by Corry Shores

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[The following is summary. All boldface in quotations, and all boldface tense operators, are not mine, unless otherwise noted. Bracketed commentary is my own. As proofreading is incomplete, you will find typos and other distracting errors. I apologize in advance.]

Summary of

John Nolt

*Logics*

Part 4: Extensions of Classical Logic

Chapter 13: Deontic and Tense Logics

13.2 A Modal Tense Logic

13.2.1 [Basic set-up and evaluation of modal tense logic]

Brief summary:

Intuitively, time involves a passage extending linearly from past to future. As it moves forward, it could follow one of many possible ramifying branches from the current now point. So the future, under this view, is multiple and undecided. The past, however, cannot become otherwise after it happens. Were we to draw this structural feature of time, we would have a tree-trunk line for history, as it has only one, and a series of branches for the future, expanding out from the present, which stands at the end of the singular past-trunk and before the branching futures. At each point in time, there are different facts and existing objects, so we can specify what holds at certain time points. But we can also see the branching futures as possible worlds in relation to our own, as these alternate future paths may or may not be realized in our own actual world. So we can speak of situations holding or not holding for certain possible worlds at certain times. Furthermore, we can use a modal tense logic to place these temporally contingent facts into temporal relations with one another. This way, we can say that some situation will hold in a future moment or in all future moments, and so on. We use these following symbols for modal tense operators:

H– it has always been the case that

P– it was (at some time) the case that

G– it will always be the case that

F– it will (at some time) be the case that(Nolt 367)

Each of these can be combined with themselves and with the others many times over, and they can also be combined with the alethic modal operators, necessity and possibility. The definition of this modal tense logic model is the following:

DEFINITION A

modelorvaluationvfor a formula or set of formulas of modal predicate logic consists of the following:1. A nonempty set ℑ of objects called the

timesofv.2. A transitive relation ℰ, consisting of a set of pairs of times from ℑ.

3. A nonempty set

Wof objects, called the_{v }worldsofv.4. Corresponding to each world

w, a set ℑof times called the_{w }times in wsuch that for any pair of timest_{1}andt_{2 }in this set, eithert_{1}ℰt_{2 }ort_{2}ℰt_{1 }ort_{1 }=t_{2}.5. For each world

wand timetinw, a nonempty setDof objects called_{(t,w) }thedomainofwatt.6. For each name or nonidentity predicate σ of that formula or set of formulas, an extension

v(σ) (if σ is a name) orv(σ,w) (if σ is a predicate andwa world inW) as follows:_{v}i. If σ is a name, then

v(σ) is a member ofDfor at least one time_{(t,w) }tand worldw. |ii. If σ is a zero-place predicate (sentence letter), and

tis inw, thenv(σ,t,w) is one (but not both) of the values T or F.iii. If σ is a one-place predicate and

tis inw,v(σ,t,w) is a set of members ofD._{(t,w)}iv. If σ is an

n-place predicate (n>1), andtis inw,v(σ,t,w) is a set of orderedn-tuples of members ofD._{(t,w)}(Nolt 370-371, boldface in the original)

Note here the ‘earlier than’ relation ℰ that orders the time points in the worlds. Possible worlds that share all moments up to a particular time point are said to be accessible to each other at that time point, meaning that the future path that one takes (the facts that become true) can be the same that the other takes, all while maintaining the same past. This accessibility relation is written ℛ. (When models are not accessible, that means they have different histories, which are not alterable, and thus what is possible for one world is no longer possible for the other.)

DEFINITION Given a model

vfor a formula or set of formulas, then for any worldsw_{1 }andw_{2 }and timetofv,w_{1}ℛw_{2}tiff1.

tis a time in bothw_{1 }andw_{2}, |2.

w_{1 }andw_{2 }contain the same times up tot; that is, for all timest′, ift′ℰt, thent′ is in w1 ifft′ is inw_{2}, and3.

w_{1 }andw_{2 }have the same atomic truths at every moment up tot; that is, for all timest′ such thatt′ℰt,D(t′,w_{1}) =D(t′,w_{2}), and for all predicates Φ,v(Φ,t′, w_{1}) =v(Φ,t′, w_{2}).(Nolt 371-372)

The valuation rules for this modal tense logic are the following:

Valuation Rules for Modal Tense LogicGiven any valuation

vof modal tense logic whose set of worlds isW, for any world_{v}winWand time_{v}tinw:1. If Φ is a one-place predicate and α is a name whose extension

v(α) is inD_{(t,w)}, thenv(Φα,t,w) = T iffv(α) ∈v(Φ,t,w).2. If Φ is an

n-place predicate (n>1) and α_{1 }... , α_{n }are names whose extensions are all inD_{(t,w)}, then

v(Φα_{1}, ... , α_{n},t,w) = T iff <v(α_{1}), ... ,v(α_{n})> ∈v(Φ,t,w)3. If α and β are names, then

v(α = β,t,w) = T iffv(α) =v(β).

For the next five rules, Φ and Ψ are any formulas:

4.

v(~Φ,t,w) = T iffv(Φ,t,w) ≠ T5 .

v(Φ & Ψ,t,w) = T iff bothv(Φ,t,w) = T andv(Ψ,t,w) = T6 .

v(Φ ∨ Ψ,t,w) = T iff eitherv(Φ,t,w) = T orv(Ψ,t,w) = T, or both7.

v(Φ → Ψ,t,w) = T iff eitherv(Φ,t,w) ≠ T orv(Ψ,t,w) = T, or both8 .

v(Φ ↔ Ψ,t,w) = T iffv(Φ,t,w) =v(Ψ,t,w)

For the next two rules, Φ

^{α}/_{β}stands for the result of replacing each occurrence of the variable β in Φ by α, andD_{(t,w) }is the domain thatvassigns to worldwat timet.9 .

v(∀_{β}Φ,t,w) = T iff for all potential names α of all objectsdinD_{(t,w)},v_{(α,d)}(Φ^{α}/_{β },t,w) = T10 .

v(∃_{β}Φ,t,w) = T iff for some potential name α of some objectdinD,_{w}v_{(α,d)}(Φ^{α}/_{β },w) = T11 .

v(□Φ,t,w) = T iff for all worldsusuch thatwℛut,v(Φ,t,u) = T12.

v(◊Φ,t,w) = T iff for some worldu,wℛutandv(Φ,t, u)= T13.

v(HΦ,t,w) = T iff for all timest′ inwsuch thatt′ℰtandv(Φ,t′,w) = T14.

v(PΦ,t,w) = T iff for some timet′ inw,t′ℰtandv(Φ,t′,w) = T15.

v(GΦ,t,w) = T iff for all timest′ inwsuch thattℰt′ andv(Φ,t′,w) = T16.

v(FΦ,t,w) = T iff for some timet′ inw,tℰt′ andv(Φ,t′,w) = T(Nolt 372-373, boldface in the original)

And important logic terms are defined in the following way for this modal tense logic:

DEFINITION A formula is

validiff it is true at all times in all worlds on all of its valuations.DEFINITION A formula is

consistentiff it is true at at least one time in at least one world on at least one valuation.DEFINITION A formula is

inconsistentiff it is not true at any time in any world on any of its valuations.DEFINITION A formula is

contingentiff there is a valuation on which it is true at some time in some world and a valuation on which it is not true at some time in some world.DEFINITION A

set of formulasisconsistentiff there is at least one valuation containing a world in which there is a time at which all the formulas in the set are true.DEFINITION A

set of formulasisinconsistentiff there is no valuation containing a world in which there is a time at which all the formulas in the set are true. |DEFINITION Two formulas are

equivalentiff they have the same truth value at every time in every world on every valuation of both.DEFINITION A

counterexampleto a sequent is a valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.DEFINITION A sequent is

validiff there is no world in any valuation containing a time at which its premises are true and its conclusion is not true.DEFINITION A sequent is

invalidiff there is at least one valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.(Nolt 373-374, boldface in the original)

Using this model of modal tense logic, we can show that one particular argument for determinism is invalid, and thus there is a philosophical usefulness for this model. However, what is philosophically at issue cannot be settled by the model, but rather, the model can only be based on philosophical assumptions and cannot always prove or disprove them. Nonetheless, the model can at least help us clarify our philosophical conceptions about time. Also, it can establish certain views as consistent or inconsistent, and it shows the deterministic view to at least be consistent.

Summary

Nolt begins with Augustine’s famous passage on time:

“What then is time?” asks St. Augustine. “I know what it is if no one asks me what it is; but if I want to explain it so someone who has asked me, I find that I do not know.”

(Nolt 365, quoting Augustine. [See Book XI, Chapter XIV, §17 of Augustine’s

Confessions.])

Nolt then describes our ordinary conceptions of time. We see time as being made of an linearly ordered series of events. There is a privileged position in time that we now occupy, called the present. All moments before this present one lie in the past and all moments after this present one lie in the future. The present is in motion, and it “constantly advances toward the future, and this advance gives time a direction” (Nolt 365). The past possibly extends infinitely behind the present, and its events are currently unalterable and thus are necessary with respect to the present. “The future, however, is not frozen into unalterability but alive with possibilities” (Nolt 365). This is because events happening now or happening later can undergo unpredictable or unforeseeable developments. “There is, in other words, more than one possible future” (365). However, even though there are many possible ways the present can unfold, only one such way will actually unfold. We might also think of alternate pasts that could have happened. But alternate possible futures are “genuinely possible” in the sense that they could be realized, but past possibilities are not genuinely possible, because there is no way they can retroactively be realized (365)

Now, if we were to make a model in the form of a diagram of this intuition about how time works and is structured, then we might design it to have the form of a tree of sorts, “with a single trunk (the past) that at a certain point (the present) begins to split and split again into ramifying branches (various possible futures). This would give us a snap-shot of the present. Were the diagram to be animated such that it alters with the flow of time in the present, then as it moves forward, it would follow one branch or another, with the others fading away, as they are no longer real parts of the temporal spectrum from past to future. If time had a finite end to it, then when we reach it, there would just be “a single path from trunk to branch tip would remain: the entire history of the actual world from the beginning to the end of time” (365). And suppose that time goes backward infinitely into the past. Then our diagram would have an endless line going downward. Similarly, if the future is infinite, the upward line(s) would be endless. Nolt then gives a diagram. [Let me quote at length, as this is excellent text.]

Ordinarily we understand time as a linearly ordered sequence of moments. We have a position in time, the present moment. All other moments lie in either the past or the future. The present constantly advances toward the future, and this advance gives time a direction. The past is a continuum of moments stretching behind us, perhaps to infinity. It is unalterable. Whatever has been is now necessarily so. The future, however, is not frozen into unalterability but alive with possibilities. Starting with the present, events could take various alternative courses. There is, in other words, more than one possible future. Though only one of these courses of events will in fact be realized (we may not, of course, know which one), still the others are genuinely possible, in a way that alternative pasts are not genuinely possible. These intuitions suggest a model on which time is like a tree with a single trunk (the past) that at a certain point (the present) begins to split and split again into ramifying branches (various possible futures) . As time moves forward, the lower branches (formerly live possibilities, lost through passage of time ) disappear. Only one path through the tree represents the actual course of time, that is, the actual world. More and more of the path is revealed as time moves on and lower branches vanish. If time were finite, eventually all the branches representing merely possible futures would disappear and only this single path from trunk to branch tip would remain: the entire history of the actual world from the beginning to the end of time. But we might also think of time as infinite-at least toward the future and perhaps also backward into the past. If time is infinite toward the past, then the tree's trunk extends endlessly downward, never touching ground; and if time is infinite toward the future, then its branches stretch endlessly upward, never touching the sky. In either case, we might picture at least a part of the tree like the diagram in Figure 13.1.

(Nolt 365)

(Nolt 366)

This is our intuition of how time works. But we cannot rush to conclude that it is the true representation of time. Relativity physics paints a different picture. It sees time as “inseparable from space and relative to motion, and though it is experienced as past, present, and future, these may not be ‘objective’ features of time itself”. And even if past, present, and future are objective features of time, our model might not work near the big bang or at the universe’s distant future where time perhaps has different features. Also, suppose that the world is purely deterministic. That means these branches are not physical realities but rather are merely illusions. Nolt says the tense logic we will look at does not purport to model some true time that we are unaware of. Rather, it models the sort of time of our everyday understanding of it. Tense logic also includes variations for other sorts of time, like “relativistic time, discrete time, circular time, | ending time, and so on,” but we do not deal with these variations here (366-367).

Nolt then defines tense logic. As we know from prior sections, it uses operators for different tense variations. Specifically there are four. Nolt says that we also include modal operators in this tense logic.

A

tense logicis a logic that includes operators expressing tense modifications. The logic we shall consider here is a modal tense logic because it contains alethic modal operators in addition to tense operators. It has four tense operators:

H– it has always been the case that

P– it was (at some time) the case that

G– it will always be the case that

F– it will (at some time) be the case that(367)

[Let us briefly recall for a moment Priest’s explanation of these operators in his *Logic: A Very Short Introduction*, chapter 8. This comes from the summary at the end of the chapter:

●

Fais true in a situation ifais true in some later situation.●

Pais true in a situation ifais true in some earlier situation.●

Gais true in a situation ifais true in every later situation.●

Hais true in a situation ifais true in every earlier situation.(Priest p.62)

]

Nolt then explains that the first and last two are duals. [So this would seem to mean that if we say “It has always been the case that there was time”, then this would be equivalent to saying “It is not that it has always been the case that there was not time.” But I am not sure. The first formulation seems to say that no matter when in the past you go, there was time. The second formulation seems to say that no matter where you look in the past, there is no part of the past where there is no time.] Thus the following are valid formulas:

HΦ ↔ ~P~Φ

PΦ ↔ ~H~Φ

GΦ ↔ ~F~Φ

FΦ ↔ ~G~Φ(Nolt 367)

Nolt then gives two sets of interpretations for various configurations of the tense operators.

GHΦ | It will always be that it has always been that Φ. [Intuitively, this means that Φ is the case at all times-past, present, and future.] |

FHΦ | It will be the case that it has always been that Φ. [Φ has always been the case and will continue to be for some time.] |

HΦ | It has always been the case that Φ. |

PHΦ | It was the case that it had always been that Φ. [There was a time before which it was always the case that Φ.] |

HPΦ | It has always been that it had (at some time) been the case that Φ. [That is, there have always been times past at which Φ was the case, but these may have occurred intermittently.] |

PΦ | It was the case that Φ. |

GPΦ | It will always have been that Φ. |

FPΦ | It will be the case that it has (at some time) been that Φ. |

(Nolt 367)

HGΦ | It has always been the case that it would always be that Φ. [Φ is the case at all times-past, present, and future.] |

PGΦ | It was (at some time) the case that it always would be that Φ. |

GΦ | It always will be the case that Φ. |

FGΦ | It will (at some time) be the case that it will always be that Φ. [That is, there will come a time after which Φ is always the case.] |

GFΦ | It will always be the case that it will sometimes be the case that Φ. [Moments at which Φ is the case will always lie in the future, though perhaps intermittently.] |

FΦ | It will be the case that Φ. |

HFΦ | It has always been that it will be the case that Φ. |

PFΦ | It was (at some time) the case that it would (later) be the case that Φ. |

(Nolt 368)

Nolt then explains that one of the many things an “adequate” tense logic would be able to do is to “determine which of these statement forms imply which others” (Nolt 368). Nolt then says that if we assume that there is no first or last moment of time [as it stretches in both directions endlessly] that means each formula from a list implies all the others below it on that same list group. [Let us take one such pairing from the second list: ‘**F**Φ It will be the case that Φ’ and ‘**PF**Φ It was (at some time) the case that it would (later) be the case that Φ.’ Perhaps the idea here is that if we say that at some point down the line of time something will be the case, and if this statement always had to be made at some moment with others before it (as there is no first moment), then there had to have been some moment in the past where it was true that later that something would be the case.] Nolt also says that under this assumption of no first and last moments of time, the first members of both lists are equivalent, as are the last members of both lists. [The two first ones are: ‘**GH**Φ It will always be that it has always been that Φ. (Intuitively, this means that Φ is the case at all times-past, present, and future.)’ and ‘**HG**Φ It has always been the case that it would always be that Φ. (Φ is the case at all times-past, present, and future)’. So as we can see, they mean basically the same thing.]

Nolt then explains that there is no special tense operator to indicate the present. So if we want to mean “that Φ is presently the case, we simply assert Φ” (368). Nolt then gives four sequents that should be understood as valid for this reason.

**FH**A ⊢ A

(‘It will be the case that it has always been that A’.)

[So if in the future it has always been the case that A, that means the present it must be A.]

A ⊢ **HF**A

(‘It has always been that it will be the case that A’.)

[So if it is now A, that means in all moments of the past it was going to be A.]

**PG**A ⊢ A

(‘It was (at some time) the case that it always would be that A’)

[So if in the past it was always going to be the case that A, that means it would have to now be A.]

A ⊢**GP**A

(‘It will always have been that A’)

[So if it is now A, that means for any time in the future, it will be the case that it was A in some moment in the past (of that future moment).]

We recall how “Our intuitive picture of time includes multiple possible futures” (368). And recall also that “each path through the tree from the base of the trunk (if it has a base) to the tip of a branch (if branches have tips) represents a complete possible world” (368). [All the development happening before the branch into the future possibility is a past shared with the actual world.] “These possible worlds share a portion of their histories with the actual world but plit off at some specific time” (368). Nolt then shows using everyday experience how all the temporal development leading up to a [present or forthcoming] decision is that of the actual world.

This is a picture we often use in decision making. Suppose I am considering whether to go to the mountains for a hike or just stay at home and relax this weekend. These are (we assume) real possibilities, though undoubtedly not the only ones. Corresponding to each is at least one possible world – that is, at least one course of events that the world might take from the beginning of time through and beyond the moment of my decision. Suppose I decide to hike and I carry out that intention. Then the world (or one of the worlds) in which I hike is the actual world, and the worlds in which I stay at home that weekend are possible but nonactual. In these nonactual worlds, everything up to the moment of my decision occurs exactly as it does in the actual world, though events depart from their actual course more or less dramatically thereafter.

(368)

Nolt continues this point by noting that when we say that we could have done something different, we mean that up until the time we decided to go hiking, there was a future alternative branch where we stayed home. But after we decided against that option, this branch disappeared. [What is important here is to relate these notions of temporal possibilities in our world with possible worlds. Before we made the decision, there was a possible world in whose future we stayed home and another possible world in whose future we went hiking. Thus we say that before the decision it was possible for us to stay home. However, now after we made that decision, the other branches representing the futures of other possible worlds that shared the same past up to that moment of decision disappeared. This means that from the perspective of now, where we live on one branch and have just a singular trunk of past behind us, there is no other possible world where in the past we stayed home. Thus we can say it is now necessary that we went hiking (even though prior to our decision it was still possible). The following is the reasoning. Something is necessary if it so in all possible worlds. Right now, in all possible worlds, we went hiking (as the others, where we did not, have ‘disappeared’ after we made the decision). Thus, it is now necessary that we went hiking.]

When the weekend is over, I may say, “Though I could have stayed home, I can’t now go back and change the past; it is now necessarily the case that I went hiking,” mixing tenses and alethic modalities in ways that our picture nicely illustrates. To say that I could have stayed home is to say that up to the beginning of the weekend a world in which I stayed home (represented by a path up the trunk through one of the thin branching lines) was possible. This branch, however, has disappeared as time has moved on. To say that my having gone hiking is now necessary is to say that I did go hiking in all currently possible worlds, a circumstance represented in our picture by the fact that all currently possible worlds have exactly the same past as the actual world (the tree has but one trunk).

(369)

[For the next idea, we need to recall Kripkean modal semantics and in particular the concept of accessibility. The basic idea we need now is that futures that were once possible but now are not, because time has taken us down an alternate path, are still logically possible, although they are no longer physically possible, given the way time works. So relative to our actual world, they are not possible. Let us now look again at what we gathered in brief summary from section 12.1 (found between ellipses):

...

Kripkean semantics allows us to model certain logical ideas and principles in modal logic that we are unable to using Leibnizian semantics. The main problem is that Leibnizian semantics will make certain arguments valid (or invalid) when they should not be for a certain type of modality. For example, physical possibility does not behave the same way as logical possibility. Take for instance the fact that accelerating an object faster than the speed of light is logically possible but not physically possible. So we need to change the way we make models for physical possibility. One way of thinking about this is by comparing what is physically possible in each world. In our world (world 1), objects in space can have either circular or elliptical orbits. Now suppose that in world 2, they only have circular orbits. So it is physically impossible in world 2 for objects to have elliptical orbits. Now suppose further we take the perspective of world 2, where elliptical orbits are impossible. Were we to consider world 1 from world 2’s perspective, we would say that world 1 is a physically impossible world, because its laws of physics do not obey our own. However, were we to look at world 2 from our perspective, we would say that it is a physically possible world, because it obeys all our physical laws (it just is more physically restricted than ours). So in order to model physical possibility, we can specify this relation of world relativity. It is called relative possibility, accessibility, or alternativeness. In our example, we would say that world 2 is possible relative to world 1, or that world 2 is an alternative to world 1, or that world 2 is accessible to world 1. However, we cannot invert these formulations. We write world *y* is accessible to world *x* as *x*ℛ*y. *We can diagram this with circles and arrows, with an arrow going from a first circle to a second meaning that the second is accessible from the first. Our orbit worlds would be diagramed as:

(Nolt 337)

As we can see, each world is accessible to itself, because each world follows its own physical laws. However, other sorts of modality, like deontic modality, does not guarantee this reflexive self accessibility. So when we use Kripkean semantics, we must stipulate the world relativities by making a set that lists ordered couples of the form <*x*, *y*> where *y* is accessible to (possible relative to) *x*. So for our example above:

ℛ = {<1, 2>, <1, 1>, <2, 2>}

...

The way this will be applied in tense logic is by making a further specification, namely, the times that the accessibilities hold. So up to a certain time, other possible worlds where the future diverges are still accessible to our actual world. But after that time point, they are no longer accessible, since they are no longer possibilities for our world.]

What we have been thinking of as the disappearance of the tree’s lower branches can also be understood in Kripkean terms as the termination of accessibility. In a sense these “vanished” branches are still there; they still represent worlds that are possible in some absolute sense. But these worlds are no longer possible relative to (i.e., no longer accessible from) the actual world. In tense logic, in other words, accessibility is time-relative. Thus to represent alethic modalities in familiar Kripkean fashion in the context of tense logic, we must add a temporal index to the accessibility relation ℛ. Instead of saying flatly that world

w_{2 }is accessible from worldw_{1}, we must specify a time relative to which accessibility is asserted. Thus we shall write, for example, ‘w_{1}ℛw_{2}t’to indicate thatw_{2 }is accessible fromw_{1 }at timet. Worlds in which I stayed home on the weekend in question are accessible from the actual world prior to my leaving, but not thereafter.(Nolt 369)

[So this allows us to specify world relativity for worlds where events unfold differently. We also need to model truth so that statements regarding possibility in temporally contingent situations can be evaluated. For this we again must specify the times for the statements, because they can be true at some times but not at others.]

Truth, already relativized to worlds in alethic modal logic, must in tense logic be further relativized to times. It is true now that I am sitting at my computer, but this will not be true a few hours hence. Thus the statement ‘I am sitting at my computer’ is true at one time and not at another within the actual world. Moreover, though it is true now in the actual world, it is not true in a world (possible until very recently) in which I got up and went for a snack a moment ago. Thus a statement may have different truth values at different times within the same world and different truth values at the same time within different worlds. Valuations for predicates (including zero-place predicates) must, accordingly, be indexed to both worlds and times. We shall write, for example, ‘

v(Φ,t,w) = T’ to indicate that formula Φ is true at timetin worldw. But we shall treat names, as before, as rigid designators, relativizing their denotations neither to worlds nor to times.(369)

So in a world there are times or ‘moments’. They “do not just occur randomly within worlds, but successively in a strict linear order. In fact, a world may simply be defined as a linear progression of times” (Nolt 369). The way that this order is specified is by means of a temporal ordering relation: ‘earlier than’, written with ‘ℰ’.

Thus ‘

t_{1}ℰt_{2}’ means that timet_{1 }is earlier than timet_{2}. To say that the times comprising a world arelinearly orderedis to say that for any timest_{1 }andt_{2 }belonging to the same world, eithert_{1}ℰt_{2 }ort_{2}ℰt_{1 }ort_{1 }=t_{2}. This implies that the moments comprising a given world can all be arrayed, as in our intuitive picture, as points along a single (possibly curved but more or less vertical) line, with each earlier moment beneath all later moments.(Nolt 370, boldface his)

As we would expect, given how linearly ordered temporal moments work, the ‘earlier than’ relation would be transitive such that were moment B to come after moment A and C after B, then C also comes after A.

ℰ, moreover, must in general be transitive – that is, for any times

t_{1},t_{2}, andt_{3}, ift_{1}ℰt_{2}andt_{2}ℰt_{3}, thent_{1}ℰt_{3}– for it violates our conception “earlier” to think oft_{1}as earlier thant_{2}andt_{2}as earlier thant_{3}but nott_{1}as earlier thant_{3}.(Nolt 384)

[Nolt’s next point seems to be that domains are also temporally relative even in one world. This is because things may exist at one time, and thus be in the domain, but not exist at other times, and thus not be in the domain.]

Finally, we must recognize that even domains, which in alethic logic were relativized to worlds, must now be relativized to times as well. Objects come into and go out of existence as time passes. Thus within a single world what exists at one time differs from what exists at another. But also at a given time what exists in one world may differ from what exists in another. I am now poised over a soap bubble, ready to pop it with my finger. If I choose to do so, then a moment afterward the actual world contains one less soap bubble than exists at the very same moment in the world that would have been actual had I not poked.

(Nolt 369)

[With these matters in mind, we will model modal tense logic. One important difference between this and the other models we have seen is that it has a set of objects called ℑ, which are the times in the model. (I am not entirely certain, however, why we use the same symbol that we will later employ for the outer domain in free logic. See section section 15.1.) The next addition is the ‘earlier than’ temporal ordering relation we mentioned above, ℰ. Recall from Suppes’ *Introduction to Logic *section 10.2 that a relation can be understood as a set of ordered *n*-tuples. A binary relation could thus be understood as a set of ordered couples. Here Nolt will define the ℰ relation as a set of pairs of times from the set of times ℑ. So it seems that ℑ establishes all the times of the world, while ℰ establishes the linear order of those times. The next part of this model will establish the set of worlds in the model, and to each world is assigned a set of ordered times. Then, we assign to both a world and some time for it a domain of objects. I presume these are the objects that are said to exist at some given time in that world. Finally, the model will establish the conditions for evaluating the truth value for statements in the modal tense model. There is a valuation function that assigns to a name a member in the domain for some world and time. A statement itself for a world and time will be either true or false. And predicates for a world and time are assigned sets of *n*-tuples from the domain at that world and time.]

DEFINITION A

modelorvaluationvfor a formula or set of formulas of modal predicate logic consists of the following:1. A nonempty set ℑ of objects called the

timesofv.2. A transitive relation ℰ, consisting of a set of pairs of times from ℑ.

3. A nonempty set

Wof objects, called the_{v }worldsofv.4. Corresponding to each world

w, a set ℑof times called the_{w }times in wsuch that for any pair of timest_{1}andt_{2 }in this set, eithert_{1}ℰt_{2 }ort_{2}ℰt_{1 }ort_{1 }=t_{2}.5. For each world

wand timetinw, a nonempty setDof objects called_{(t,w) }thedomainofwatt.6. For each name or nonidentity predicate σ of that formula or set of formulas, an extension

v(σ) (if σ is a name) orv(σ,w) (if σ is a predicate andwa world inW) as follows:_{v}i. If σ is a name, then

v(σ) is a member ofDfor at least one time_{(t,w) }tand worldw. |ii. If σ is a zero-place predicate (sentence letter), and

tis inw, thenv(σ,t,w) is one (but not both) of the values T or F.iii. If σ is a one-place predicate and

tis inw,v(σ,t,w) is a set of members ofD._{(t,w)}iv. If σ is an

n-place predicate (n>1), andtis inw,v(σ,t,w) is a set of orderedn-tuples of members ofD._{(t,w)}(Nolt 370-371, boldface his)

Nolt notes that different worlds can share the same times, and also something true at a time in one world can be false at the same time in another world. This would be the case for example when we think of our own world going down one path of development rather than some other, with that other path in another alternate world. Also, one world does not need to have all the times in the model, because “One world may begin or end sooner than another, and some might be temporally infinite – having no beginning and no end” (Nolt 371). Also two worlds may have two distinct sets of times. But even in that case, both worlds will be drawing from the same sets of times, which have an ordered relation [and so one world could be said to have times that come after all those of the other world]: “the earlier-than relation transcends worlds in the sense that if *t*_{1}ℰ*t*_{2 }, *t*_{1 }is earlier than *t*_{2 }in any world that includes both of these times” (Nolt 371). Also we can have a world that somehow skips over times, although we may not find ourselves needing to model such situations (371).

[Recall, from above, the notion that when a world diverges down one path of development, the alternate path that it could have gone down (but did not) is understood as a possible world that our own actual world does not have access to after that divergence. Nolt now says that in our modal tense logic model, we can define the accessibility relation ℛ by saying that the relation holds between two worlds at a particular time when both worlds share the same history up to that time. And having the same history is defined as sharing exactly the same moments (up to that time), with every formula being true for certain moments in one world also being true in the other at those same times. The technical definition will have three stipulations for one world having access to another up to a certain time. The first is that both worlds share this certain time in question. The second is that they share all prior times up to that certain time in question. The third says that for all times up to that certain time in question, the domain for each prior time is the same in each world, and the truth valuations for the predicates at each time are the same in each world.]

Notice, finally, that our definition of a model does not include a specification of the alethic accessibility relation ℛ. This is because ℛ is definable in terms already available to us. Specifically, we may say that world

w_{2 }is accessible fromw_{1 }at timetiffw_{2 }has exactly the same history asw_{1 }up to timet. Ifw_{2 }differs in any respect fromw_{1 }beforet, thenw_{2 }is no longer possible relative tow_{1}, for otherwise the past would not be necessary. (Note, however, that in order to be accessible fromw_{1 }at timet,w_{2 }need not diverge fromw_{1 }precisely at timet; the divergence of the two worlds may yet lie some distance into the future.) Two worldsw_{1 }andw_{2 }have the same historyiff they consist of the same moments up to timetand every atomic formula that is true at a given moment beforetin one is true at the same moment in the other, that is, if they meet conditions 1-3 of the following definition:

DEFINITION Given a model

vfor a formula or set of formulas, then for any worldsw_{1 }andw_{2 }and timetofv,w_{1}ℛw_{2}tiff1.

tis a time in bothw_{1 }andw_{2}, |2.

w_{1 }andw_{2 }contain the same times up tot; that is, for all timest′, ift′ℰt, thent′ is in w1 ifft′ is inw_{2}, and3.

w_{1 }andw_{2 }have the same atomic truths at every moment up tot; that is, for all timest′ such thatt′ℰt,D(t′,w_{1}) =D(t′,w_{2}), and for all predicates Φ,v(Φ,t′, w_{1}) =v(Φ,t′, w_{2}).(Nolt 371-372, boldface in the original)

[Recall the truth evaluation procedures. We will want to evaluate whether some formula, which contains names and a predicate function, is true or not. There will be a function that assigns to the name some object. Consider first one-place predicates, for example, ‘is red’. There will be another function that assigns a set of objects to the predicate, in this case, a set of red objects. So when we have a complete formula with this predicate, like ‘*x *is red’, then it is true if the object assigned to name *x* is in the set of objects assigned to ‘is red’, and it is false otherwise. Now we are dealing with worlds and times. We are saying also that a name in one world is assigned the same object as it is in all worlds and times. If we say further that all the times up to a certain time in question have the same domains and that all the predicates for these times in both worlds have the same extensions (the same set of objects assigned to them), then this means that all atomic formulae that are true in one world are true in the other. This is because, were we to perform the truth evaluation procedure to test them, we would find that the names, which are assigned to the same objects, will, when placed in a formula, yield the same sets of objects for the predicates, and thus in both cases fulfill the truth evaluation in the same way. Nolt’s next point is that this does not always hold for non-atomic formula. It seems here he is referring specifically to formulas that are modified by tense operators that refer to the future. For, two worlds with the same histories up to a certain point do not necessarily have the same futures after that point.]

Since names are not relativized to either worlds or times, stipulating that all times up to

thave the same domains and give predicates the same extensions in both worlds insures that the truth values of atomic formulas are the same in both worlds up tot. It also guarantees that most nonatomic formulas have the same truth values inw_{1 }andw_{2 }– but not that all nonatomic formulas do. We should expect, for example that a formula of the formFΦ might have different truth values inw_{1 }andw_{2 }even at times beforet, since though the two worlds’ histories up totare the same, their futures need not be.(Nolt 372)

[Nolt’s next point is that the accessibility relation is reflexive, symmetric, and transitive. It would be reflexive for some world if that world shared with itself the same moments and truths as itself (prior to some moment within it), which of course it does. The accessibility relation for two worlds would be symmetric if it both holds for the first to the second and as well for the second to the first. Since as we said the times (and names and extensions) will be the same for the worlds up to a point, that means both will be accessible to each other up to that point. And finally, the accessibility relation between three worlds will be transitive if the first’s being accessible to the second and the second to the third implies that the first is accessible to the third. This would be the case, because the common basis for the accessibility of the first pair would be the same basis for the second pair, meaning that the first and third would also share the same basis for accessibility, namely, the same times (and referents and extensions for names and predicates) up to a certain time.]

It is not difficult to see from our definition of the accessibility relation that ℛ is reflexive, symmetric, and transitive – in the sense that, for any worlds

w_{1},w_{2}, andw_{3 }and any timetin them:

w_{1 }ℛw_{1}tif

w_{1 }ℛw_{2}t, thenw_{2 }ℛw_{1}tif

w_{1 }ℛw_{2}tandw_{2 }ℛw_{3}t, thenw_{1 }ℛw_{3}t.(Nolt 372)

[For the next idea, recall the following from section 12.1 (found between ellipses):

...

[Recall that the rules of Leibnizian modal logic include all the rules from propositional logic along with the identity rules and the seven additional rules given in section 11.4, namely, DUAL, K, T, S4, B, N, and □=. And the rules other than =I, =E, and □= (that is, the set of purely propositional rules) makes up a logic called S5. Nolt now says that reflexivity, transitivity, and symmetry in the accessibility relation “define the logic of S5, which is characterized by Leibnizian semantics”.]

these three characteristics together define the logic S5, which is characterized by Leibnizian semantics. That is, making the accessibility relation reflexive, transitive, and symmetric has the same effect on the logic as making each world possible relative to each.

(Nolt 343)

...

]

Nolt says that given the properties of our tense logic, it is associated with the alethic logic S5. However, here we are concerned especially with the tense operators and not just with the alethic operators (necessity and possibility). Nolt will now give the valuation rules for modal tense logic. He explains that the first twelve are simply the Kripkean evaluation rules for alethic modal logic, except they have been modified to incorporate times as well as worlds. The last four rules will establish the valuations for the tense operators. Nolt explains that he will not give the rules for falsity, because the list is already quite long. Normally a formula is false if and only if it is not true. The exceptional cases are nondenoting names, but we are putting them aside here. [Nolt does not elaborate further on these cases. He discusses them later in section 15.1 on free logics. See also Graham Priest’s *In Contradiction* section 4.7.] [For rules 13 through 16, recall the following translations for the symbols:

**H **– it has always been the case that

**P **– it was (at some time) the case that

**G **– it will always be the case that

**F **– it will (at some time) be the case that

]

Valuation Rules for Modal Tense LogicGiven any valuation

vof modal tense logic whose set of worlds isW, for any world_{v}winWand time_{v}tinw:1. If Φ is a one-place predicate and α is a name whose extension

v(α) is inD_{(t,w)}, thenv(Φα,t,w) = T iffv(α) ∈v(Φ,t,w).2. If Φ is an

n-place predicate (n>1) and α_{1 }... , α_{n }are names whose extensions are all inD_{(t,w)}, then

v(Φα_{1}, ... , α_{n},t,w) = T iff <v(α_{1}), ... ,v(α_{n})> ∈v(Φ,t,w)3. If α and β are names, then

v(α = β,t,w) = T iffv(α) =v(β).

For the next five rules, Φ and Ψ are any formulas:

4.

v(~Φ,t,w) = T iffv(Φ,t,w) ≠ T5 .

v(Φ & Ψ,t,w) = T iff bothv(Φ,t,w) = T andv(Ψ,t,w) = T6 .

v(Φ ∨ Ψ,t,w) = T iff eitherv(Φ,t,w) = T orv(Ψ,t,w) = T, or both7.

v(Φ → Ψ,t,w) = T iff eitherv(Φ,t,w) ≠ T orv(Ψ,t,w) = T, or both8 .

v(Φ ↔ Ψ,t,w) = T iffv(Φ,t,w) =v(Ψ,t,w)

For the next two rules, Φ

^{α}/_{β}stands for the result of replacing each occurrence of the variable β in Φ by α, andD_{(t,w) }is the domain thatvassigns to worldwat timet.9 .

v(∀_{β}Φ,t,w) = T iff for all potential names α of all objectsdinD_{(t,w)},v_{(α,d)}(Φ^{α}/_{β },t,w) = T10 .

v(∃_{β}Φ,t,w) = T iff for some potential name α of some objectdinD,_{w}v_{(α,d)}(Φ^{α}/_{β },w) = T11 .

v(□Φ,t,w) = T iff for all worldsusuch thatwℛut,v(Φ,t,u) = T12.

v(◊Φ,t,w) = T iff for some worldu,wℛutandv(Φ,t, u)= T13.

v(HΦ,t,w) = T iff for all timest′ inwsuch thatt′ℰtandv(Φ,t′,w) = T14.

v(PΦ,t,w) = T iff for some timet′ inw,t′ℰtandv(Φ,t′,w) = T15.

v(GΦ,t,w) = T iff for all timest′ inwsuch thattℰt′ andv(Φ,t′,w) = T16.

v(FΦ,t,w) = T iff for some timet′ inw,tℰt′ andv(Φ,t′,w) = T

[Rule 9 is saying that the ‘for all’ quantification of a formula is true in a world at a certain time if it holds for all possible substitutions of the variables for that world and time. Rule 10 says that the ‘for some’ or ‘there is a’ quantification of a formula is true at a world and time if it holds for at least one substitution of the variables. Rule 11 says that the necessity operation on a formula is true for a world at a time if it is true for all worlds accessible to that world. That means in our modal tense context that the formula does not need to be true for worlds whose histories are not the same as the one in question. So suppose two worlds at a certain time have the same pasts and the same possible futures. What is possible in one world will be possible in another, since what happens in one could happen in the other. And this is most notable here with reference to what is possible to happen in the future. In other words, at that time point, both could go down the same path. Then suppose a moment later that one world has gone down one path and the other world has gone down another path. This means that the further future branches of one world are no longer available to those of the other (at least as attained by means of the same series of events). So since they lost accessibility, what is possible in one is not necessarily possible in the other. (Also, since pasts cannot be altered, one world will no longer be able to have the same past as the other, and in that sense perhaps they are not longer accessible temporally speaking.) Rule 13 says that a formula modified by the ‘it has always been the case that’ operator (**H**) is true for a world and time only if it holds for all times preceding that one in question. Rule 14 says that a formula modified by the ‘it was (at some time) the case that’ (**P**) is true for a world and time only if prior to that time in that world there is at least one moment where it is true. Rule 15 says that a formula modified by the ‘it will always be the case that’ operator (**G**) is true at a world and time only if it holds for all times coming after that one, in that world. Rule 16 says that a formula modified by the ‘it will (at some time) be the case that’ operator (**F**) is true for some world and time only if there is at least one moment in the future where it holds in that world.]

Nolt then revises some of the basic semantic definitions to suit this modal tense logic:

DEFINITION A formula is

validiff it is true at all times in all worlds on all of its valuations.DEFINITION A formula is

consistentiff it is true at at least one time in at least one world on at least one valuation.DEFINITION A formula is

inconsistentiff it is not true at any time in any world on any of its valuations.DEFINITION A formula is

contingentiff there is a valuation on which it is true at some time in some world and a valuation on which it is not true at some time in some world.DEFINITION A

set of formulasisconsistentiff there is at least one valuation containing a world in which there is a time at which all the formulas in the set are true.DEFINITION A

set of formulasisinconsistentiff there is no valuation containing a world in which there is a time at which all the formulas in the set are true. |DEFINITION Two formulas are

equivalentiff they have the same truth value at every time in every world on every valuation of both.DEFINITION A

counterexampleto a sequent is a valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.DEFINITION A sequent is

validiff there is no world in any valuation containing a time at which its premises are true and its conclusion is not true.DEFINITION A sequent is

invalidiff there is at least one valuation containing a world in which there is a time at which its premises are true and its conclusion is not true.(Nolt 373-374)

Nolt will now show some applications of modal tense logic. He will focus on the problem of determinism. [He seems to be saying that according to the determinist view, there would never be alternate possibilities but rather just one solid trunk.]

The primary application of modal tense logic is in clarifying our understanding of the relation between time and possibility. One of the perennial philosophical issues concerning that relation is the question of determinism. Determinism is the thesis that at any given time the only possible world is the actual world – that, in terms of our picture, the tree of time has no thin branches.

(Nolt 374).

Nolt says there have been a number of arguments for determinism. One of them holds that “since God knows everything that will happen, the course of events cannot deviate from what God foresees and is therefore determined” (374). Nolt notes that this argument is based, however, “on a dubious theological premise” (374).

The more cogent arguments are ones “that aim to deduce determinism not from assumptions about God’s foreknowledge, but from assumptions about the structure of time itself” (374). He then gives an example of such an argument. [Here the relevant structural feature of time seems to be that after events happen, you cannot rewrite history. The inference from this structural feature of time then seems to be the following. We take the perspective of the present. What happened before this moment cannot be altered. We also cannot alter this moment, as it is actually what is happening now. So the prior moment cannot be altered, and this moment following it cannot be altered. Were we to go back in time, any event coming after and leading up to the present will likewise follow a course that cannot be altered. Since the present is a moment in time which is past in relation to future moments (which will later be present), that means also we cannot alter the course of events into the future. (See Priest’s *Logic: A Very Short Introduction* chapter 6 and chapter 8 for similar discussions.) The structure of the argument begins with the fact that something is the case now. From this we infer that it was the case in the past that in the future this would be so (now). Then, from this we infer that it was always the case that it necessarily would be so that later in the future (now) this would be so. Thus the present moment was predetermined in the past.]

consider the following argument, which purports to show that anything that happens has always been predetermined (i.e., has always necessarily been going to happen):

Suppose that as a matter of fact, a certain event happens – for example, that you read this logic book. Then it has always been the case that you would read this logic book. Therefore it was always necessary that you would read this book. Since the same reasoning can be applied to any actual event, anything that happens was always destined to happen.

The core of this argument consists of two inferences, which, using ‘R’ for ‘You read this book’, we may formalize as follows:

R ⊢

HFR

HFR ⊢H□FR(374)

Nolt then shows that the first inference is valid. [Nolt first does the proof intuitively. We suppose for reductio the negation of the conclusion, and we will see if we can find a contradiction. So we suppose that we are reading this book, but in fact it was not the case that we were always going to be reading the book now. The next part of the reasoning seems to be that since in all moments before this one we were not going to be reading the book, that means there is at least one such past moment. Let us now look closer at this past moment and what we are saying about it. We are saying that for this past moment, we were not going to be reading the book in any moment coming after it. That means in the present moment in question, we are not reading the book. But we said already that we are reading that book. This means that the inference is valid.]

Analysis of these inferences provides a good illustration of the uses of our semantics. The first is valid. Intuitively, we can see this as follows. Suppose for reductio that this inference is invalid – that you are now at this moment,

t, reading this book, but that it is not the case that you have always been going to read this book. Since you have not always been going to read this book, there was a time earlier thant, call itt′, at which it is not the case that you were going to read this book. | But then it was true att′ that at all later moments you would not read this book. Since in particular the current momenttis later thant′, it follows that you are not reading this book now att– which contradicts what we said above. Hence the first inference is valid. Here is the same reasoning in strict meta theoretic terms:

METATHEOREM: ‘R ⊢HFR’ is valid.

PROOF: [see page 375. Nolt performs a reductio like he did informally above].(374-375, boldface in the original)

So the first inference is valid. But what about the second one, **HF**R ⊢ **H**□**F**R? Nolt will show it is invalid. For this, we need to “construct a model on which the premise is true and the conclusion is false” (375). [So it would need to be true that there was a moment in the past for which the present moment of reading is true, but also that there was a moment in the past in which it is not *necessarily* the case that in the future it will be true. Thus for that past moment, there needs to be a possible world where reading the book does not happen in the future. Nolt says that in this alternate possible world, reading the book does not happen in the present. (He will construct a model without any moments beyond the present, so that will eliminate the possibility that in the other world, we are reading in a moment past the present.) Nolt says that our model will need two worlds. Both have the same past up to this moment, but the alternate world does not have us reading in the present moment. This means that in this alternate world, during the past moment, it was not going to be the case that in the future we would read the book. Therefore, it is not necessary in our own world that in the past we would be reading it in the future. In the formal proof, we say there are two worlds and two times. In world 1, we are not reading at the past time 1 but we are reading in the current time 2. In world 2, we are not reading at the past time 1 and we also are not reading in the present time 2. In world 1 at time 1 (the past time), **F**R is true, because it is true that at a later time R (we are reading) holds. The model then infers that **HF**R (it has always been the case that it will be the case that R) at time 2 is true. This is because there is only one time prior to the present time 2, so it holds for all such past moments. In world 2 at (past) time 1, however, **F**R is not true, because in the only moment coming after it, R is false. Now, at time 1, both worlds have access to each other, because they have the same pasts. This means that at time 1, it is not true that it is necessary in world 1 for R to hold in time 2, because in the alternate (and accessible) world it does not. Now recall rule 13: “*v*(**H**Φ, *t*, *w*) = T iff for all times *t*′ in *w* such that *t*′ℰ*t* and *v*(Φ, *t*′, *w*) = T”. Now since in world 1, □**F**R is false for all times coming before time 2, that means it is not true (at world 1 and time 2) that it was always necessary in the past that in the future R would hold. So this model shows the premise that **HF**R is true for world 1 and time 2, and this is because in world 1, it always was going to be the case that R will hold at time 2. However, the conclusion **H**□**F**R was shown to be not true, because in the model there is another world for which in the past moment 1, R will not hold for time 2.]

To make this informal reasoning rigorous, we must formalize this counterexample. We aim to make it simple – though that also makes it unrealistic. Our model need contain only two times, a past time

t_{1 }and the presentt_{2}, and two worlds,w_{1 }andw_{2}, each containing both times. (The future plays no role in this example, nor do changes in the past that would require more than one past time in either world.) The sentence letter ‘R’ will be false att_{1 }and true att_{2}in the actual worldw_{1 }and false at botht_{1 }andt_{2}in the merely possible worldw_{2}. [...] | [...] These stipulations define the model. Having done that, the only work that remains is to apply the truth clauses for the operators to verify that this model does indeed make the premise ‘HFR’ true and the conclusion ‘H□FR; false:

METATHEOREM: ‘HFR ⊢H□FR’ is invalid.

PROOF: [See page 376. The proof follows the reasoning outlined above in brackets.](Nolt 375-376, boldface in the original)

Nolt then acknowledges that this proof does not refute determinism itself but rather just one argument for it. In fact, since it is modeled on non-deterministic time where there are multiple possible futures,

That it yields counterexamples to deterministic arguments is therefore no wonder. The determinist could well retort that we have used the wrong semantics and hence the wrong logic, that the true semantics represents time not as a branching tree but as a single nonbranching line, and that our semantics simply begs the question.

(376)

Nolt then notes the need to incorporate metaphysical assumptions into logical modeling in order to make the findings philosophically interesting and fruitful, but this makes them more contestable.

Logic alone cannot settle this issue. For any purported solution, one can always ask whether the correct semantics, and hence the correct logic, has been used. But here opinions will differ. One can be extremely conservative, allowing into one’s semantics only the most strictly logical presuppositions or one can be more venturesome, adopting presuppositions with a metaphysical tinge. (I have been somewhat venturesome, for example, in assuming that ℰ is transitive and, within worlds, linear, and also in assuming that accessibility amounts to shared | history; the determinist, who makes the strong assumption that only one world is possible at any given time, is more venturesome still.) Conservative logics, which operate with fewer presuppositions, are less controversial. But, because they validate fewer inferences, they are also less interesting. The most interesting tense logics venture some way into the hazy borderland between logic and metaphysics.

(Nolt 377)

But even though these debates over metaphysical assumptions call into question our proofs, we still can “clarify our conceptions if we take the time to formalize them and relate them to various models” (377). Nolt adds that there is still something we can gain from making a working model of deterministic time, namely it shows that deterministic time is a consistent concept and thus

a nondeterministic universe cannot be ruled out on logical grounds alone.

There is a logic of nondeterministic time, whether or not time itself is deterministic.(Nolt 377, boldface mine)

**[[I note here that to say there is a ‘logic’ of something means that there is a consistent model for it.]]**

[Nolt now turns to a description of the general features of modal tense logic, which I leave for another summary, even though there is no formal section division within section 13.2.]

From:

Nolt, John. *Logics*. Belmont, CA: Wadsworth, 1997.

Or if otherwise noted:

Priest, Graham. *Logic: A Very Short Introduction*. Oxford: Oxford University, 2000.

[See http://piratesandrevolutionaries.blogspot.com/2015/12/priest-ch8-of-logic-very-short_18.html

]

.

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