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Table 5.6. The readings of corresponding to each reading of .
φ |
φ |
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It is necessarily true that φ |
It is possibly true that φ |
It will always be true that φ |
Sometime in the future φ |
It ought to be that φ |
It is permitted to be that φ |
Agent Q believes that φ |
φ is consistent with Q’s beliefs |
Agent Q knows that φ |
For all Q knows, φ |
After any execution of program P, φ holds |
After some execution of P, φ holds |
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5.3.1 The stock of valid formulas
We saw in the last section some valid formulas of basic modal logic, such as instances of the axiom scheme K: (φ → ψ) → ( φ → ψ) and of the schemes in (5.3). Many other formulas, such as
p → p
p → p
¬ p → ¬ p
are not valid. For example, for each one of these, there is a world in the Kripke model of Figure 5.3 which does not satisfy the formula. The world x1 satisfies p, but it does not satisfy p, so it does not satisfy p → p. If we add R(x2, x1) to our model, then x1 still satisfies p but does not satisfyp. Thus, x1 fails to satisfy p → p. If we change L(x4) to {p, q}, then x4 does not satisfy ¬ p → ¬ p, because it satisfies ¬ p, but it does not satisfy ¬ p – the path R(x4, x5)R(x5, x4) serves as a counter example. Finally, x6 does not satisfy , for this formula states that there is an accessible world satisfying , which is not the case.
If we are to build a logic capturing the concept of necessity, however, we must surely have that p → p is valid; for anything which is necessarily true is also simply true. Similarly, we would expect p → p to be valid in the case that p means ‘agent Q knows p,’ for anything which is known must also be true. We cannot know something which is false. We can, however, believe falsehoods, so in the case of a logic of belief, we would not expectp → p to be valid.
Part of the job of logic engineering is to determine what formula schemes should be valid and to craft the logic in such a way that precisely those ones are valid.
Table 5.7 shows six interesting readings for and eight formula schemes. For each reading and each formula scheme, we decide whether we should expect the scheme to be valid. Notice that we should only put a tick if the
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in Chapter 3 on CTL since this resulted in an easier presentation of our model-checking algorithms, but we might choose to model it otherwise, as in Table 5.7.
Ought. In this case the formulas φ → φ and φ → φ state that the moral codes we adopt are themselves forced upon us by morality. This seems not to be the case; for example, we may believe that ‘It ought to be the case that we wear a seat-belt,’ but this does not compel us to believe that ‘It ought to be the case that we ought to wear a seatbelt.’ However, anything which ought to be so should be permitted to be so; therefore, φ → φ.
Belief. To decide whether , let us express it as ¬ , for this is semantically equivalent. It says that agent Q does not believe any contradictions. Here we must be precise about whether we are modelling human beings, with all their foibles and often plainly contradictory beliefs, or whether we are modelling idealised agents that are logically omniscient – i.e. capable of working out the logical consequences of their beliefs. We opt to model the latter concept. The same issue arises when we consider, for example, φ → φ, which – when we rewrite it as ¬ ψ → ¬ ψ – says that, if agent Q doesn’t believe something, then he believes that he doesn’t believe it. Validity of the formula φ ¬φ would mean that Q has an opinion on every matter; we suppose this is unlikely. What aboutφ ψ → (φ ψ)? Let us rewrite it as ¬ (φ ψ) → ¬( φ ψ), i.e. (¬φ ¬ψ) → ( ¬φ ¬ψ) or – if we subsume the negations into the φ and ψ – the formula (φ ψ) → ( φ ψ). This seems not to be valid, for agent Q may be in a situation in which she or he believes that there is a key in the red box, or in the green box, without believing that it is in the red box and also without believing that it is in the green box.
Knowledge. It seems to di er from belief only in respect of the first formula in Table 5.7; while agent Q can have false beliefs, he can only know that which is true. In the case of knowledge, the formulas φ → φ and ¬ ψ → ¬ ψ are called positive introspection and negative introspection, respectively, since they state that the agent can introspect upon her knowledge; if she knows something, she knows that she knows it; and if she does not know something, she again knows that she doesn’t know it. Clearly, this represents idealised knowledge, since most humans – with all their hang-ups and infelicities – do not satisfy these properties. The formula scheme K is sometimes referred to as logical omniscience in the logic of knowledge, since it says that the agent’s knowledge is closed under logical consequence. This means that the agent knows all the