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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
11.2.1 [basic set-up and evaluation of Leibnizian possible worlds]
Brief summary:
Evaluating formulas in modal logic involves possible world modeling. A possible world is, intuitively speaking, an alternate world-situation where things are slightly or drastically different from our own. Formally, a possible world is a combination of a domain of objects along with valuative functions that assign names to these objects, assign ordered n-tuples to n-place predicates, and assign truth values to well-formed formulas. When evaluating modal operators, we consider a set of worlds, each with its own valuation schemes and domains. When a formula is true in all such worlds under consideration, it is necessary. And when it is true in some worlds, it is possible. The following defines possible worlds and their valuations.
DEFINITION A Leibnizian valuation or Leibnizian model v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds of v.
2. For each world w in Wv a nonempty set Dw of objects, called the domain of w.
3. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of the domain of at least one world.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F. |
iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.
(Nolt 314-315)
Valuation Rules for Leibnizian Modal Predicate Logic
Given any Leibnizian valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T; |
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u in Wv, v(Φ, u) ≠ T;
12 .
v(◊Φ, w) = T iff for some world u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T.
(Nolt 315-316)
Summary
Leibniz “was among the first to investigate the logic of alethic operators” (Nolt 310). This logic was based on the interesting metaphysical claim that ours is just one of many possible worlds.
His semantics for modal logic was founded upon a simple but metaphysically audacious idea: Our universe is only one of a myriad possible universes, or possible worlds. Each of these possible worlds comprises a complete history, from the beginning (if there is a beginning) to the end (if there is an end) of time.
(Nolt 310)
Nolt then has us consider how possible worlds can be understood in terms of the different possibilities for how our day may unfold, depending on the decisions we make along with other contingent factors.
Such immodest entities may rouse skepticism, yet we are all familiar with something of the kind. I wake up on a Saturday; several salient possibilities lie before me. I could work on this book, or weed my garden, or take the kids to the park. Whether or not I do any of these things, my ability to recognize and entertain such possibilities is a prominent feature of my life. For ordinary purposes, my awareness of possibilities is confined to my doings and their immediate effects on the people and things around me. Yet my choices affect the world. If I spend the day gardening, the world that results is a different world than if I had chosen otherwise. Leibnizian metaphysics, then, can be seen as a widening of our vision of possibility from the part to the whole, from mere possible situations to entire possible worlds.
(310)
[A great cartoonist, Ruben Bolling, depicted this structure many years ago with his “Infinite Universes of Bob”.
(Rubin Bolling, Tom the Dancing Bug, 10-May-1998)
]
In Leibniz’ theodicy, God calculates all the [infinitely?] many possible worlds. [Many of them are logically impossible, because certain facts are logically incompatible, that is to say, they are incompossible. There are a certain number of worlds which do not contain logical impossibilities.] Among these God chose the best of all possible worlds, which is ours (310).
Logicians use Leibniz’ notion “to adumbrate an alethic modal semantics” (Nolt 310). [Something is necessarily true if it is true in all possible worlds, and it is possibly true if it is true in at least one possible world.]
On Leibniz view:
□Φ is true if and only if Φ is true in all possible worlds.
| and
◊Φ is true if and only if Φ is true in at least one possible world.
(Nolt 310-311)
Nolt then relates the necessity and possibility operators to universal and existential ones, since
to say that it is necessary that 2 + 2 = 4 is to say that in all possible worlds 2 + 2 = 4; and to say that it is possible for the earth to be destroyed by an asteroid is to say that there is at least one possible world (universe) in which an asteroid destroys the earth.
(Nolt 311)
Nolt says that we can extend what Leibniz says to other sorts of modal operators.
Deontic operators are like quantifiers over morally possible (i.e., permissible) worlds – worlds that are ideal in the sense that within them all the dictates of morality are obeyed (exactly which morality is a question we shall defer!). Epistemic operators are like quantifiers over epistemically possible worlds – that is, over those worlds compatible with our knowledge (or, more specifically, with the knowledge of a given person at a given time). And tense operators act like quantifiers too – only they range, not over worlds, but over moments of time.
(Nolt 311)
[Nolt continues to show analogies between the different types of modalities. See p.311.]
Nolt will now “reformulate Leibniz’ insight about alethic operators in contemporary metatheoretic terms” (311).
[Before we continue, let us first review some ideas about models from Agler’s Symbolic Logic, section 6.4. A set is a collection of members. In the example below, set M is the set of odd integers from 1 to 5.
M = {1, 3, 5}
(Agler 263)
We can build “models”, which have two parts, namely, a domain, which tells us what objects we are dealing with, as in the sets above, and an interpretation function, which interprets formulas in relation to the domain (Agler 264). The interpretation function {1} assigns objects in the domain to names, {2} assigns sets of n-tuples to n-place predicates, and {3} assigns truth values to sentences. Let us consider an example.
Domain:
D = {Alfred, Bill, Corinne}
Interpretation Function assigning objects in the domain to names:
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
Interpretation Function assigning sets of n-tuples to n-place predicates:
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>, etc.}
Interpretation Function assigning truth values to sentences:
v(Lca) = T
v(Lcb) = F
etc.
In light of Nolt’s rule 2 below, let me construe the situation with Agler’s example in the following way. First, let me just call up Nolt’s formulation for reference:
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
The idea is that we determine valuations like v(Lca) = T by doing two things. First we look at the assignments of n-tuples to the predicate:
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>}
(Note, we will treat interpretation and truth-valuation functions similarly, so these Is from Agler can be thought of as v’s too.) Then we look at some formulas we want to evaluate for truth, say for example Lca and Lcb. We then form n-tuples by using the name assignment valuations. In our case, they were:
I (a) = {Alfred}
I (b) = {Bill}
I (c) = {Corinne}
So let us start with Lca. First I will structure it like in Nolt’s formulation, which was: <v(α1), ... , v(αn)> .
<v(c), v(a)> or
<Corinne, Alfred>
We then look again at the n-tuple assignments for the predicate:
I (Lxy) = {<Bill, Corinne>, <Corinne, Alfred>}
We see that <Corinne, Alfred> is among them. Therefore, v(Lxy) is true. What about Lcb? Let us follow the same procedure.
v(Lcb) = ?
<v(c), v(b)> or
<Corinne, Bill>
v(Lcb) = F
For, <Corinne, Bill> is not in the extension for the predicate L.
So, Nolt will be building models, except now the valuations are different for different worlds. What will essentially make one possible world distinct from another is that different things are true in that world.] Consider some particular valuation for predicate logic [take for example the one from Agler above.] This “in effect models a single world” (Nolt 311) [, as it gives one particular set of truths and falsities that define some world.] Such a model for a single world “consists of a domain and assignments of appropriate extensions to predi-|cates and names within that domain” (Nolt 311-312). But since in modal logic we have many possible worlds, each one will have its own domain, and each world should have its own assignments to predicates (312). Nolt will illustrate with a model for three worlds, called w1, w2, and w3. Each one contains the following objects.
| World | Domain |
| w1 | {α, β, γ, δ} |
| w2 | {β, γ} |
| w3 | {α, δ, ε} |
(Nolt 312)
Nolt will have us suppose that we have a one-place predicate ‘B’, which can mean ‘is blue’. [The idea seems to be like the following. We have these objects in our domain. Perhaps we might think of one thing being a fruit like a banana. In our world, bananas are not blue. But in another possible world, they could be. In our world, certain things are in the set of blue things. And were we to consider any one thing, we can say that it is true or it is false that it is blue.] Since in each world the predicate may apply to different things, we need to assign separate sets for the predicate ‘B’, one for each world.
For each world w, the set assigned to ‘B’ at w then represents the things that are blue in w. Suppose we assign to ‘B’ the set {α, β} in w1, { } in w2, and {α, δ, ε} in w3. Then, according to our model there are two blue things in w1 and none in w2, and in w3 everything is blue.
(Nolt 312)
This means that we need a different valuation assignment for each one. So in our model:
v(‘B’, w1) = {α, β}
v(‘B’, w2) = { }
v(‘B’, w3) = {α, δ, ε}
(Nolt 312)
We similarly evaluate truth for each world. There are blue things in w1 but not in w2. So
v(‘∃xBx’, w1) = T
but
v(‘∃xBx’, w2) = F
(Nolt 312)
We also can assign truth values to sentence letters for each world.
Let ‘M’, for example, mean “there is motion.” We might let ‘M’ be true in w1 but not in w2 or w3. Thus v(‘M’, w1) = T, but v(‘M’, w2) = v(‘M’, w3) = F.
(Nolt 312)
[The next idea is a little more complex. Regardless of the world, each name will be assigned to the same object. Nolt gives the example of names over time in one world. Our name refers to our child self long ago, to our current self, and to our future self. In that way, we are spread out over time and we have the same name. By analogy, a name for a thing could be understood as being “spread out” across different worlds.]
We shall assume, however, that names do not change denotation from world to world. Thus we shall assign to each name a single object, which may inhabit the domains of several possible worlds, and this assignment will not be world relative. This models the metaphysical idea that people and things are “spread out” through possibilities, just as they are “spread out” through time. With respect to time, for example, the name ‘John Nolt’ refers to me now, but also to me when I was a child and to the old man whom (I hope) I will become. I occupy many | moments, and my name refers to me at each of these moments. Analogously, I have many possibilities, and my name refers to me in each. When I consider that I could be a farmer, part of what makes this possibility interesting to me is that it is my possibility. It is I, John Nolt, who could be a farmer; my name, then, refers not only to me as I actually am, but to me as I could be. I am a denizen of possibilities (that is, possible worlds), as well as times, and my name tracks me through these possibilities, just as it does through the moments of my life.
(312-313, boldface and underlining mine)
Since names “refer to the same object in each world in which they refer to anything at all,” they have been called rigid designators, “due to Ruth Marcus and Saul Kripke” [footnote: See Kripke’ s Naming and Necessity (Cambridge: Harvard University Press, 1972)] (Nolt 313). Nolt then explains how we would model rigid designation:
In our semantics we shall model rigid designation by representing the value assigned to a name a simply as v(α), rather than as v(α, w), which would represent the value assigned to α at a world w. The omission of the world variable indicates that the denotations of names are not world-relative.
(Nolt 313)
[In other words, I think, when we use v(α, w), we mean the object assigned to in some particular word, but when we use v(α), we mean that the name gets the same object in all worlds. I am not sure about that, however.]
[So we have understood that a name can track something that is spread out over different worlds (or times). And in each world, that thing being tracked may have such different properties as to call into question whether or not they both deserve the same name. What is important here is a concept of identity that is at work. The thing in different worlds is not necessarily qualitatively identical in each world. Is it numerically identical? I am not sure, but I would think not, as we have something in another world with different properties, so how can it be numerically identical? But then, what kind of indenticality does it have among its possible world variations? Perhaps what makes them identical is simply the fact that we have assigned the same name to them. Or perhaps the identity comes from their differential relations to all the other things; so in other words, an object in many worlds is like a place-holder that can be determined very differently in each case, but still maintains a distinctness of a relational kind to the other things. The specific concept Holt uses for this sort of identity is transworld identity, which is “the idea that the same object may exist in more than one possible world”. But this I still find confusing, because we are saying it is the “same” object in each world, but we have not established what makes it the same yet, and the definition for its transworld identity is based on that undefined notion of “same”. Maybe this requires a substance-property notion of objects, where somehow an object in different worlds is substantially the same even if it has utterly different properties. But even in that case I do not know how to understand two objects within one world being substantially different from one another, while they are substantially the same with their other possible-world counterparts that may not resemble them at all.]
The concept of rigid designation harbors a metaphysical presupposition: the doctrine of transworld identity. This is the idea that the same object may exist in more than one possible world. It is modeled in our semantics by the fact that we allow the same object to occur in the domains of different worlds. Most logicians who do possible worlds semantics take transworld identity for granted, though there are exceptions.
Though a rigidly designating name refers to the same object in different worlds, that object need not be “the same” in the sense of having the same properties. I would have quite different properties in a world in which I was a farmer, but I would still be the same person – namely, me.
[Recall again our model:
World Domain w1 {α, β, γ, δ} w2 {β, γ} w3 {α, δ, ε}
v(‘B’, w1) = {α, β}
v(‘B’, w2) = { }
v(‘B’, w3) = {α, δ, ε}
v(‘∃xBx’, w1) = T
v(‘∃xBx’, w2) = F
]
These ideas are reflected in the model introduced above. Object β, for example, exists in w1 and w2. It therefore exhibits transworld identity. Moreover, it is in the extension of the predicate ‘B’ in w1, but not in w2. Thus, though it is the same object in w1 as it is in w2, it is blue in w1 but not in w2. If we think of w1 as the actual world, this models the idea that an object that is actually blue nevertheless could be nonblue (it is capable, for example, of being dyed or painted a different color, yet retaining its identity).
(313)
Suppose that ‘n’ denotes the object β. This means that ‘Bn’ or ‘n is blue’ is true in w1 but not in w2.
Suppose now that we use the name ‘n’ to denote object β, that is, let v(‘n’) = β. (Note the absence of a world-variable here; the denotation of a rigidly designat- | ing name, unlike truth or the denotation of a predicate, is not world-relative.) Then we would say that the statement ‘Bn’ (“n is blue”) is true in w1, but not in w2, that is, v(‘Bn’, w1) = T, but v(Bn’, w2) = F.
(Nolt 313-314)
When the referred object does not exist in the world, it would seem arbitrary whether a formula predicating that object is true or false.
But what are we to say about the truth value of ‘Bn’ in w3, wherein β does not exist? Consider some possible (but nonactual) stone. Is it blue or not blue in the actual world? Both answers are arbitrary. Similarly, it seems arbitrary to make ‘Bn’ either true or false in a world in which ‘n’ has no referent.
(Nolt 314)
To solve this problem will require concepts that we deal with later, so for now we will put the matter aside.
This problem cannot be satisfactorily resolved without either abandoning bivalence (so that ‘Bn’, for example, may be neither true nor false) or modifying the logic of the quantifiers. The first approach is perhaps best implemented by means of supervaluations, which are discussed in Section 15.3; the second by free logics, which are covered in Section 15.1. Discussion of either method now would perhaps complicate things beyond what we could bear at the moment. We shall therefore leave the question unsettled.
(Nolt 314)
So when we look at valuation rules, particularly 1 and 2 below, we are assuming that the “the extensions of the names contained in those
formulas are in the domain of that world” (Nolt 314).
So a valuation or model consists of the following things: {1} worlds that each have their own domain of objects, and {2} functions that assign an extension to each predicate and world. Objects may belong to many worlds, but they need not belong to all of them.
A valuation, or model, then, consists of a set of things called worlds, each with its own domain of objects. In addition, it assigns to each name an object from at least one of those domains, and it assigns to each predicate and world an appropriate extension for that predicate in that world. An object may belong to the domain of more than one world, but it need not belong to domains of all worlds. Two different worlds may have the same domain. The full definition is as follows:
DEFINITION A Leibnizian valuation or Leibnizian model v for a formula or set of formulas of modal predicate logic consists of the following:
1. A nonempty set Wv of objects, called the worlds of v.
2. For each world w in Wv a nonempty set Dw of objects, called the domain of w.
3. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
i. If σ is a name, then v(σ) is a member of the domain of at least one world.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F. |
iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.
(Nolt 314-315)
Nolt then provides the valuation rules for this system of modal logic.
Valuation Rules for Leibnizian Modal Predicate Logic
Given any Leibnizian valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
5 .
v(Φ & Ψ, w) = T iff both v(Φ, w) = T and v(Ψ, w) = T;
v(Φ & Ψ, w) = F iff either v(Φ, w) ≠ T or v(Ψ, w) ≠ T, or both.
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
7.
v(Φ → Ψ, w) = T iff either v(Φ, w) ≠ T or v(Ψ, w) = T, or both;
v(Φ → Ψ, w) = F iff both v(Φ, w) = T and v(Ψ, w) ≠ T.
8 .
v(Φ ↔ Ψ, w) = T iff either v(Φ, w) = T and v(Ψ, w) = T, or v(Φ, w) ≠ T and v(Ψ, w) ≠ T;
v(Φ ↔ Ψ, w) = F iff either v(Φ, w) = T and v(Ψ, w) ≠ T, or v(Φ, w) ≠ T and v(Ψ, w) = T.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) ≠ T;
10 .
v(∃βΦ, w) = T iff for some potential name α of some object d in Dw, v(α,d) (Φα/β , w) = T;
v(∃βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T; |
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for some world u in Wv, v(Φ, u) ≠ T;
12 .
v(◊Φ, w) = T iff for some world u in Wv, v(Φ, u) = T;
v(◊Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T.
(Nolt 315-316)
[Nolt will not explain these rules in this section. For now, let us try to put a little more sense into them, although I might not have it right in all cases.
DEFINITION A Leibnizian valuation or Leibnizian model v for a formula or set of formulas of modal predicate logic consists of the following:
Note that in the original text, the v looks like:
I am not quite sure, but my impression with the idea of the valuation (also called a model) is the following. We use the term “valuation” in the singular, even though it normally involves a number of valuation/assignment functions, and as we will see, those functions are different for various worlds. So “valuation” I think means the conglomerate of valuation functions that apply to the world or worlds. Returning to the definition, the Leibnizian valuation or model consists of:
1. A nonempty set Wv of objects, called the worlds of v.
I am not exactly sure of the significance of it being non-empty, but it would seem not to be any interest for us if there were no worlds in the model. In the text the Wv looks like:
So we have one conglomerate of valuation functions v applying to a set of worlds W, which is why we notate it as: Wv .
2. For each world w in Wv a nonempty set Dw of objects, called the domain of w.
Here the Dw looks like:
Notice that the subscripted ‘w’ does not have the scripty-looking decorations on it. So it would not seem to be the set of all worlds but rather of some particular world. So it seems we call any domain D and when it is the domain of some world we call it: Dw.
3. For each name or nonidentity predicate σ of that formula or set of formulas, an extension v(σ) (if σ is a name) or v(σ, w) (if σ is a predicate and w a world in Wv) as follows:
So we firstly are talking about names and non-identity predicates. I am not sure yet why we are not talking about identity predicates. We call such a name or predicate σ. [Recall from Agler section 6.4.2 that the extension of a valuation on a predicate will be an ordered n-tuple.] We will now say more about the extension of a valuation function on that name or non-identity predicate.
i. If σ is a name, then v(σ) is a member of the domain of at least one world.
So this should be obvious. When we use a valuation function on a name, it should assign to the name some object in some world’s domain (or in some worlds’ domains). Note also here that we seem to be using the rigid designation sense of the name, as we have not yet specified the world.
ii. If σ is a zero-place predicate (sentence letter), v(σ, w) is one (but not both) of the values T or F.
I might be wrong here, but it seems in this case we are talking about propositions without predicate structures, as in propositional logic. To object names of course we will assign some object. (And to n-place predicates we will assign n-tuples). But to simple propositions, which will take the form of zer0-place predicates, we will only assign a truth value. And notice also the formulation v(σ, w). I think this means the valuation of σ in some particular world, as that valuation can be different for the various worlds.
iii. If σ is a one-place predicate, v(σ, w) is a set of members of Dw .
So here let us try to be clear about something. Refer again if needed to Agler Symbolic Logic section 6.4.2. We would not assign to a bare non-zero-place predicate a truth value. “is red” can be neither true nor false until we specify what is being said to be red (and in what model, etc.). However, in some world, there will be many things taking that predicate. In our example, there will be some set of things that are red. So this is why the valuation function will assign to a one-place predicate a set of members of some world’s domain. Those members, again, are the things taking that predicate in that world.
iv. If σ is an n-place predicate (n>1), v(σ, w) is a set of ordered n-tuples of members of Dw.
(And again, see Agler section 6.4.2. for more on the assignments of ordered n-tuples to n-place predicates.) So consider if we have a two-place predicate “is taller than”. In some world, it will correspond to ordered couples like ⟨John, Mary⟩, ⟨Jane, Jill⟩ or whatever.
Now for the valuation rules. They will tell us how to assign true and false values to formulas in Leibnizian modal predicate logic.
Given any Leibnizian valuation v, for any world w in Wv:
1. If Φ is a one-place predicate and α is a name whose extension v(α) is in Dw, then
v(Φα, w) = T iff v(α) ∈ v(Φ, w);
v(Φα, w) = F iff v(α) ∉ v(Φ, w).
So begin with one-place predicates and names taking those predicates. Let us begin with the right side of the biconditionals, starting with the first biconditional, namely, v(α) ∈ v(Φ, w). So in some world, we do not assign a truth value to a non-zer0-place predicate taking a variable; rather we assign to it the set of items which in fact take that predicate. So that is the second valuation function at work there. The first one assigns objects to a name. Now, what we are doing is taking the predicate Φ, to which is assigned a set of objects, and the name α, to which is assigned a set of object(s), and we are asking, in some certain world, is the extension that is assigned to the name α also in the extension assigned to the predicate Φ? If so, then v(Φα) in that world is true. If not, it is false.
2. If Φ is an n-place predicate (n>1) and α1 ... , αn are names whose extensions are all in Dw, then
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
v(Φα1, ... , αn, w) = F iff <v(α1), ... , v(αn)> ∉ v(Φ, w).
So now we are dealing with predicates whose place value is two or greater. But the basic structure we established in rule 1 holds here, only now we must keep in mind there will be more terms involved. So suppose our world has these members {John, Mary, Jane, Jill, Bob}. And we again have the n-place predicate “is taller than”. To make it more concrete, let us establish some assignments:
v(a) = John
v(b) = Mary
v(c) = Jane
v(d) = Jill
v(e) = Bob
v(Lxy) = ⟨John, Mary⟩, ⟨Jane, Jill⟩
Here “is taller than” we are symbolizing as ‘L’. So look again at the formulation for truth here.
v(Φα1, ... , αn, w) = T iff <v(α1), ... , v(αn)> ∈ v(Φ, w);
Suppose we want to know v(Lab) in some world. That conforms to the structure on the far left, namely, v(Φα1, ... , αn, w). The next thing we do is we make an ordered couple on the basis of the assignments for ‘a’ and ‘b’. So in our case, <John, Mary>. This conforms to this structure: <v(α1), ... , v(αn)>. We next ask, is <John, Mary> one of the members assigned to the predicate L? The answer is yes. Therefore, it is true. However, v(Lae) would be false.
3. If α and β are names, then
v(α = β, w) = T iff v(α) = v (β);
v(α = β, w) = F iff v(α) ≠ v (β).
This rule seems to establish the relation of equality between names. So when to two names are assigned the same item, then it is true that they are equal, and false otherwise.
For the next five rules, Φ and Ψ are any formulas:
4.
v(~Φ, w) = T iff v(Φ, w) ≠ T;
v(~Φ, w) = F iff v(Φ, w) = T.
This and the next few rules will explain how to determine truth and falsity for sentence operators. They work as we would expect. But we should note that the uppercase Greek symbols here no longer stand for predicates but rather for full wffs I think. So we are not now concerned with the extensions of assigned members but rather with the truth assignments for whole wffs. These truth assignments perhaps are establishable by means of rules 1-3 by dealing with the formula’s predicate and constant extensions. As there is little to explain in these cases, as they seem to follow the normal way to define the operators, let us skip to the next set of rules. They will explain how to evaluate quantified formulas.
For the next two rules, Φα/β stands for the result of replacing each occurrence of the variable β in Φ by α, and Dw is the domain that v assigns to world w.
So now we should be clear about symbols. Even though both α and β are lowercase Greek letters, β is to be understood as a variable and α as a constant. I am not entirely sure about Φ. One possibility is that it is the symbol for a predicate that takes variables or constants, and so it will always be written with variables or constants. The other possibility is that it would be written without those constants and variables, but implying that they are contained in it. It may not matter really, because we are describing the substitution.
9 .
v(∀βΦ, w) = T iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) = T;
v(∀βΦ, w) = F iff for all potential names α of all objects d in Dw, v(α,d) (Φα/β , w) ≠ T;
This situation is more complex, and I may get it wrong. Let us continue working with our model, but we will use a simpler predicate.
v(a) = John
v(b) = Mary
v(c) = Jane
v(d) = Jill
v(e) = Bob
v(Lx) = {<John>, <Mary>, <Jane>, <Jill>}
Here L can mean “is tall”. We might ask, is (∀x)(Lx) true? Look again at the first part of the formula:
v(∀βΦ, w) = T
We said that the β is a variable. So considering our example, in the world in question, is the valuation of “for all x, Lx is true”. Suppose we are in world 1. The formulation might be:
v(∀xΦ, w1) = T
I am not sure if the subscript for β is still needed, so perhaps something like:
v[∀x(Φx), w1] = T
The next part reads:
iff for all potential names α of all objects d in Dw
So in our example, the objects d in our domain for world 1 are: John, Mary, Jane, Bill, Bob. The names are a, b, c, d, e.
v(α,d) (Φα/β , w) = T;
What does the v(α,d) mean? I am not sure, but I suppose it means the valuation for names α with respect to their own valuation assignments to objects in the domain. Then we have (Φα/β , w) = T. We said this means we substitute name α for variable β. The formulation says for all potential names. So we might do an evaluation like:
v(La) = T
v(Lb) = T
v(Lc) = T
v(Ld) = T
v(Le) = F
As we can see, for all possible names for objects in our domain, the valuations are not all true, and thus the valuation for the universal quantification on the variable being substituted is false.
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;
Similar reasoning holds here, so let us pass to the last set of rules.
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T;
So while before we were looking at the domains and interpretation/valuation functions of each world taken on its own, we now consider all the worlds together. We consider a formula that is made in all these worlds. We are saying that when the formula is true in all worlds, then it is necessarily true. Otherwise it is false.
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.
Similarly, when the proposition is true in at least one world, it is possibly true, and it is false otherwise.]
Since all these rules are a lot to handle at once, Nolt will “take the propositional fragment of the semantics by itself first and come back to the full modal predicate logic later” (316). [I think this means that he will only deal with full propositions without consideration for predicates and variables]
This simplifies the definition of a valuation considerably:
DEFINITION A Leibnizian valuation or Leibnizian model v for a formula or set of formulas of modal propositional logic consists of
1. A nonempty set Wv of objects, called the worlds of v.
2 For each sentence letter σ of that formula or set of formulas and each world w in Wv , an extension v(σ, w) consisting of one (but not both) of the values T or F.
Here worlds are like the (horizontal) lines on a truth table, in that each is distinguished by a truth-value assignment to atomic formulas-though not all lines of a truth table need be represented in a single model.
(Nolt 316)
Nolt gives an example. There will be four worlds, named 1, 2, 3, and 4. We are concerned with the valuation for the formula (V ∨ W). He will first give the truth valuations for V and W, for each world.
Wv = {1 , 2, 3 , 4}
v(‘V’, 1) = T
v(‘W’, 1) = F
v(‘V’, 2) = F
v(‘W’, 2) = F
v(‘V’, 3) = F
v(‘W’, 3) = T
v(‘V’, 4) = F
v(‘W’, 4) = T
(Nolt 316)
Here we take V to mean Sam is virtuous and W to mean Sam is wicked. As we can see, there is no world where he is both virtuous and wicked, and thus that combination is impossible. And given that there is one line where both terms are false, we see that (V ∨ W) is possible in just three worlds. [Let me quote just to be sure we have that right.]
The “worlds” here are the numbers 1 , 2, 3 , and 4. (In a model, it doesn’t matter what sorts of objects do the modeling.) In world 1, ‘V’ is true and ‘W’ is false – that is, Sam is virtuous, not wicked. In world 2, Sam is neither virtuous nor wicked. And in worlds 3 and 4, Sam is wicked, not virtuous. Our model represents the situation in which Sam is both virtuous and wicked as impossible, since this situation occurs in none of the four possible worlds. In other words, only three of the four lines of the truth table for ‘V ∨ W’ are regarded as possible. This is arguably appropriate, given the meanings we have attached to ‘V’ and ‘W’.
(Nolt 316)
So (V ∨ W) has the value true in some world if either V or W is true in that world:
6 .
v(Φ ∨ Ψ, w) = T iff either v(Φ, w) = T or v(Ψ, w) = T, or both;v(Φ ∨ Ψ, w) = F iff both v(Φ, w) ≠ T and v(Ψ, w) ≠ T.
Nolt then says that the “real novelty [...] and the heart of Leibniz’s insight, lies in rules 11 and 12” (317). [Again, they are:
11 .
v(□Φ, w) = T iff for all worlds u in Wv, v(Φ, u) = T;
v(□Φ, w) = F iff for all worlds u in Wv, v(Φ, u) ≠ T;
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 315-316)
] He has us consider the following statement, “it is necessarily the case that Sam is not both virtuous and wicked” (317), or:
□~(V & W)
(Nolt 317)
Rule 11 tells us that this formula will be true if and only iff ~(V & W) is true in all worlds. In fact, there is no world where this formula is true, and thus □~(V & W) is true in every world.
And what about “it is possible that Sam is virtuous” or ◊V? As we see, it is true in world 1, and thus ◊V is true in all worlds. “For consider any given world w. Whichever world w is, there is some world u (namely, world 1) in which ‘V’ is true. Hence by rule 12, ‘◊V’ is true in w” (317).
[And when something is true in all worlds, it is necessarily true. Thus it is necessarily true that it is possible Sam is virtuous. In fact, we can keep compounding the necessity operator.]
Notice also that since ‘◊V’ is true in all worlds, it follows by another application of rule 11 that ‘□◊V’ (“it is necessarily possible that Sam is virtuous”) is true in all worlds. In fact, repeated application of rule 11 establishes that ‘□□◊V’, ‘□□□◊V’, and so on are all true at all worlds in this model. The following metatheorem exemplifies the formal use of modal semantics; [...]
METATHEOREM: For any world w of the model just described, v(‘□◊V’, w) = T.
PROOF: Let u be any world of this model, that is, u ∈ Wv. Since v(‘V’, 1) = T, it follows by rule 12 that V(‘◊V’, u) = T. Thus, for all u in Wv, v(‘◊V’, u) = T. Now let w be any world in Wv. It follows by rule 11 that v(‘□◊V’, w) = T. QED
(Nolt 317)
From:
Nolt, John. Logics. Belmont, CA: Wadsworth, 1997.
Or if otherwise noted:
Agler, David. Symbolic Logic: Syntax, Semantics, and Proof. New York: Rowman & Littlefield, 2013.
Bolling, Rubin. “The Infinite Universes of Bob”. 10-May-1998. Available at:
http://www.gocomics.com/tomthedancingbug/1998/05/10
.
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