by Corry Shores
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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]
Summary of
Graham Priest
An Introduction to Non-Classical Logic: From If to Is
Part I:
Propositional Logic
3.
Normal Modal Logics
3.6a
The Tense Logic Kt
Brief summary:
(3.6a.1) We will now examine tense logic. (3.6a.2) The semantics of tense logic are the same as modal logic, only with some modifications to reflect certain temporal senses. The notion of succession is modeled with the accessibility relation such that w1Rw2 has the intuitive sense: ‘w1 is earlier than w2’. “□A means something like ‘at all later times, A’, and ◊A as ‘at some later time, A’,” but “we will now write □ and ◊ as [F] and ⟨F⟩, respectively. (The F is for ‘future’)” (49). (3.6a.3) The tense logic operators for the past are [P] and ⟨P⟩, which correspond semantically to □ and ◊. (3.6a.4) We evaluate the tense operators in the following way:
vw([P]A) = 1 iff for all w′ such that w′Rw, vw′ (A) = 1
vw(⟨P⟩A) = 1 iff for some w′ such that w′Rw, vw′ (A) = 1
vw([F]A) = 1 iff for all w′ such that wRw′, vw′ (A) = 1
vw(⟨F⟩A) = 1 iff for some w′ such that wRw′, vw′ (A) = 1
(50, with the future operator formulations being my guesses.)
(3.6a.5) “If, in an interpretation, R may be any relation, we have the tense-logic analogue of the modal logic, K, usually written as Kt” (50). (3.6a.6) The tableaux rules for the tense operators is much like for necessity and possibility only we need to keep in mind the order of r formulations for the different tenses. Priest provides the following tableau rules for the tense operators.
| Full Future Development ([F]D) |
| [F]A,i irj ↓ A,j
(For all j) |
| Partial Future Development (⟨F⟩D) |
| ⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
| Negated Full Future Development (¬[F]D) |
| ¬[F]A,i ↓ ⟨F⟩¬A,i |
| Negated Partial Future Development (¬⟨F⟩D) |
| ¬⟨F⟩A,i ↓ [F]¬A,i |
| Full Past Development ([P]D) |
| [P]A,i jri ↓ A,j
(For all j) |
| Partial Past Development (⟨P⟩D) |
| ⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) |
| Negated Full Past Development (¬[P]D) |
| ¬[P]A,i ↓ ⟨P⟩¬A,i |
| Negated Partial Past Development (¬⟨P⟩D) |
| ¬⟨P⟩A,i ↓ ⟨P⟩¬A,i |
(50, with my added names and other data at the bottoms)
(3.6a.7) Priest then gives a tableau example. (3.6a.8) Priest then shows how to construct a counter-model in tense logic, using an example. (We use the same procedure given in section 2.4.7.) (3.6a.9) We can think of time going in reverse, from the future, moving backward through the past, by taking the converse R relation (yRx becomes xŘy) (and/or by converting all F’s to P’s and vice versa ).
[Turning to Tense Logic]
[Tense Logic Notation. Future Operators: [F] and ⟨F⟩.]
[Past Operators: [P] and ⟨P⟩]
[Tense Operator Semantic Evaluation]
[Tense Logic as Kt]
[Tense Logic Tableau Rules]
[Tableau Example]
[Counter-Models]
[Mirror Images and Converse Temporal Relations]
Summary
[Turning to Tense Logic]
[We will now examine tense logic.]
[In previous sections, we have been learning normal modal logics K. Now Priest will now look at tense logic.]
In the last two sections of this chapter, we will look at another interpretation of modal logics: tense logic.
(49)
[Tense Logic Notation. Future Operators: [F] and ⟨F⟩.]
[The semantics of tense logic are the same as modal logic, only with some modifications to reflect certain temporal senses. The notion of succession is modeled with the accessibility relation such that w1Rw2 has the intuitive sense: ‘w1 is earlier than w2’. “□A means something like ‘at all later times, A’, and ◊A as ‘at some later time, A’,” but “we will now write □ and ◊ as [F] and ⟨F⟩, respectively. (The F is for ‘future’)” (49). ]
[Tense logic uses the same semantics as for normal modal logic, only we give a different intuitive sense to the R accessibility relation for worlds. We now think of a world as being a world at some time, and the R relation as meaning that the first indicated world is at a time prior to the second one. With that in mind, the necessity operator would be like “at all later times” while the possibility operator means “at some later time”. We now write □ as [F] and ◊ as ⟨F⟩. (At this point I am a little confused. I am guessing that the notion of “at all later times” means that we are thinking that the future is a singular linear series, and we are saying that the proposition holds for all future moments. I would guess an example would be something like, at all later times, water is wet. And for, “at some later time,” maybe an example could be, at some later time it will be raining. I am a little uncertain, because I am wondering about how the future can also be seen as branching out along divergent possibilities, and thus to say “at all later times” would mean something like, regardless of which way things go, in all cases something will hold. For example, at all later times the sun is on the path to dying. But as far as I can tell, that is not the meaning, even though I still cannot say for sure yet.)]
The semantics of a tense logic are exactly the same as those for a normal modal logic. Intuitively, though, one thinks of the worlds of an interpretation as times (or maybe states of affairs at times), and the relation w1Rw2 as ‘w1 is earlier than w2’. Hence □A means something like ‘at all later times, A’, and ◊A as ‘at some later time, A’. For reasons that will become clear in a moment, we will now write □ and ◊ as [F] and ⟨F⟩, respectively. (The F is for ‘future’.)
(49)
[Past Operators: [P] and ⟨P⟩]
[The tense logic operators for the past are [P] and ⟨P⟩, which correspond semantically to □ and ◊.]
[In tense logic there are also operators for the past, [P] and ⟨P⟩, semantically corresponding to □ and ◊.]
What is novel about tense logic is that another pair of operators, [P] and ⟨P⟩, is added to the language. (The P is for ‘past’.)5 Their grammar is exactly the same as that for [F] and ⟨F⟩. So we can write things such as ⟨P⟩[F](p∧¬[P]q).
(49)
5. Traditionally, the operators ⟨F⟩, [F], ⟨P⟩ and [P], are written as F, G, P and H, respectively.
(49)
[Tense Operator Semantic Evaluation]
[We evaluate the tense operators in the following way: vw([P]A) = 1 iff for all w′ such that w′Rw, vw′ (A) = 1; vw(⟨P⟩A) = 1 iff for some w′ such that w′Rw, vw′ (A) = 1; vw([F]A) = 1 iff for all w′ such that wRw′, vw′ (A) = 1; vw(⟨F⟩A) = 1 iff for some w′ such that wRw′, vw′ (A) = 1.]
[Priest now gives the truth conditions for the tense operators. (Recall from section 3.6a.2 above that the R relation indicates successive order of times: “one thinks of the worlds of an interpretation as times (or maybe states of affairs at times), and the relation w1Rw2 as ‘w1 is earlier than w2’” (p.49, section 3.6a.2). So here, the first world in the wRw sequence comes before the second world, meaning really the first moment of a world comes before the second moment of that world. Now recall from section 3.6a.3 above that [P] means something like, “at all prior times...” and that it is equivalent to the necessity operator, except with the additional temporal qualification of holding for just a certain set of other worlds (times), namely, the ones in the past. As such, [P] would be true if for all worlds (times) in the past, the formula it modifies holds. And so [P], which means something like, “at some prior time...” and which is equivalent to the possibility modifier, would be true if there is at least one prior world (time) where the formula holds. The same would go for the future tense operators, which I will add to Priest’s explicit formulations for the past ones. But I may have them wrong, so please trust your own formulations over mine.
vw([P]A) = 1 iff for all w′ such that w′Rw, vw′ (A) = 1
vw(⟨P⟩A) = 1 iff for some w′ such that w′Rw, vw′ (A) = 1
vw([F]A) = 1 iff for all w′ such that wRw′, vw′ (A) = 1
vw(⟨F⟩A) = 1 iff for some w′ such that wRw′, vw′ (A) = 1
(50, with the future operator formulations being my guesses.)
]
The truth conditions for ⟨P⟩ and [P] are exactly the same as those for ⟨F⟩ and [F], except that the direction of R is reversed:
vw(⟨P⟩A) = 1 iff for some w′ such that w′Rw, vw′ (A) = 1
vw([P]A) = 1 iff for all w′ such that w′Rw, vw′ (A) = 1
(50)
[Tense Logic as Kt]
[“If, in an interpretation, R may be any relation, we have the tense-logic analogue of the modal logic, K, usually written as Kt” (50).]
[I am not sure about the next point, so please consult the quotation below. I am guessing it is saying the following. We have distinguished R relations which relate a world to a past world from R relations that relate a world to a future world. But if we make it such that we consider all R relations alike, then we have a system no different from normal modal logic K. But maybe this is not what is meant. The footnote perhaps clarifies this matter, but I am not sure. The point might be the following, but I am guessing. We could distinguish two sorts of R relations, one for the past and one for the present. However, this is unnecessary, because we only would need to switch the places of the w and w′ in the formulation w′Rw to designate the other temporality. Overall I am confused, because still it would seem that we have different semantic rules than K, as we have two sets of the same rule. Let us look again at the rules in K for necessity and possibility and compare them with the tense rules:
For any world w ∈ W:
vw(◊A) = 1 if, for some w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
vw(□A) = 1 if, for all w′ ∈ W such that wRw′, vw′(A) = 1; and 0 otherwise.
(Priest p.22, section 2.3.5)
vw([P]A) = 1 iff for all w′ such that w′Rw, vw′ (A) = 1
vw(⟨P⟩A) = 1 iff for some w′ such that w′Rw, vw′ (A) = 1
vw([F]A) = 1 iff for all w′ such that wRw′, vw′ (A) = 1
vw(⟨F⟩A) = 1 iff for some w′ such that wRw′, vw′ (A) = 1
(50, with the future operator formulations being my guesses.)
The necessity/possibility rules of K have the same structure as the future operator. But the ones for the past reverse the relation, and so I still am not grasping how it is an analogue of K. Given what is said later, I am going to make the following guess, which is likely wrong. Tense logic is called Kt, because it is structurally the same as K except it is endowed with an intuitive temporal sense by making specific semantic rules corresponding to the tense operators, which otherwise function exactly the same as necessity and possibility. Or maybe the idea is that an analogue need not be identical but simply be structured analogously.]
If, in an interpretation, R may be any relation, we have the tense-logic analogue of the modal logic, K, usually written as Kt.6
(50)
6. Generally speaking, modal logics with more than one pair of modal operators are called ‘multimodal logics’, and in an interpretation for such a logic there is an accessibility relation, RX, for each pair of operators, ⟨X⟩ and [X]. In tense logic, however, it is unnecessary to give an independent specification of RP, since this is just the converse of RF. That is, w1RPw2 iff w2RFw1.
(50)
[Tense Logic Tableau Rules]
[The tableaux rules for the tense operators is much like for necessity and possibility only we need to keep in mind the order of r formulations for the different tenses.]
[Priest now provides the following tableau rules for the tense operators. (See quotation below).]
Appropriate tableaux for Kt are easy. The rules for ⟨F⟩ and [F] are exactly the same as those for ◊ and □, and those for ⟨P⟩ and [P] are the same with the order of r reversed appropriately. Thus, we have:
| Full Future Development ([F]D) |
| [F]A,i irj ↓ A,j
(For all j) |
| Partial Future Development (⟨F⟩D) |
| ⟨F⟩A,i ↓ irj A,j
(j must be new: it cannot occur anywhere above on the branch) |
| Negated Full Future Development (¬[F]D) |
| ¬[F]A,i ↓ ⟨F⟩¬A,i |
| Negated Partial Future Development (¬⟨F⟩D) |
| ¬⟨F⟩A,i ↓ [F]¬A,i |
| Full Past Development ([P]D) |
| [P]A,i jri ↓ A,j
(For all j) |
| Partial Past Development (⟨P⟩D) |
| ⟨P⟩A,i ↓ jri A,j
(j must be new: it cannot occur anywhere above on the branch) |
| Negated Full Past Development (¬[P]D) |
| ¬[P]A,i ↓ ⟨P⟩¬A,i |
| Negated Partial Past Development (¬⟨P⟩D) |
| ¬⟨P⟩A,i ↓ ⟨P⟩¬A,i |
(50, with my added names and other data at the bottoms)
In the first rule of each four, this is for all j; in the second, j is new.
(50)
[Tableau Example]
[Priest then gives a tableau example.]
[Priest now gives an example for a tableau. (See section 2.4 for how to construct modal tableaux.) We see from the example the way that the two tense operators interact.]
The main novelty in Kt is in the interaction between the future and past tense operators. Thus, for example, A⊢[P]⟨F⟩A:
A⊢[P]⟨F⟩A
1.
.
2.
.
3.
.
4a.
4b.
.
5.
.
6.
.
.
A,0
↓
¬[P]⟨F⟩A,0
↓
⟨P⟩¬⟨F⟩A,0
↓
1r0
¬⟨F⟩A,1
↓
[F]¬A,1
↓
¬A,0
×
P
.
P
.
2¬[P]
.
3⟨P⟩
3⟨P⟩
.
4b¬⟨F⟩
.
4a,5[F]
(1×6)
Valid
(50, enumerations and step accounting are my own and are not to be trusted)
We have the last line, since 1r0.
(50)
[Counter-Models]
[Priest then shows how to construct a counter-model in tense logic, using an example. (We use the same procedure given in section 2.4.7.)]
[Recall from section 2.4.7 how we construct counter-models in K:
Counter-models can be read off from an open branch of a tableau in a natural way. For each number, i, that occurs on the branch, there is a world, wi; wiRwj iff irj occurs on the branch; for every propositional parameter, p, if p, i occurs on the branch, vwi(p) = 1, if ¬p, i occurs on the branch, vwi(p) = 0 (and if neither, vwi(p) can be anything one wishes).
(p.27, section 2.4.7)
Priest says that we formulate counter models in Kt in the same way.]
Counter-models are read off from tableaux just as they are for K. For example, ⊬p ⊃ ([F]p ∨ [P]p). The tableau for this is:
⊬p ⊃ ([F]p ∨ [P]p)
1.
.
2.
.
3.
.
4.
.
5.
.
6.
.
7.
.
8a.
8b.
.
9a.
9b.
.
¬(p ⊃ ([F]p ∨ [P]p)),0
↓
p,0
↓
¬([F]p ∨ [P]p),0
↓
¬[F]p,0
↓
¬[P]p,0
↓
⟨F⟩¬p,0
↓
⟨P⟩¬p,0
↓
0r1
¬p,1
↓
0r2
¬p,2
P
.
1¬⊃
.
1¬⊃
.
3¬∨
.
3¬∨
.
4¬[F]
.
5¬[P]
.
6⟨F⟩
6⟨F⟩
.
7⟨P⟩
7⟨P⟩
(open)
(51, enumerations and step accounting are my own and are not to be trusted)
This gives the counter-model which may be depicted as follows:
w2 xx→xxw0 xx→xx w1
¬p xx→xxxpxxx→x x p
(51)
[Mirror Images and Converse Temporal Relations]
[We can think of time going in reverse, from the future, moving backward through the past, by taking the converse R relation (yRx becomes xŘy) (and/or by converting all F’s to P’s and vice versa ).]
[Priest next explains the notion of mirror image, where you replace all F’s with P’s and vice versa. I am not sure if the next idea is a result, another different thing, or an addition to that, but we can also say that the converse of any R relation is the reverse order of its terms, noted with Ř, such that yRx becomes xŘy. I am not sure what the intuitive notion is here, but I would guess that we might consider some timeline going from past to future and instead think of it as going from future to past, as if time reversed itself. I am guessing very wildly. At any rate, Priest’s next point is that in any such converse interpretation, whatever was valid or invalid in it will also be valid or invalid in the converse interpretation. Please read the quote, as I did not follow this one very well.]
If A is any formula, call the formula obtained by writing all ‘P’s as ‘F’s, and vice versa, its mirror image. Thus, the mirror image of [F]p ⊃ ¬ ⟨P⟩q is [P]p ⊃ ¬ ⟨F⟩q. Given any binary relation, R, let its converse, Ř, be the relation obtained by simply reversing the order of its arguments. Thus, xŘy iff yRx. It is clear that if we have any interpretation for Kt, the interpretation that is exactly the same, except that R is replaced by Ř, is just as good an interpretation. Moreover, in this interpretation, ⟨F⟩ and [F] behave in exactly the same way as ⟨P⟩ and [P] do in the original interpretation, and vice versa. Hence any inference is valid/invalid in Kt just if the inference obtained by replacing every formula by its mirror image is valid/invalid. So, for example, by3.6a.7, A⊢[F]⟨P⟩A.
(51)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
.
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