Determinación de carga microbiana de los productos desarrollados

J1, 3A∧3B 3A, whence by J23A∧3BA. Similarly, 3A∧3BB. By axiomS,3A∧3BA?B, and 5 follows by J4. 2 Recall that according to Theorem 33 the sentence3A∧3B→3(A_?B) is true, and according to Theorem 37 it is not provable inILMS. In the context ofGL, we get a logic that proves all true sentences by simply taking all theorems ofGL, adding the reflection principle2A→A (or by contraposition,A→3A) and allowing modus ponens as the only rule of inference. The resulting system GLS is the provability logic of sentences that are true (in the standard model)1. One might wonder whether the same strategy gives all true sentences for the system ILMS. Item 5 in Lemma 1.1 suggests that this could indeed be the case. By reflection principle, we have

3A∧3B →3(3A∧3B), whence using 5, we get3A∧3B →3(A_?B). 2. Axiom S and Structural Properties of Models

In this section, we will try to find a structural characterization of ILM–frames or

ILM–models validating axiom S. As the possibility of such a characterization will be ruled out in Section 4 below, the reader should not get too attached to the material presented in this section. Instead, it should be seen as preliminary to the negative result of Section 4.2, and as an illustration for how the natural strategies for finding a modal semantics break down in the case ofILMS.

2.1. Minimal conditions. When encountering a new axiom in the context of modal logic, it is natural to ask whether it characterizes some nicely describable class of frames. We will now try to answer this question for the axiomS.However, as we do not have any conditions for a node to satisfy a formula of the formA?B (to investigate whether such conditions can be found is the purpose of the current chapter!), we will not arrive at fully fledged frame conditions. Instead, we will obtain certain minimal structural conditions that are sufficient for an ILM–model to validate axiomS.

Let K be the class ofILM–frames. Since we want our hypothetical ILMS–frames also to satisfy ILM, our question is whether there is some reasonable classK? of

frames withK?⊆ K, and such that

F ∈ K? ⇒ FS.

Without having a structural truth condition for_?–formulas, it is not clear how one should proceed to investigate the other direction of frame characterizability. In this section, as well as in Chapter 5, we shall only be concerned with the direction of frame characterizability depicted above2_{. So letMbe anILM–model. We will see} what structural characteristics Mshould possess in order for us to conclude that axiomS is valid inM.

First, consider the direction ofSwhich says that (A_?BA)∧(A_?BB). Suppose that M, xA_?B for somex∈ M. Using the semantics for , we see that for 1_{For more about the systemGLS, see [Boo93].}

2_{Under some circumstances, this might be the only direction we need. Suppose that we have a} class of framesK?withF ∈ K?⇒ F S, butF S⇒ F ∈ K0 for someK0 6=K?, where of

courseK?⊂ K0. Now if there is no modal formula that is valid inK?but not inK0, we might

2. AXIOMSAND STRUCTURAL PROPERTIES OF MODELS 48

anyws.t.wRx, there should bey, y0 withxSwyAandxSwy0B. Thus we get

the first minimal condition:

(69) xA_?B⇒ ∀w[wRx⇒ ∃y, y0(xSwyA ∧ xSwy0B)].

Suppose that we are able to find a condition C(A, B) for a node x in a model to satisfy the formula A_?B. Then the first minimal condition would become: if

xC(A, B), then for allwwithwRx, there arey, y0withxSwyAandxSwy0 B.

For the other direction of axiom S, let w (CA)∧(CB), and let xbe s.t.

wRx C. Then we have y, y0 with xSwy A and xSwy0 B. We want to

conclude that there is some z withxSwz andz A?B. Thus we get the second minimal condition3:

(70) xSwyA∧xSwy0B⇒ ∃z(xSwzA?B).

Using our hypothetical condition C(A, B) again, (70) becomes: if xSwy A and xSwy0 B, then there existsz withxSwz andzC(A, B).

IfMis anILM–model satisfying the minimal conditions (69) and (70), then clearly MS. We would want to find a conditionCwith two free variables, and such that ifMis anILM–model and we define for allx∈ M,

M, xA_?B:⇔xC(A, B),

thenMsatisfies the minimal conditions (69) and (70) above. It would follow that M S, and thus we would have a notion of an ILMS–model (i.e. an ILM–model where the truth values of _?–formulas are defined using the condition C). In the following, we will look more closely at what such a conditionC could be.

2.2. Relational semantics. In the context of modal logic, it is natural to expect the conditionC(A, B) to mention the truth ofA andB at nodes bearing a certain (fixed) relation toxin the underlying frame of the model M. In this way, the truth value of A_?B at xis completely determined by the truth values of A

andB at nodes that stand in the relevant relation tox. If we are able to find such a conditionC, we say that we have arelational semanticsfor?. The semantics for

2and foris relational in this sense.

The only thing we require fromCis that it only uses the relations of the underlying frame of the model. These could be R- or S-relations. However, we also allow

new relations to be added toILM–frames in order to deal with the new connective ?. We will see in Chapter 5 how such a freedom could in principle be used to get a relational semantics forILMS. We explore a semantics that arises when adding to the ILM–frames a new relation Q, and using it to define the truth values of ?–formulas.

Given a condition C as described above, the minimal conditions from Section 2.1 can be transformed into actual frame conditions. The first one would say that if

xstands in the specified relation to nodes y and y0, then for all wwith wRx, we have that xSwy and xSwy0. The second one would say that if xSwy and xSwy0,

then there is somezwithxSwz, andz stands in the specified relation toy andy0.

3_{See Section 4.1 for a discussion of the argument by which the second minimal condition is} obtained.