PART 2. TIME AND LOGIC REUNITED

'The rule is, jam tomorrow and jam yesterday but never jam today.' 'It must come sometimes to ~am today", objected. 'No it can't', said the Queen. 'It's jam every other day: today isn't any other day, you know.' 'I don't understand you', said Alice, 'It is dreadfully confusing. '. , Alice in [p. 150]

According to the great Danish logician and philosopher JCrgen Jcrgensen the history of logic can be divided into three periods: Classical Logic, covering Ancient and Medieval times; the algebra of logic founded by George Boole (1815-1864); and finally logicism as introduced by Gottlob Frege (1848-1925). This view on the history of logic is typical of the first part of the twentieth century, and reflects a depreciation of the importance of medieval logic - characteristic for that period. The analyses in part 1 of this book should have made it clear that such views were quite misguided, a point which is particularly evident if one focuses on the rSle of the concept of time in logic. We have seen that considerations of the temporal content were an integral element of logic in ancient and medieval times, but they were removed from the realm of logic during the Renaissance. Not until last century was time reintroduced into logic. One of the earlier intimations that logicians had again become aware of time in logic can be found in Lewis Carroll's Alice stories. The quote above is one of the many examples of the awareness of the relationship between time and logical reasoning which pervades his stories. Lewis Carroll (alias Charles Lutwidge Dodgson: 1832-1898) was a fellow in mathematics at Christ Church, Oxford. His scientific work was concerned primarily with logic. It can hardly be doubted that a thorough analysis of his entire corpus would reveal that he was one of the first 18th century logicians to rediscover the relation between logic and time. This 118 TIME AND LOGIC REUNITED 119

credit being given, it is nonetheless clear that Lewis Carroll's works have had little direct impact on the reintroduction of time into logic. We shall instead concentrate on the Irish mathema- tician and logician George Boole (1815 - 64), whose works proved crucial for the development of modern symbolic logic. Boole was obviously aware of some of the logical aspects of time and tense, and we shall give a short commentary on his works with regard to the these aspects. We also intend to show that Charles Sanders Peirce (1839 - 1914) understood that it would be possible to (re)introduce time into logic, and that this view played an important r61e for the founding father of modern tense logic, Arthur Normann Prior (1914-1969). At the beginning of this century logic was, generally speaking, considered to be a timeless discipline, although there were a few exceptions. In the chapters 2.1 and 2.2 we shall substantiate that George Boole as well as C. S. Peirce in the previous century both recognised the importance of time in their pioneering works in logic. Their observations in these respects were little noted at that time, however. Regarding logical reasoning about time the philosophers active in the last decades of the 19th century were more aware of others attitudes such as the opinion expressed by Henri Bergson in 1889:

... even in cases where the action is freely performed, we cannot reason about it without setting out its conditions ex- ternally to one another, therefore in space and no longer in pure duration. [Bergson 1950, p.240]

According to this view our reasoning about crucial temporal phenomena, like human freedom, will necessarily make us speak of time in terms of a spatial language. That is, speaking of possible acts at different times can easily give rise to an idea of these acts 'existing simultaneously' in some artificial or theore- tical space. From a systematic point of view, tense-logic may be said to provide at least a part of the answer to the Bergsonian problem, since it shows how one can speak of time in a formal manner without using the spatial metaphors. 120 CHAPTER 2.0

Tense-logic has been rediscovered during the twentieth cen- tury, particularly within the last few decades. The main impe- tus to this development is due to A. N. Prior, who in the early 1950's formulated some important ideas regarding temporal and modal logic, and ensuingly developed formal calculi for such systems. Prior's rediscovery was at first motivated by studies in ancient and medieval logic. Prior also found great in- spiration in his studies of the ideas of Boole and Peirce as well in the works of Polish logicians, first and foremost Jan Lukasiewicz. The relation between time and modality consti- tutes a prime example of rediscovered and reconstrued me- dieval concepts, which have proved fruitful in modern logic. The study of the medieval considerations on time and modality, together with a letter from Saul Kripke in 1958, led Prior to the formulation of a new idea of time, the so called 'branching time'. In the chapters 2.3, 2.4, and 2.5 we shall deal with Prior's re= discovery in its historical context. An informal but interesting literary anticipation of the concept of branching time can be found in Borges' writings as early as 1941. As far as we know Prior was not aware of these writings, but later tense-logicians have been. In chapter 2.6 we shall deal with the concept of branching time using Borges' writings as a point of departure. The notion of branching time is certainly relevant to discussions of freedom and determinism, but that part of the matter will mainly be postponed until part 3 of the book. The study of tense-logic is related to studies in physics as well as philosophy and psychology. Firstly, tense-logic with its em- phasis on the present seems to stand in contradiction of Special Relativity Theory. Prior, himself, was early aware of this ten- sion. He paid much attention to it. This is evident not only from his books and papers, but also from his letters and papers he left. We shall deal with the apparent contradiction in chapter 2.7 and we shall argue that on a closer study it can be shown to be spurious in the sense that the basic assumptions of even the strongest tense logic do not contradict any of the empirical con- sequences of relativistic physics. TIME AND LOGIC REUNITED 121

Prior dealt with a number of tense-logical systems and he made some very interesting discoveries in formal logic. He also tried to classify the various possible positions in relation to tem- poral and modal logic. According to one of his most interesting ideas temporal instants can be viewed as a special type of propo- sitions. In the chapters 2.8, 2.9, and 2.10 we shall study the essential features of his theories and some of the most important results.