, six are thus missing.
The connective T may be said to co-ordinate two “worlds”, viz. the world which is now and the world which will be next. The view of T as an asymmetrical co-ordinator of worlds opens up interesting possibilities of generalizing the theory.
T, as we have interpreted it, is a “forward-looking” temporal connective. We could, of course, just as well interpret it as “backward-looking”. Then it co-ordinates the world of a certain moment with the world which was just before. We could distinguish the two connectives by means of two symbols, say →T and ←T, and construct a joint calculus containing both connectives.
Academy of Finland
1 A. N. Prior, Time and Modality.Oxford at the Clarendon Press, 1957.
2 In The Philosophical Review 73, 1964.
3 Routledge and Kegan Paul, London, 1963. See especially Ch. ii, Sections 7—10, pp. 28—34.
4 See below p. 297.
5 For further remarks on the notion of a generic proposition see Norm and Action Ch. ii, Sect. 4, pp. 22—25.
6 Cf. Norm and Action, p. 29.
7 An earlier version of the present system contained an axiom of associativity (pTq)Tr ↔ pT(qTr). I am much indebted to Dr L. Åqvist for having drawn my attention to the fact that this axiom leads to counterintuitive results and therefore must be omitted.