by Corry Shores
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[The following is summary. All boldface in quotations are in the original unless otherwise noted. Bracketed commentary is my own.]
Summary of
John Nolt
Logics
Part 4: Extensions of Classical Logic
Chapter 11: Leibnizian Modal Logic
11.2 Leibnizian Semantics
11.2.2 [Possible worlds, actual worlds, redefining logical concepts for possible worlds, and some important theorems in possible world semantics]
Brief summary:
In our possible world semantics, no one possible world is uniquely actual. Rather, from the perspective of a world, it is itself actual and the others possible. However, likewise, from the perspective of a second possible world, the first along with the others are possible, and second is itself actual. This is called the indexicality of actuality. For our possible world semantics, we need to revise our definitions for many concepts to incorporate the element of possible worlds.
DEFINITION A formula is valid iff it is true in all worlds on all of its valuations.
DEFINITION A formula is consistent iff it is true in at least one world on at least one valuation.
DEFINITION A formula is inconsistent iff it is not true in any world on any of its valuations.
DEFINITION A formula is contingent iff there is a valuation on which it is true in some world and a valuation on which it is not true in some world.
DEFINITION A set of formulas is consistent iff there is at least one valuation containing a world in which all the formulas in the set are true.
DEFINITION A set of formulas is inconsistent iff there is no valuation containing a world in which all the formulas in the set are true.
DEFINITION Two formulas are equivalent iff they have the same truth value at every world on every valuation of both.
DEFINITION A counterexample to a sequent is a valuation containing a world at which its premises are true and its conclusion is false. |
DEFINITION A sequent is valid iff there is no world in any valuation on which its premises are true and its conclusion is not true.
DEFINITION A sequent is invalid iff there is at least one valuation containing a world at which its premises are true and its conclusion is not true.
(Nolt 319-320)
Some important metatheorems for possible world semantics are:
METATHEOREM: Any sequent of the form □Φ⊦Φ is valid.
(Nolt 320)
[From something being necessarily true we can conclude that it is true in the actual world.]
METATHEOREM: Any formula of the form ◊Φ↔~□~Φ is valid.
(Nolt 321)
[If something is possible then it is not the case that it is necessarily not so. This structure works for necessity to: □Φ↔~◊~Φ.]
METATHEOREM: Every sequent of the form α = β ⊦ □α = β
(Nolt 323)
[Because of rigid designation and transworld identity, if two names designate the same object in the actual world, then they designate the same object in every possible world.]
METATHEOREM: The sequent ‘∃xFx ⊦ ∃x◊Fx’ is valid.
(Nolt 323)
[If there is something with a certain property, then we can conclude that there is something for which it possibly has that property. In other words, whatever actually has a property also possibly has it.]
METATHEOREM: The sequent ‘◊∃xFx ⊦ ∃x◊Fx’ is invalid.
(Nolt 324)
[Suppose it is possible that there is something that takes some predicate. That means in some world, not necessarily our actual one, something does take that predicate. From this we cannot conclude that in our actual world there is something that possibly takes that predicate. For, were it for example the case that nothing in our actual world in fact takes that predicate, it is not possible that something can take it.]
Summary
[Previously we learned how to set up possible world models for evaluation.] Nolt explains that our possible world semantics is “democratic” in the sense that it “treats all possible worlds as equals; none is singled out as uniquely actual” (Nolt 318). [The next idea is a bit odd, and I will quote the text. It seems to be that there is a view in modal metaphysics which says that from the perspective within any possible world, it itself is the actual one, and the others are possible. Thus our world, which we think is actual, is only possible from the perspective of the other possible worlds.]
Our semantics is democratic: It treats all possible worlds as equals; none is singled out as uniquely actual. This models another prominent idea in modal metaphysics: the thesis of the indexicality of actuality. According to this doctrine, no world is actual in an absolute sense; each is actual from a perspective within that world but not from any perspective external to it. For those whose perspective (consciousness?) is rooted in other possible worlds, our world is merely possible, just as their worlds are merely possible for us. Actuality, then, is indexed to worlds (world-relative) in just the way truth is.
(Nolt 318, underline mine)
[Nolt’s next point is a little complex, so I might get it wrong. He notes that some logicians do not subscribe to the thesis of the indexicality of actuality. They instead think that there should always be one of the possible worlds designated as actual to the absolute exclusion of the others. But Nolt notes a problem with this, which I may misconstrue. The problem seems to be the following. Such logicians who designate one world as actual seem to be saying that all things that are true in that world really are true. They are not necessarily true on account of what is going on or not going on in other worlds. However they still define possibility and necessity using the relative notion of truth that things in other worlds can be understood as being true too. Let me quote, as I am not following well enough.]
The thesis of the indexicality of actuality is much disputed. Logicians who think that actuality is not indexical may incorporate this idea into their semantics by designating exactly one world of each model as actual. But this bifurcates their concept of truth. They have, on the one hand, a notion of nonrelative or actual truth – that is, truth in the actual world – and, on the other, the same relative notion of truth (truth-in-a-world) that we use in defining possibility and necessity. I use the indexical conception of actuality here because it requires only one sort of truth (world-relative) and hence yields a simpler semantics.
(318)
[We still might have trouble understanding how actuality can be indexical. How can all worlds be equally actual to themselves but possible for other worlds? Does not just one actuality get to be the actual actuality? How can some actuality also be just a possibility? To help us understand this, he has us think of how God stands outside the flow of time. The question of “what is the present moment?” does not make sense from God’s perspective. For, “Presentness is indexed to moments of time – that is, relative to temporal position.” We then use this idea analogically. For a God who exists outside all worlds, none is the actual actual one in an absolute sense. (Now, suppose God were to enter into one particular present somehow. That would make all other moments not present in relation to that moment. In the same way, when we take the perspective of one world being actual, the others become possible.)
Those who find the indexicality of actuality dizzying may appreciate the following analogy. Imagine you are a transcendent God, perusing the actual universe from creation to apocalypse. As you contemplate this grand spectacle, ask yourself: Which moment is the present?
In your omniscience you should see at once that this question is nonsensical. There is a present moment only for creatures situated within time, not for a God who stands beyond it. The present moment for me at noon on my third birthday is different from the present moment for me as I write these words, which is different from the present moment for you as you read this. None of these is the present moment, for there is no absolute present. Presentness is indexed to moments of time – that is, relative to temporal position. If I have lived or will live at a given moment, then that moment is present to the temporal part of me that intersects it but not present to other temporal parts of me. |
Analogously, according to the understanding that grounds our semantics, there is an actual world only for creatures situated within worlds, not for a God – or a modal semanticist – standing beyond them. A world in which I become a farmer is just as actual for that farmer (i.e., for that possible “part” of me) as the world I am currently experiencing is for the professorial portion of me that inhabits it. Neither of these, nor any other, is the actual world in some absolute sense, because actuality is always relative to a perspective within some possible world.
That, at any rate, is one way of understanding the “democratic” semantics presented here: Models do not single out an actual world, because our model theory operates from a perspective beyond worlds from which no world is uniquely actual.
(Nolt 318-319)
Now that we have relativized truth to worlds, we need to refine our definitions for valuations, as that truth or falsity is now world relative.
Having relativized truth to worlds, we must make compensatory adjustments in those metatheoretic concepts that are defined in terms of truth. Consistency, for example, is no longer merely truth on some valuation (model), for formulas are no longer simply true or false on a valuation; they are true or false at a world on a valuation. Thus we must revise our definitions of metatheoretic concepts as follows :
DEFINITION A formula is valid iff it is true in all worlds on all of its valuations.
DEFINITION A formula is consistent iff it is true in at least one world on at least one valuation.
DEFINITION A formula is inconsistent iff it is not true in any world on any of its valuations.
DEFINITION A formula is contingent iff there is a valuation on which it is true in some world and a valuation on which it is not true in some world.
DEFINITION A set of formulas is consistent iff there is at least one valuation containing a world in which all the formulas in the set are true.
DEFINITION A set of formulas is inconsistent iff there is no valuation containing a world in which all the formulas in the set are true.
DEFINITION Two formulas are equivalent iff they have the same truth value at every world on every valuation of both.
DEFINITION A counterexample to a sequent is a valuation containing a world at which its premises are true and its conclusion is false. |
DEFINITION A sequent is valid iff there is no world in any valuation on which its premises are true and its conclusion is not true.
DEFINITION A sequent is invalid iff there is at least one valuation containing a world at which its premises are true and its conclusion is not true.
(Nolt 319-320)
Nolt will now use these concepts to work through a series of metatheorems. [Whenever I quote a metatheorem, I will normally exclude the proofs, which can be found at the cited pages.] The first metatheorem “confirms the truism that what is necessary is the case.”
METATHEOREM: Any sequent of the form □Φ⊦Φ is valid.
(Nolt 320)
[I am not exactly sure what it means for something to be the case in this situation, when we are not referring to possible worlds. I am guessing the idea is that if something is necessarily true, then it is true in any given world. (From what is said later, I also get the impression that it means true in the actual world.)] The converse of this theorem does not hold, because whatever is the case [in some world or in the actual world] is not necessarily the case [that is, true in all worlds.] Hence the next metatheorem:
METATHEOREM: The sequent ‘P⊦□P’ is invalid.
(Nolt 320)
[The next metatheorem says that when something is necessarily true in one world, it is necessarily true in all worlds. And when it is necessarily true in all worlds, it is necessarily true that it is true in all worlds. But I might not be following, so see the quote.]
On Leibnizian semantics what is necessary at one world is necessary at all; therefore, what is necessary is necessarily necessary. This is because necessity itself | is truth in all worlds, and if something is true in all worlds, then it is true in all worlds that it is true in all worlds. The following metatheorem gives the details:
METATHEOREM: Any sequent of the form □Φ⊦□□Φ is valid.
(Nolt 320-321)
[Nolt then notes something about the prior proof, which I excluded. He used other world variables like w, u, x, and y. He says that “Each such variable should be introduced with a metalinguistic quantifier to indicate whether it stands for all worlds or just some” (321). As this concerns proofs, I am leaving it out, but see p.321 more details.]
[Recall duals from section 11.1: Alethic modifiers are often duals meaning that one can be converted into another by adding negations around the symbols, as for example:
□Φ↔~◊~Φ
◊Φ↔~□~Φ
] The next metatheorem [and its proof, here excluded] “proves one of the two biconditionals expressing the idea that ‘□’and ‘◊’ are duals.
METATHEOREM: Any formula of the form ◊Φ↔~□~Φ is valid.
(Nolt 321)
Nolt now further develops the notion of rigid designation. He says that one important consequence of it is the necessity of identity. This is the metatheorem.
METATHEOREM: Every sequent of the form α = β ⊦ □α = β
(Nolt 323)
[The idea I think is that if there is some designation in one world, that designation holds for all worlds. Nolt explains that this combines the concepts of rigid designation and trans-world identity. Perhaps what is meant by this is that in all worlds, both something’s identity and its designations are the same. Nolt gives Kripke’s example of Hesperus = Phosphorus. Venus had two names, depending on whether it was seen as the first visible star at dawn or the last at dusk (there is a diagram from this at the entry for Mates’ Stoic Logic §6). Possibly the ancients could have regarded them as two stars, Nolt observes. But now we know that the two names are for the same object. Now if names are rigid designators, meaning they will designate the same object in each world, and if the object which is Venus has transworld identity, making it the same object in each world, then it is necessarily true that for all worlds Hesperus = Phosphorus. Nolt then considers another view, which would say that this equality of terms is not necessary. For, the ancients may not have regarded the two as the same. I am not sure how this applies to the way necessity works for possible worlds however. Nolt seems to clear that up by saying that in fact it does not matter, because we are dealing with different sorts of possibilities. It is not possible that on another world Hesperus would not equal Phosphorus. I am not sure why, but maybe it is for the following reason. It would require that in the solar system there are two planets like Venus. But then we are not talking about the same object any more, so the comparison becomes impossible between a world where there is just one Venus. I am really not sure. At any rate, Nolt clears up the problem by saying that in the case of the ancients making the distinction, we would say that it is epistemically possible for the names to not equal one another, but it is not genuinely (that is, alethically) possible.]
One of the most important consequences of the doctrine that names are rigid designators is the thesis expressed in the next metatheorem: the necessity of identity. Kripke, who popularized this thesis in its contemporary form, illustrates it with the following example. ‘Phosphorus’ is a Latin name for the morning star; ‘Hesperus’ is the corresponding name for the evening star. But the morning star and the evening star are in fact the same object, the planet we now call Venus. Hence the statement
Hesperus = Phosphorus
is true. Now if names are rigid designators, then since this statement is true, the object designated by the name ‘Hesperus’ in the actual world is the very same object designated by ‘Hesperus’ in any other world, and the object designated by the name ‘Phosphorus’ in the actual world is the same as the object designated by that name in any other world. Thus in every world both names designate the same object they designate in the actual world: the planet Venus. So ‘Hesperus = Phosphorus’ is not only true in the actual world but necessarily true.
Yet this conclusion is disturbing. So far as the ancients knew, Hesperus and Phosphorus could have been separate bodies; it would seem, then, that it is not necessary that Hesperus = Phosphorus. But this reasoning is fallacious. The sense in which it was possible that Hesperus was not Phosphorus is the epistemic sense; it was possible so far as the | ancients knew – that is, compatible with their knowledge – that Hesperus was not Phosphorus. It was not genuinely (i.e., alethically) possible. The planet Venus is necessarily itself; that is, it is itself in any possible world in which it occurs. And if names are rigid designators, then the names ‘Hesperus’ and ‘Phosphorus’ both denote Venus in every world in which Venus exists. Hence, given the dual doctrines of transworld identity and rigid designation (both of which are incorporated in our semantics), it is alethically necessary that Hesperus is Phosphorus, despite the fact that it is not epistemically necessary.
(Nolt 322-323)
[In the following, I find that I am unsure what it means when a formula does have a modal operator in the context of others that do. From what Nolt says, it seems that when there are no operators, it means we are making a statement about the “actual world”. But that is something relative, so does it mean for some world? At any rate, the idea will be that if something has the property F, then that something is possibly F.] The next two metatheorems will illustrate the interaction between quantifiers and modal operators.
METATHEOREM: The sequent ‘∃xFx ⊦ ∃x◊Fx’ is valid.
(Nolt 323)
[Let us work gradually through the proof.]
PROOF: Suppose for reductio that ‘∃xFx ⊦ ∃x◊Fx’ is not valid.
[We will prove the theorem by assuming its negation. We will then show that this leads to a contradiction, which will allow us to confirm the original formulation.]
Then there is some valuation v and world w of v such that v(‘∃xFx’, w) = T and v(‘∃x◊Fx’, w) ≠ T.
[So in order for this inference to be invalid, the premises would need to be true and the conclusion false. I am not sure, but since the conclusion involves existential quantification and the possibility operator, perhaps that means there cannot be any world where it is true. For if there is at least one world where it is true, then it is possibly true on any world.]
Since v(‘∃xFx’, w) = T, it follows by rule 10 that for some potential name α of some object d in Dw, v(α,d) (Fα, w) = T.
[First recall rule 10 from section 11.2.1.
10 .
v(∃βΦ, w) = T iff for some potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for some potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T;
(Nolt 315)
So as we can see, because we already have v(‘∃xFx’, w) = T, that means we have the first part of the biconditional being true, namely, v(∃βΦ, w) = T. That means the right side of the biconditional is true. Thus what is true is: for some potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T. Using the terms we have in our proof, that means the following is true: for some potential name α of some object d in Dw, v(α,d) (Fα, w) = T. As we can see, we have substituted name α in (which designates object d) for the variable x. In other words, we are saying that because in a world there is some thing that is F, that means some specific thing with some name is F in that world.]
So for some world u (namely w) in Wv, v(α,d) (Fα, u) = T.
[This may seem redundant. But I think the idea is that the w was the name for some specific world, while the u is a variable for worlds. But I am not sure. So it seems that we are saying that since on some specific world a specific object is F, that means there is some world where it is so.]
But then by rule 12, v(α,d) (◊Fα, w) = T.
[So rule 12 is:
12 .
v(◊Φ, w) = T iff for some worlds u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for some worlds u in Wv, v(Φ, u) ≠ T.
(Nolt 316)
In this case, we seem to have affirmed the right side of the biconditional by saying that for some world u (namely the world named w) there is a valuation that makes a proposition true in that world. It therefore is possibly true in some given specific world (in this case, w). So in other words, because it is true in some world, it is possibly true in some specific world.]
Hence, since d is in Dw, by rule 10, v(‘∃x◊Fx’, w) = T, contrary to what we had supposed above.
Thus we have established that ‘∃xFx ⊦ ∃x◊Fx’ is valid. QED
[Rule 10 again was:
v(∃βΦ, w) = T iff for some potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T
Here it seems the basic idea is that since it is true for some specific object, then we can say there is an object such that it is F. But this goes against our assumption that it is false that there is some thing for which it is possible that it is F. So we have shown that the negation of our assumption is false, and thus our original theorem is true.]
The sequent says that given that something is F, it follows that something (that very same thing, if nothing else) is possibly F. This is a consequence of the fact that the actual world, which we may think of as w – and also u – in the proof, is also a possible world, so that whatever actually has a property also possibly has | it. In the proof, the object which actually has the property F is object d. Since d has F in w, d has F in some possible world, i.e., possibly has F. It follows, then, that something possibly has F. This enables us to contradict the reductio hypothesis.
(323-324)
The next metatheorem is:
METATHEOREM: The sequent ‘◊∃xFx ⊦ ∃x◊Fx’ is invalid.
(Nolt 324)
[The idea here seems to be the following. Suppose there are two possible worlds, and we are oriented on the first one. And suppose further that there is something true on the other possible world that is not true on ours. For example, bananas are blue there but they are not here. Since they are true on at least one world, it is possibly true that there is a blue banana. However, that does not mean that any bananas in our world are possibly blue. I am not sure I follow, but maybe this is like saying, “it is not possible that any bananas in our world are blue”; for, in fact, none of them are. So we can look and look, but we will never find one.]
Our final metatheorem shows that from the fact that it is possible something is F, it does not follow that the world contains anything which itself is possibly F. Suppose, for example, that we admit that it is (alethically) possible that there are such things as fairies. (That is, there is a possible world containing fairies.) From that it does not follow that there is in the actual world anything which itself is possibly a fairy.
(Nolt 324)
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
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