In this section we present the syntax of our logic, explain all the connectives: unary and binary, and also the unit constants. We explain each of these, with focus on the connectives that have epistemic or dynamic significance, leaving the other ones to our next chapter. The syntax of IDEAL is generated as follows
m ::= ⊥ | > |s|p|m∧m|m∨m|2MAm|fAM(m)|[q]m|m.q q ::= ⊥ |1|σ|q•q|q∨q |q∧q|q/q |q\q |2AQq|fAQ(q)
whereσ is in a setVQ of atomic actions,sis in a setVM of atomic propositions,Ais in the setAof
agents, andpis in the setΦof facts. 3.2.1 Binary Connectives
We denote propositions bymand actions byq. The logic consists of three sorts of binary connectives: 1. Between two propositions.These are the usual binary connectives on propositions, that is clas- sical conjunctionm1∧m2and disjunctionm1∨m2. Note that we do not have the propositional
implication connective.
2. Between two actions.These are also the usual action operations, that is sequential composition
q1 •q2 and non-deterministic choice of actions q1 ∨q2. There are two implications between
programs, referred to asresidualsin the literature,q1/q2 andq1 \q2, these are Galois adjoints
to the sequential composition discussed in the previous chapter. For the reader familiar with Linear Logic notation, we present the following table for the correspondence between our binary connectives and the connectives in Intuitionistic Non-Commutative Linear Logic:
Q-system Linear Logic
1 1 > > ⊥ 0 • · / ◦− \ −◦ ∨ ⊕ ∧ &
3. Between a proposition and an action. We have two connectives of this sort that are strongly related by adjunction. The first one is thePDLdynamic modality[q]mor the weakest precon- dition of PDL. Given an actionq and a propositionm, it returns the weakest proposition that should have been true before runningq so thatmbecomes true afterwards. This proposition is theweakestin the sense that it is entailed by any other propositionm0that has the same property. Dynamic modality can be seen as abeforeoperator with a backward nature: given the outputs, it provides with the weakest input that was truebefore the action and that caused this certain output. We have another binary connective, a new one, that complements the backward nature of dynamic modality[q]m. It is called updateand denoted asm.q, given a propositionm and an actionq, it tells you what proposition will be true afterq, and thus can be seen as anafter
operator. More explicitly, it says ifmis true at the input of actionq, thenm.qwill be true at the output ofq. The relation between the dynamic modality and update is as follows:
(m.q entails m0) iff (m entails [q]m0). (3.1) We call this relationdynamic adjunction; it says if an updated proposition m.qimplies another propositionm0, thenmimplies that after runningq, propositionm0holds and vice versa. 3.2.2 Unary Connectives or Modalities
We start with our epistemic modality on propositions2Am. This modality, which is the same as in
DEL, stands for knowledge or belief of agents. The knowledge modality is based on a more primitive notion, that ofappearancefA(m), which a new connective of our logic. It stands for all the propositions
that agentAconsiders possible when in factmholds in the real world. fAis monotone, that is if we
havem entails m0, then we also have thatfA(m) entails fA(m0). Knowledge can be derived from
appearance as follows: if the appearance ofmto an agentfA(m)entailsm0, then whenever proposition
mholds, thenAknows thatm0. The other direction also holds, if inmagentAknows thatm0, thenm
appears toAasm’, or the appearance ofmtoAentailsm0. Thus we have
(fA(m) entails m0) iff (m entails 2Am0). (3.2)
We call this relationepistemic adjunction, it is a Galois adjunction denoted asfA(m) a 2Am. This
relation is the base of our epistemic reasoning and motivates a slogan: ‘Our knowledge is the left adjoint to how the worldappearsto us.’
The same unary connectives are defined on actions, that is the appearance map of an agentA of an actionfA(q), that is all the actions that an agent thinks are running when in reality actionq is hap-
pening. The agent’s knowledge about the current action is denoted as2Aq. Defining having epistemic
modalities on actions is another novelty of our logic and will prove to be very useful in applications discussed in chapter five. When the context is not clear, we distinguish between the appearance of
the propositions and actions by denoting the former asfAQ and the latter asfAM, and similarly for the knowledge2QAand2MA. Note that these are our only unary connectives: we do not have negation. 3.2.3 Constants and Facts
True and False propositions are denoted by constants>and⊥. The true proposition>is the unit of conjunction and makes any disjunction true
m∧ >=m m∨ >=>
The false proposition⊥has more properties, it is the unit of disjunction and makes the conjunction false
m∨ ⊥=m , m∧ ⊥=⊥
Moreover, it appears as it is to all the agents
fA(⊥) =⊥
that is, if a contradiction holds in the real world, every agent knows about it. also makes every update false:
⊥. q=⊥
So if we live in a contradictory world, i.e.. one in which a contradiction⊥holds, then there is no action that can save us from this contradiction1.
On the action side, the ⊥or the false action is the most deterministic action and has properties similar to the false proposition. That is, it is the unit of choice of actions
⊥ ∨q=q
and makes the sequential composition of actions false
q• ⊥=⊥
and also its appearance as itself to every agent
fA(⊥) =⊥
The dual notion of most non-deterministic action is the denoted as>; this is the choice of all the actions and is the unit of conjunction of actions. But since conjunction of actions does not have any intuitive
1This goes against the para-consistent approach to logic e.g. Graham Priest’s logic, that say the world contains contradic-
tion. Since in our system the contradiction is contagious and will transfer from reality to all agents’ minds, reasoning about which will have no added value.
meaning, we will not discuss it in this chapter.
The action that does not do any thing or theskipaction is denoted as 1, it is the unit of sequential composition and update, that is
1•q =q•1 =q , m·1 =m
Not every proposition in an epistemic setting is epistemic. Some of propositions are objective in the sense that no epistemic action can change their truth value. These objective propositions are referred to asfactsand we denote them by letterp, the set of all the facts is denoted asΦ. No update can change the truth value of a factp ∈Φ, that is ifpis true before running an actionq, it will remain true after running the actionq. Thus for any factpand any actionq, we have:
p entails [q]p ,
or equivalently by dynamic adjunction:
(p.q) entails p .
We have now explained all of our connectives, so we move on to our sequent rules.