The extensions of sequent calculi by rules, presented in the previous chapter, share the good structural properties of the purely logical G3-calculi, i.e., the rules of weakening, contraction, and cut are admissible. In addition to being admissible, weakening and contraction are height-preserving admissible. The usual consequence of cut elimination, the subformula property, holds in a weaker form, because all the formulas in the derivations in such extensions are subformulas of the endsequent or atomic formulas. However, by analysing, analogously to natural deduction, minimal derivations in specific theories, we can establish a subterm property, by which all terms in a derivation can be restricted to terms in the endsequent.

This chapter gives proofs of the subterm property for partial and linear order, the latter not an easy result. To make its presentation manageable, a system of rules that act on the right part of multisuccedent sequents is used. Further, it is shown through a proof-theoretical algorithm how to linearize a partial order, a result known as Szpilrajn's theorem. The extension is based on the conservativity of the rule system for linear order over that for partial order for sequents that have just one atom in the succedent. Finally, the proof-theoretical solution of the word problem for lattices of Chapter 4 is extended to linear lattices, i.e., lattices in which the order relation is linear.

Having laid the groundwork of semantic information and agents’ knowledge in Chapter 2, we will now study dynamic scenarios where information flows and knowledge changes by acts of observation or communication. This chapter develops the simplest logic in this realm, that of public announcement or observation of hard information that we can trust absolutely. This simple pilot system raises a surprising number of issues, and its design will be a paradigm for all that follows in the book. We start with motivating examples, then define the basic logic, explore its basic properties, and state the general methodology coming out of this. We then sum up where we stand. At this stage, the reader could skip to Chapter 4 where richer systems start. What follows are further technical themes, open problems (mathematical, conceptual, and descriptive), and a brief view of key sources.

Intuitive scenarios and information videos

Recall the Restaurant scenario of Chapter 1. The waiter with the three plates had a range of six possibilities for a start, this got reduced to two by the answer to his first question, and then to one by the answer to the second. Information flow of this kind means stepwise range reduction.

While we have looked extensively at individual agents and their interaction, a further basic feature in rational agency is the formation of collective entities: groups of agents that have information, beliefs, and preferences, and that are capable of collective action. Groups can be found in social epistemology, social choice theory, and the theory of coalitional games. Some relevant notions occurred in the preceding chapters, especially common knowledge – but they remained a side theme. Indeed, the logical structure of collectives is quite intricate, witness the semantics of plurals and collective expressions in natural language, which is by no means a simple extension of the logic of individuals and their properties. This book develops no systematic theory of collective agents, but this chapter collects a few themes and observations, connecting logical dynamics to new areas such as social choice.

Collective agents in static logics

Groups occur in the epistemic logic of Chapter 2 with knowledge modalities such as CGϕ or DGϕ. But the logic had no explicit epistemic laws for natural group forming operations such as G1 ∪ G2, G1 ∪ G2. Actually, two logics in this book did provide group structure. One is the epistemic version E-PDL of propositional dynamic logic in Chapter 4, where epistemic program expressions defined complex ‘collective agents’ such as i ; (?p ; j∪k)*. Another was mentioned in Chapter 2: the combined topologies of van Benthem & Sarenac (2005). Even so, epistemic logic still needs a serious extension to collective agents: adding common or distributed knowledge is too timid. Here, group structure and information may be intertwined: for instance, membership of a group seems to imply that one knows this. Moreover, groups also have beliefs and preferences, and they engage in collective action. In all these cases, generalization may not be straightforward. Collective attitudes or actions may reduce to behaviour of individual group members, but they need not. One sees this variety with collective predicates in natural language, such as ‘Scientists agree that the Earth is warming’, or ‘The sailors quarrelled.’ There is no canonical dictionary semantics for what these things mean in terms of individual predication. Finally, there is a temporal aspect. Groups may change their composition, so their structure may be in flux, with members entering and leaving – and then, dynamic and temporal logics come into play.

Proof theory, one of the two main directions of logic, has been mostly concentrated on pure logic. There have been systematic reasons to think that such a limitation of proof theory to pure logic is inevitable, but about twelve years ago, we found what appears to be a very natural way of extending the proof theory of pure logic to cover also axiomatic theories. How this happens, and how extensive of our method is, is explained in this book. We have written it so that, in principle, no preliminary knowledge of proof theory or even of logic is necessary.

The book can be profitably read by students and researchers in philosophy, mathematics, and computer science. The emphasis is on the presentation of a method, divided into four parts of increasing difficulty and illustrated by any examples. No intricate constructions or specialized techniques appear in these; all methods of proof analysis for axiomatic theories are developed by analogy to methods familiar from pure logic, such as normal forms, subformula properties, and rules of proof that support root-first proof search. The book can be used as a basis for a second course in logic, with emphasis on proof systems and their applications, and with the basics of natural deduction and sequent calculus for pure logic covered in Part I, Chapter 2, and Part II, Chapter 6.

The axiomatizations of plane projective and affine geometry include the axiom of non-collinearity, i.e., of the existence of at least three non-collinear points. It is shown that this axiom, when converted into a suitable rule, is conservative over the other rules in the following sense: if an atomic formula is derivable by all the rules from a given finite number of atomic formulas used as assumptions, it is derivable without the rule of non-collinearity. (Thus, a proper use of existential axioms requires existential conclusions.) By the subterm property for the rules with non-collinearity excluded, derivability by the rules of projective and affine geometry is decidable.

As an immediate application of the decision method, we conclude that any finite set of atomic formulas is consistent. As a second application, we prove the independence of the parallel postulate in affine geometry: a very short proof search is exhaustive but fails to give a derivation. Thus, we see, within the system of geometry, that no derivation can lead to the parallel postulate.

It should be noted that the solution to the decision problem for projective and affine geometries applies only to derivations by the geometric rules. When logical rules are applied, to conclude logically compound formulas, the decision problem is known to have, by a result announced first in Tarski (1949), a negative solution. Finally, it should be noted that the decision methods presented here are provably terminating algorithms of proof search.

The preceding chapters took our study of rational agency from single update steps to mid-term activities like finite games that mix agents' actions, beliefs, and preferences. In the limit, this leads to long-term behaviour over possibly infinite time, that has many features of its own. In particular, in addition to information about facts, agents can now have procedural information about the process they are in. This chapter makes a junction between dynamic epistemic logic and temporal logics of discrete events, occurring in philosophy, computer science, and other disciplines. We prove semantic representation theorems, and show how dynamic-epistemic languages are fragments of temporal ones for the evolution of knowledge and belief. Amongst other things, this gives a better understanding of the balance between expressive power and computational complexity for agent logics. We also show how these links, once found, lead to merges of ideas between frameworks, proposing new systems of PAL or DEL with informational protocols.

The analysis of information and interactive agency in this book crucially involved ideas from computer science. Conversation and games are processes with stepwise changes of information states. And even without a computer around – only hearts and minds – these updates resemble acts of computation: sequential for individuals, and parallel for groups. Conversation as computation turned out a fruitful metaphor – and we recall some illustrations below. But ideas also flow the other way. One can take the logical dynamics stance to computer science, inverting the metaphor to computation is conversation. We illustrate the latter direction, too, ‘epistemizing’ and ‘gamifying’ various algorithmic tasks. This fits with how modern computer systems have moved away from single physical devices to societies of agents engaged in a wide variety of tasks. Thus, in the end, our two directions belong together. Humans and machines mix naturally in logical theory, witness the dynamics of email users in Chapter 4. And joint agency by humans and machines is also a cognitive reality in practice, as we will emphasize in Chapter 16.

Plato was not present on the day that Socrates drank hemlock in the jail at Athens and died. Phædo, who was, later related that day's conversation to Echecrates in the presence of a gathering of Pythagorean philosophers at Phlius. Once again, Plato was not around to hear what was said. Yet he wrote a dialog, “Phædo,” dramatizing Phædo's retelling of the occasion of Socrates' final words and death. In it, Plato presents to us Phædo and Echecrates' conversation, though what these two actually said he didn't hear. In Plato's account of that conversation, Phædo describes to Echecrates Socrates' conversation with the Thebian Pythagoreans, Simmias and Cebes, though by his own account he only witnessed that conversation and refrained from contributing to it. Plato even has Phædo explain his absence: “Plato,” he tells Echecrates, “I believe, was ill.”

We look to Socrates' death from a distance. Not only by time, but by this doubly embedded narrative, we feel removed from the event. But this same distance draws us close to Socrates' thought. Neither Simmias nor Cebes understood Socrates' words as well as Phædo did by the time he was asked to repeat them. Even Phædo failed to notice crucial details that Plato points out. Had we overheard Socrates' conversation, we would not have understood it. We look to Socrates' death from a distance, but to understand Socrates, we don't need to access him—we need Plato.

§ 1.Some historical background. The theorem known as “Tennenbaum's Theorem” was given by Stanley Tennenbaum in a paper at the April meeting in Monterey, California, 1959, and published as a one-page abstract in the Notices of the American Mathematical Society [28]. It is easily stated as saying that there is no nonstandard recursive model of Peano Arithmetic, and is an attractive and rightly often-quoted result.

This paper celebrates Tennenbaum's Theorem; we state the result fully and give a proof of it andother related results later. This introduction is in the main historical. The goals of the latter parts of this paper are: to set out the connections between Tennenbaum's Theorem for models of arithmetic and the Gödel–Rosser Theorem and recursively inseparable sets; and to investigate stronger versions of Tennenbaum's Theorem and their relationship to some diophantine problems in systems of arithmetic.

Tennenbaum's theorem was discovered in a period of foundational studies, associated particularly with Mostowski, where it still seemed conceivable that useful independence results for arithmetic could be achieved by a “handson” approach to building nonstandard models of arithmetic.

Husserl.

Conversation March 3, 1972. Husserl's philosophy is very different before 1909 from what it is after 1909. At this point he made a fundamental philosophical discovery, which changed his whole philosophical outlook and is even reflected in his style of writing. He describes this as a time of crisis in his life, both intellectual and personal. Both were resolved by his discovery. At this time he was working on phenomenological investigation of time.

There is a certain moment in the life of any real philosopher where he for the first time grasps directly the system of primitive terms and their relationships. This is what had happened to Husserl. Descartes, Schelling, Plato discuss it. Leibniz described it (the understanding or the system?) as being like the big dipper — it leads the ships. It was called understanding the absolute.

The analytic philosophers try to make concepts clear by defining them in terms of primitive terms. But they don't attempt to make the primitive terms clear. Moreover, they take the wrong primitive terms, such as “red”, etc., while the correct primitive terms would be “object”, “relation”, “well”, “good”, etc.

The understanding of the system of primitive terms and their relationships cannot be transferred from one person to another. The purpose of reading Husserl should be to use his experience to get to this understanding more quickly. (“Philosophy As Rigorous Science” is the first paper Husserl wrote after his discovery.)

Perhaps the best way would be to repeat his investigation of time. At one point there existed a 500-page manuscript on the investigation (mentioned in letters to Ingarden, with whom he wished to publish the manuscript).