"if you think childlike, you'll stay young. If you keep your energy going, and do everything with a little flair, you're gunna stay young. But most people do things without energy, and they atrophy their mind as well as their body. you have to think young, you have to laugh a lot, and you have to have good feelings for everyone in the world, because if you don't, it's going to come inside, your own poison, and it's over" Jerry Lewis "I don’t believe in the irreversibility of situations" Deleuze
by Corry Shores
[Bracketed commentary and boldface are my own, unless otherwise noted (for example, there are some symbols that are boldface in the original). Please forgive my typos, as proofreading is incomplete. I am not trained in logic, so at times my summaries may be unhelpful or misleading. Please consult and trust the original text, which is absolutely wonderful.]
Summary of
Graham Priest
Logic: A Very Short Introduction
Ch.8
The Future and the Past: Is Time Real?
Very Brief Summary:
Tense logic allows us to examine the validity of inferences that refer to situations happening at different times. For a statement h referring to a state of affairs that could happen in one or in many temporally distinct situations, we can modify it to refer to a particular situation in the past (Ph) or in the future (Fh) or to all situations in the past (Hh) or in the future (Gh). We can construct a model of a chronological succession of situations and use the tense-modifiers to determine which temporally modified propositions are true and which are false. This helps for example in analyzing the validity of McTaggart’s argument that time is unreal. His reasoning is that an event happening only at one moment must be understood in two contradicting ways. It did not happen both in the past and the future, since it only happened one singular time. But since time flows, it once was true for the future and later was true for the past. Thus contrary to our prior claim, the event does happen both in the past and future. The tense-logic model shows the mistake in this reasoning. Never in one same temporally located situation does it happen both in the past and future, no matter how many additional tense modifiers we add to our formulations. Nonetheless, since this schema models the flow of time spatially, it is perhaps useless for this particular argument which is about the non-spatial flow of time.
Brief Summary:
We can use tense logic to analyze the validity of inferences that are based on statements referring to different moments in time. We first think of a one-dimensional series of situations arranged in their proper chronological sequence. We then think of statements of fact. They may or may not be true for one temporalized situation or another. Suppose a statement h is true only for the temporally situated moment s0. This statement refers to an instantaneous state of affairs, like the moment the first bullet entered Czar Nicolas’ heart. It will be false for all situations coming before and after that temporalized situation, since the event did not happen at those other moments. However, at a succeeding moment in the future, we can say truly that the event happened in the past. And likewise for a preceding moment in the past, we can say it will be happening in the future. We use the modifier P for past (“it was the case that”) and F for future (“it will be the case that”). So in moment s1, Ph is true, and for moment s-1, Fh is true. We can further designate temporal relations by compounding the modifiers. PPh would apply h to a situation coming before some other situation that is already in the past. FPh would apply h then to some situation coming after some other situation that is already in the past. Now, P and F refer to some determinate situation in the past or future. We can instead refer to all future situations with the modifier G (“it is always Going to be the case that”) and all past ones with the modifier H (“it Has always been the case that”). We can also make a model for this tense logic by arranging in sequence a number of s’s, placing s0 in the middle, and counting up and down the subscripts on both sides. This allows us to evaluate inferences based on tense modifiers. One example is McTaggart’s argument against the reality of time. If time is real, then the past and future are real, and thus they do not present logical contradictions. We then consider a sentence that is true just for the situation at one time-point. This means it did not happen in two temporally distinct time-points, and thus it did not happen both in the past and in the future: ¬(Ph&Fh). However, time flows, and so before it happened, it was in the future, and after it happened, it was in the past: Ph&Fh. The concepts of past and future present a contradiction, and thus time is unreal. One may object to the second formulation and say that it pretends that, for one situation that is located at one time point, the event can be both in the past and in the future. So to clarify the problem, we might then compound the modifiers and write ¬(PPh&FFh) to mean that the event did not happen at some determinate point coming before another in the past and at the same time happen at some determinate point coming after another in the future. Those following McTaggart’s reasoning can then say that still, because of the flow of time, PPh will be true and FFh was true, and thus, in contradiction with the prior, negated conjunction, PPh&FFh. But, by using the tense-logic model, we can display visually that the McTaggart argument is mistaken. There is never a singular temporalized situation where both terms in the past&future parings are true. Nonetheless, as this is a model that spatializes the flow of time, it might not be adequate for dealing with this argument about time’s non-spatial flow.
Summary
Time is something we deal with in our everyday behavior and thinking, and we often make intuitively valid inferences about time, like the following ones:
It is raining.
It will have been raining.
It will be true that it has always been raining.
It is raining.
But as soon as we begin to reflect more on time, we get confused. One problem is that time itself seems to change, but it is the seemingly constant form that measures how much everything else changes (56). This issue lies “at the heart of several conundrums concerning time,” and one of which was posed by John McTaggart Ellis McTaggart. “Like many philosophers, McTaggart was tempted by the view that time is unreal – that, in the ultimate order of things, time is an illusion” (56d).
We now need to fashion some symbols for our analysis of temporal relations in logic. We begin with the past tense designation, “it was the case that,” which we will notate in boldface as P (for past). So when we mean “The sun was shining” or “it was the case that the sun is shining” we write “P the sun is shining”. If we further abbreviate “The sun is shining” as s, then we can write for these sentences: Ps. We do something very similar for future time. For “The sun will be shining” or “It will be the case that the sun is shining,” we will write Fs, since F (future) will mean “it will be the case that”.
Priest writes, “P and F are operators, like ◻ and ◇, that affix to whole sentences to make whole sentences. Moreover, like ◻ and ◇, they are not truth functions” (56). To illustrate, he shows how by adding the future operator to two true sentences, in one case it stays true but in the other it becomes false.
‘It is 4 p.m.’ and ‘It is 4 p.m. on August 2nd, 1999’ are both true (at the instant I write); ‘It will be 4 p.m.’ is also true (at the present instant) – it is 4 p.m. once every day – though ‘It will be 4 p.m. August 2nd, 1999’ is not.
(56)
In logic, P and F are called tense operators. They can be iterated, meaning that they can be compounded. So, using our above example, we could have FPs, meaning, “The sun will have been shining” or put more formally, “It will be the case that it was the case that the sun is shining.” Or we could have PPs, meaning, “The sun had been shining” or “It was the case that it was the case that the sun is shining” (56). Priest also notes that we can iterate the modal operators. [I am not exactly sure how yet. What does it mean that it is possible that it is possible that the sun is shining, or necessary that it is necessary?] We might however have difficulty rendering iterated tense operators into standard English tenses. FPFs for example would have to be “It will be the case that it was the case that the sun will be shining”. Priest notes however that “The iterations, though, make perfectly good grammatical sense. We can call iterations of P and F, like FP, PP, FFP, compound tenses” (56).
One part of McTaggart’s argument is that without past and future, you would not have time, since they are of the essence of time. “Yet pastness and futurity, he argued, are inherently contradictory; so | nothing in reality can correspond to them” (56-57). One reason that past and future are contradictory is that they are incompatible. [The reasoning is a bit tricky. Perhaps the basic insight is that we can regard temporal relations from the perspective of a singular determinate position in the flow of time, or we can see time as one continuous flow that binds together all moments of time. In the first case, past and future mutually exclude one another. In the second case, they intertwine. If we orient ourselves within the flow of time by placing ourselves at one particular moment or another in the succession, we would see that a singular instantaneous event cannot happen both in the past and in the future. However, if we stand outside the flow of time and see all moments lined up in a series, we can see that each moment is past or future, relative to other moments. But no such moment is a point of origin for comparison under this view, so each moment is both past and future, because we have not oriented that moment by giving it a privileged position for comparison.]
If some instantaneous event is past, it is not future, and vice versa. Let e be some instantaneous event. It can be anything you like, but let us suppose that it is the passing of the first bullet through the heart of Czar Nicholas in the Russian Revolution. Let h be the sentence ‘e is occurring’. Then we have:
¬(Ph&Fh)
But e, like all events, is past and future. Because time flows, all events have the property of being future (before they happen) and the property of being past (after they happen):
Ph&Fh
So we have a contradiction.
(57)
[Someone might note that the second formulation, Ph&Fh, is confused about the structure of the series of moments. We might observe that, yes, there is one situation at one moment where Ph is true, and there is a prior situation where Fh is true, but there is never a situation where both is true. There was a moment in the past before it happened when it was still to happen in the future, FPh, then after this past event happened, it was in the past, PPh. On this basis, Priest will then make the formulation ¬(PPh&FFh), but I am not sure I follow how we get there. Why is this not ¬(PPh&FPh)? Perhaps it is ¬(PPh&FFh) to speak more generally, rather than to refer only to past historical events. But again, I do not know how yet. PPh&FFh seems to be referring to a point in the past, before which an event is said to happen, and a point in the future, after which an event is said to happen. Thus we are interposing a duration of time between the past and future moments. But maybe that is not what it means.]
This argument isn't likely to persuade anyone for very long. An event can’t be past and future at the same time. The instant the bullet passed through the Czar's heart was past and future at different times. It started off as future; became present for a painful instant; and then was past. But now – and this is the cunning part of McTaggart's argument – what are we saying here? We are applying compound tenses to h. We are saying that it was the case that the event was future, PFh; then it was the case that it was past, PPh. Now, many compound tenses, like simple tenses, are incompatible. For example, if any event will be future, it is not the case that it was past:
¬(PPh&FFh)
|
But, just as with the simple tenses, the flow of time suffices to ensure that all events have all compound tenses too. In the past, Fh; so in the distant past FFh. In the future, Ph; so in the distant future, PPh:
PPh&FFh
And we are back with a contradiction.
(57-58)
[It is still not entirely clear to me how to conceptualize the way this problem regresses, since like before, I do not understand how we go from PFFh in the example to FFFh in the conjunction. But the basic idea seems to be that the objector keeps designating a point before and a point after the event which each independently have their own different temporal relation to the event, when really we should be orienting our our statements about the event to one singular moment when it happens.]
Those who have kept their wits about them will reply, just as before, that h has its compound tenses at different times. It was the case that FFh; then, later on, it was the case that PPh. But what are we saying here? We are applying more complex compound tenses to h: PFFh and PPPh; and we can run exactly the same argument again with these. These compound tenses are not all consistent with each other, but the flow of time ensures that h possesses all of them. We may make the same reply again, but it, too, is open to the same counter-reply. Whenever we try to get out of the contradiction with one set of tenses, we do so only by describing things in terms of other tenses that are equally contradictory; so we never escape contradiction. That is McTaggart’s argument.
(58)
We will now see why we have this problem by examining “the validity of inferences concerning tenses” (58). We begin with a schema that does not use temporal modal operators like P and F. Rather, we begin with a series of temporally various situations s that follow one another along the one-dimensional extension of time. We orient the series around a middle point s0, with left-most being past-most, like so:
[We might have a statement, like “Czar Nicholas is shot in the heart”. We might call it a. Then, a is true in a situation s if it happened at that situation’s designated time-point. Suppose it really happens at s0. Then a is true for s0. This also means that Pa is true for all situations happening after s0.]
As usual, each s provides a truth value, T or F, for every sentence without tense operators. What about sentences with tense operators? Well, Pa is T in any situation, s, just if a is true in some situation to the left of s; and Fa is true in s just if a is true in some situation to the right of s.
(58)
[In our prior example where a meant “Czar Nicholas is shot in the heart”, the statement a was true only for one situation s. Now we are to think of a proposition being true in all situations in the past or in the future. I do not know how to exemplify this. Perhaps we might make a statement like, “the cosmos exists in one form or another”. And supposing that the cosmos has no beginning or end, then this statement will hold for all s’s coming before and after some given one.]
While we are doing all this, we can add two new tense operators, G and H. G can be read ‘It is always Going to be the case that’, and Ga is true in any situation, s, just if a is true in all situations to the right of s. H can be read as ‘It Has always been the case that’, and Ha is true in any situation, s, just if a is true in all situations to the left of s. (G and H correspond to F and P, respectively, in just the way that ◻ corresponds to ◇.)
(59)
[Recall our inferences from the beginning of the chapter. 1) It is raining. Therefore, it will have been raining. 2) It will be true that it has always been raining. Therefore, it is raining.] Using these new tense operators, we can formulate the two inferences from the beginning of the chapter like this:
r
FPr
FHr
r
[We look initially at the first one. r means “it is raining”. We assume it is true in some situation s0. I do not follow so well, but it seems we then add P, which then orients us at some point in the future, where r is now true in the past. Then we add F, which takes us back to the present, where it is true that in the future Pr is true. But there must be a much better way to restate the following:]
The first inference is valid, since if r is true in some situation, so, then in any situation to the right of so, say s1, Pr is true (since so is to its left). But then FPr is true in so, since s, is to its right. We can depict things like this:
(59)
[The second inference again was: It will be true that it has always been raining. Therefore, it is raining. The H modifier to r will orient us somewhere in the future, and it says that r is true in all prior moments. Since so is one such prior moment, the inference is true.]
The second inference is valid, since if FHr is true in so, then in some situation to the right of so, say s2, Hr is true. But then in all situations to the left of s2, and so in particular so, r is true.
(59)
Priest then notes that certain combinations of tenses are impossible. [If an event happens just within one time-point, then it cannot happen in more than one time-point. Thus it cannot be both past and future at any point in time.]
if h is a sentence that is true in just one situation, then Ph&Fh is false in every s. Both conjuncts are false in so, the first conjunct is false to the left of so; the second conjunct is false to the right. Similarly, e.g., PPh&FFh is false in every s.
(60a)
We return now to McTaggart’s argument. [We were dealing with an event holding exclusively for a determinate time-point. The contradiction we found was between two sets of seemingly equally valid claims. The first claim is that the event cannot happen both in the past and in the future, ¬(Ph&Fh), and yet, since time flows, the event would have to have been in the future before it happened and in the past after it happened, Ph&Fh. The next step was an objection to the second claim. It implies that the event is future and past at the same time. But really, we need to be clear that it was past at one time and future at another. For this, we add another tense marker to make it clear that the McTaggart argument is oriented around two temporally separate moments. And thus it is not saying that in one same moment the event is in both the past and the future. The one taking the McTaggart view will then say that because of the flow of time, any more specific designations will still be both past and future, for the same reason as before. (In fact I am not exactly sure how things work with these compounded tenses in the McTaggart debate, since, as I mentioned before, the literal translation of the formulations is hard for me to situate in the way it is presented here.)]
Now, how does all this bear on McTaggart's argument? The upshot of McTaggart's argument, recall, was that, given that h has every possible tense, it is never possible to avoid contradiction. Resolving contradictions in one level of complexity for compound tenses only creates them in another. The account of the tense operators that I have just given, shows this to be false. Suppose that h is true in just so. Then any statement with a compound tense concerning h is true somewhere. For example, consider FPPFh. This is true in s2, as the following diagram shows:
Clearly, we can do the same for every compound tense composed of F and P, zigzagging left or right, as required. And all this is perfectly consistent. The infinitude of different situations allows us to assign h all its compound tenses in appropriate places without violating the various incompatibilities between them, e.g., by having Fh and Ph true in the same situation. McTaggart's argument, therefore, fails.
(60)
Priest then notes that we have disproved McTaggart’s argument of the reality of time by building a model of time and of the relations between temporally various moments of time. But models might distort what they model, or leave out important features. In this case, we have modeled time using space. But perhaps the flow of time is not analogous to the flow of space.
Now, it is exactly the flow of time that produces the supposed contradiction that McTaggart was pointing to. No wonder this does not show up in the model! Exactly what, then, is missing from the model? And once that is taken into account, does the contradiction reappear?
(62)
[The following is quotation.]
Main Ideas of the Chapter
● Every situation comes with an associated collection of earlier and later situations.
● Fa is true in a situation if a is true in some later situation.
● Pa is true in a situation if a is true in some earlier situation.
● Ga is true in a situation if a is true in every later situation.
● Ha is true in a situation if a is true in every earlier situation.
(quoted from Priest, 62, boldface his, with F, P, G, and H in extra bold)
From:
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.
[Central Entry Directory]
[Terence Blake, entry directory]
Terence Blake has written a very thoughtful and interesting piece on the role of obscurity in the writing style of continental philosophy.
Terence Blake, ‘ON THE “OBSCURITY” OF CONTINENTAL PHILOSOPHY: the case of Foucault’
[Central Entry Directory]
[Terence Blake, entry directory]
Terence Blake has yet another great post up. I had not taken much interest in Badiou before, but Terence’s excellent presentation now has me very curious.
Following after that post are some excellent summaries of Badiou seminars. I very highly recommend Blake’s Agent Swarm, if you have not seen it yet. It is a great resource, and I think, it is one of the very best philosophy blogs around.
Terence Blake, ‘CROSSINGS OF TRUTH PROCEDURES: Badiou’s new lesson’
https://terenceblake.wordpress.com/2015/11/22/crossings-of-truth-procedures-badious-new-lesson/
by Corry Shores
[Bracketed commentary and boldface are my own. Please forgive my typos, as proofreading is incomplete. I am not trained in logic, so please consult and trust the original text, which is wonderful.]
Summary of
Graham Priest
Logic: A Very Short Introduction
Ch.7
Conditionals: What’s in an If?
Brief Summary:
Conditionals are of the form, “if a then c,” or a→c. The first term is the antecedent, and the second, the consequent. Conditionals are false only if the antecedent is true and the consequent false, and they are true for all other value assignments. But there are many difficulties regarding conditionals, and some of which call into question the universal applicability of these value-assignments. For example, according to the truth table for conditionals, when the antecedent is false, then the whole conditional is true, regardless of whether or not the consequent is true. This means that the following two conditionals should both be true: “If Italy is part of France, Rome is in France” and “If Italy is part of France, Beijing is in France”. But intuitively, the second one seems false. So conditionals are not truth-functional, since a lot depends on the meanings of the terms. In order to evaluate them, we can use possible worlds, like with modal operators: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” Since Rome is by definition in Italy, that means in no possible world would it not be in France, were Italy to be in France. So that is why the first sentence is true. However, since Beijing is by definition a city in China and not a city of Italy, then in some possible worlds Beijing will not be in France, were Italy to be in France. And that is why the second sentence is false. Another problem with conditionals has to do with ¬(a&¬c), which has the same truth table as a→c, and in fact is called the material conditional and is symbolized as a⊃c. But although we might think that we can infer a→c from ¬(a&¬c), this is not in fact a valid inference, and we can show this using the possible worlds analysis. The important difference between a→c and ¬(a&¬c) is that a→c involves the relevance of a to c, where there is no such relevance implied in ¬(a&¬c). For this reason we can think of situations where a→c will be false but ¬(a&¬c) will technically true, thereby invalidating the inference. There are other cases too of inferences using conditionals that seem valid, and yet there are troubling counter-examples that call their validity into question.
Summary
[In the previous chapter, we discussed modal logic, and we used formulations like, “if it is true to say that I will be involved in an accident then it cannot fail to be the case that I will be involved”. Also recall the inference called “modus ponens”, where you say, “if a then b,” then you assert a (is true), which allows you to infer b (is true).]
Recall that a conditional is a sentence of the form ‘if a then c’, which we are writing as a→c. Logicians call a the antecedent of the conditional, and c the consequent. We also noted that one of the most fundamental inferences concerning the conditional is modus ponens: a, a→c/c.
(47)
Priest explains that although conditionals are “fundamental to much of our reasoning” and “[t]hey have been studied in logic ever since the earliest times,” they nonetheless are also “are deeply puzzling” (47).
Priest will now show one reason why conditionals can be so puzzling. We will take an example of a conditional: “if you miss the bus, you will be late.” [Now, imagine that you miss the bus. You will be late. Think of this now as a conjunction of facts. Imagine two things: you miss the bus and you are late. It would then seem to be false to make the following conjunction: you miss the bus and you are still on time (that is, not late). We can formulate this as, “it is not the case that both you miss the bus and you are not late”. And we can symbolize this as:
¬(a&¬c)
What is interesting is that although this is a conjunction and in English uses ‘and’, we can also consider it as a sort of conditional with another symbol.]
Suppose, for example, that someone informs you that if you miss the bus, you will be late. You can infer from this that it is false that you will miss the bus and not be late. Conversely, if you know that a→c, it would seem that you can infer ¬(a&¬c) (it is not the case that a and not c). Suppose, for example, that someone informs you that if you miss the bus, you will be late. You can infer from this that it is false that you will miss the bus and not be late. Conversely, if you know that ¬(a&¬c), it would seem that you can infer a→c from this. Suppose, for example, that someone tells you that you won’t go to the movies without spending money (it’s not the case that you go to the movies and do not spend money). You can infer that if you go to the movies, you will spend money. |
¬(a&¬c) is often written as a⊃c, and called the material conditional. Thus, it would appear that a→c and a⊃c mean much the same thing.
(47-48)
[For reference, this again is the truth table for conditionals that we saw already.
Priest then shows the truth table for a⊃c. and says, “it is a simple exercise, which I leave to you, to show that this is as follows:”
(based on Priest 48)
Perhaps we might work out the steps for ¬(a&¬c) in the following way.
As we can see, the material conditional ⊃ and the conditional → have the same truth tables.
] [The next idea is a bit complicated, and I may misconstrue it. Priest will note some oddities in the truth table. We first consider the first and third rows. They say that if c is T, then the whole implication is true. But that would suggest it does not matter what you say for the a, since it can be false and yet the implication will be true. We begin by giving a strange example for the third row. Our c value will be T, which means it does not matter what we put for the a value. Regardless, the whole implication will be true. But Priest gives an example where the a sentence is directly contradictory to the c sentence. So how can such a contradiction still be true, even if all we need is for c to be true? The third and fourth rows suggest that if the a value is false, then it does not matter what we say for the c value, since in all cases the whole implication will be true. He then gives an example for row four where both the a and the c are false, which makes the implication seem false rather than true.]
But this is odd. It means that if c is true in a situation (first and third rows), so is a→c. This hardly seems right. It is true, for example, that Canberra is the federal capital of Australia, but the conditional ‘If Canberra is not the federal capital of Australia, Canberra is the federal capital of Australia’ seems plainly false. Similarly, the truth table shows us that if a is false (third and fourth rows), a→c is true. But this hardly seems right either. The conditional ‘If Sydney is the federal capital of Australia, then Brisbane is the federal capital’ also appears patently false. What has gone wrong?
(48)
[Recall what we said about modal operators. We noted that when we add them to sentences, they may or may not alter their truth value. It depends on the situation, and so it is not a change that happens with mechanical regularity. Thus modal operators are not truth functions. Priest will then give two examples of conditionals. In the first, the a will be false and the c will be false, but because of the meanings of the terms, the conditional will still be true. In the second example, likewise the a will be false and the c will be false, but this time the conditional will be false. This demonstrates that conditionals also are not truth functional, since the same input values do not always give the same output values.]
What these examples seem to show is that → is not a truth function: the truth value of a→c is not determined by the truth values of a and c. Both of ‘Rome is in France’ and ‘Beijing is in France’ are false; but it’s true that:
If Italy is part of France, Rome is in France.
While it’s false that:
If Italy is part of France, Beijing is in France.
(48)
Priest then wonders, how do conditionals work [since we do not know in a mechanically consistent way what determines the output values]?
He says that one way we can explain the workings of conditionals is by using the “machinery of possible worlds” that we saw in the prior chapter. [It seems the reasoning is as follows. Consider the first sentence: “If Italy is part of France, Rome is in France.” We consider all possible worlds where Italy becomes a part of France. In all of them, Rome would have to also become a part of France. For, it is a part of Italy and thereby becomes a part of France when Italy becomes a part of France. (So perhaps we might say, the conditional is true if in no possible world can it be untrue). What about “If Italy is part of France, Beijing is in France”? We can imagine some possible worlds where Beijing does in fact become a part of France in conjunction with Italy’s becoming a part of France. But surely we can think of possible worlds where China is entirely unaffected by Italy’s incorporation into France. Thus, it is not necessarily true. (And so we might say, the conditional is false if in some possible world it is false).]
Consider the last two conditionals. In any possible situation in which Italy had become incorporated into France, Rome would indeed have been in France. But there are possible situations in which Italy was incorporated in France, but this had no effect on China at all. So Beijing was still not in France. This suggests that the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.
(49)
[The next idea I do not follow adequately. Let us first work on the point just made: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” We first note that we are not determining that a→c is necessarily true, but just that it is true. And we are saying that we can determine that a→c is true in a situation if c is true in all associated worlds where a is true. On this basis, we will say that the inference modus ponens is valid. Priest will also say, “suppose that a and a→c are true in some situation, s. Then c is true in all situations associated with s in which a is true.” This seems to follow from the stipulation. We are saying that a→c is true in a situation, and the stipulation says that c will be true in all associated situations where a is true. But we still have not yet gotten to modus ponens. For, we know that in this situation a is true and a→c. But we do not yet know that c is true. The next step in the reasoning is tricky. We do not yet know that c is true in our ‘actual’ situation. But we do know it is true in the other possible ones. The next step in the reasoning is to say that an actual situation should be counted among the possible ones. And this “possible” rendition of an actual situation is identical to the actual one in all respects regarding its facts and truth value assignments. Whatever we say of the possible version we can say of the actual, since they are identical. Now, if c is true in all possible situations related to the actual one, including the possible situation that is identical to the actual one, then it must also be true for the actual version as well.]
This gives a plausible account of →. For example, it shows why modus ponens is valid – at least on one assumption. The assumption is that we count s itself as one of the possible situations associated with s. This seems reasonable: anything that is actually the case in s is surely possible. Now, suppose that a and a→c are true in some situation, s. Then c is true in all situations associated with s in which a is true. But s is one of those situations, and a is true in it. Hence, so is c, as required.
(49)
[We already established that ¬(a&¬c) and a→c have the same truth tables. But the question we had was, can we make the following inference:
¬(a&¬c)
a→c
? We then test for validity it seems by seeing if in our situation it can be that the premise is true and the conclusion false. But we will make those determinations by way of possible worlds which can determine the values in our world. Now, one way that the premise can be true is if a is false. That makes the conjunction false, regardless of c’s value. And the negation of that false conjunction then becomes true. Moreover, if we stick just to the truth tables, the conclusion would be true, since, if you recall from the table, any time the a value is false, the whole implication will be true. But just on the basis of a true conjunction we do not know what the a value is in the other worlds. (Things were different for the conditional, where by knowing that a→c is true in a situation means that we know c is true in associated situations where a is true. For conjunctions we do not seem to be able to draw any further conclusions about the terms’ values in the other words, and thus they can be assigned either way.) If in another world a is true but c is false, then ¬(a&¬c) will be true, but a→c will be false.
But, recall our stipulation, “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” So even in our own world, a→c is false in accordance with this criteria.
Thus since in our own world the premise is true but the conclusion false, it is not a valid inference.]
Going back to the argument with which we started, we can now see where it fails. The inference on which the argument depends is:
¬(a&¬c)
a→cAnd this is not valid. For example, if a is F in some situation, s, this suffices to make the premiss true in s. But this tells us nothing about how a and c behave in the possible situations associated with s. It could well be that in one of these, say s′, a is true and c is not, like this: |
So a→c is not true at s.
(49-50)
[But recall our prior example where this inference seemed to hold. “Suppose, for example, that someone tells you that you won’t go to the movies without spending money (it’s not the case that you go to the movies and do not spend money). You can infer that if you go to the movies, you will spend money.” Priest then gives a counter example, but the reasoning is tricky for me. It seems the idea is the following. We begin by assuming: it is not the case that you will go to the movies, and also, you will not spend money: ¬(g&¬m). We next make two other assumptions, namely, that tonight the movies are free and also that we will not go to them anyway. We might now say: ‘but then the original statement, “you cannot go to the movies without spending money” no longer seems to apply. For, now the movies are free’. But it can still apply. If we do not go to the movies, then the g is false in ¬(g&¬m). That means the whole negated conjunction is true. So even if the movies are free, if we do not go, it is still true that: “you cannot go to the movies without spending money” (“it’s not the case that you go to the movies and do not spend money”). Although this negated conjunction is true, the conditional is not, namely, that “if you go to the movies you will spend money”. For, as we said, it is a free movie night.]
What about the example we had earlier, where you are informed that you won’t go to the movies without spending money. Didn’t the inference seem valid in this case? Suppose you know that you won’t go to the movies without spending money: ¬(g&¬m). Are you really entitled to conclude that if you go to the movies you will spend money: g→m? Not necessarily. Suppose you are not going to go to the movies, come what may, even if admission is free that night. (There is a programme on the television that is much more interesting.) Then you know that it is not true that you will go (¬g), and so that it is not true that you will go and not spend money: ¬(g&¬m). Are you then entitled to infer that if you go you will spend money? Certainly not: it may be a free night.
(50)
The next point seems to be about relevance. Recall the above situation where we said that “it’s not the case that you go to the movies and do not spend money” was true because actually we are not going to the movies anyway. A person would not normally make such a statement if they knew you were not going, because then who cares whether or not you spend money? Of course you will not anyway. Instead, if someone tells you this, it matters that there be an important connection between g being true and m being true. So therefore, if someone tells us this, even though we cannot logically infer g→m, we can still conclude that this is what is meant by the statement (50-51).
Priest then notes how we often make correct inferences based on context and relevance, even though the inferences are not made deductively. Unlike implication in the sense of conditionals, this inductive sort of inferring is called “conversational implicature.”
Suppose, for example that I ask someone how to get my computer to do something or other, and they reply ‘There is a manual on the shelf’. I infer that it is a computer manual. This does not follow from what was actually said, but the remark would not have been relevant unless the | manual was a computer manual, and people are normally relevant in what they say. Hence, I can conclude that it is a computer manual from the fact that they said what they did. The inference is not a deductive one. After all, the person could have said this, and it not be a computer manual. But the inference is still an excellent inductive inference. It is of a kind usually called conversational implicature.
(51-52)
Priest now addresses another problem with how we have so far characterized conditionals. He will first give two arguments that are variations on a certain form. The examples will seem valid intuitively, and thus the forms will seem valid. Then afterward he gives other examples which fit the form, but they seem intuitively invalid. He leaves it to the reader to think about the matter, and it is left unsettled.
The first form is:
a→b b→c
a→c
And its example is:
If you go to Rome you will be in Italy.
If you go to Italy, you are in Europe.
Hence, if you go to Rome, you will be in Europe.
(52)
But then he gives this counter-example:
If Smith dies before the election, Jones will win.
If Jones wins the election, Smith will retire and take her pension.
Hence, if Smith dies before the election, she will retire and take her pension.
(53)
[Perhaps the problem here has to do with the relevance between the two premises, but I am not sure.]
The second form is:
a→c
(a&b)→c
And its example is:
If x is greater than 10 then x is greater than 5.
Hence, if x is greater than 10 and less than 100, then x is greater than 5.
(52)
But its counter-example is:
If Smith jumps from the top of a tall precipice, she will die from the fall. Hence, if Smith jumps from the top of a tall precipice and wears a parachute, she will die from the fall.
(53)
These tricky examples demonstrate just how contentious the topic of conditionals is in logic (54).
[The following is quotation.]
Main Idea of the Chapter
● a→b is true in a situation, s, just if b is true in every situation associated with s where a is true.
(quoted from Priest, 54, boldface his)
From:
Priest, Graham. Logic: A Very Short Introduction. Oxford: Oxford University, 2000.
[Central Entry Directory]
[Terence Blake, entry directory]
This is a post by Terence Blake regarding Deleuze’s philosophies of difference and multiplicity. I had previously thought difference to be a more fundamental concept, but now I am no longer so sure.
“I argue that Deleuze’s philosophical evolution involves a passage from a problematic of difference to one of multiplicity. There is nothing explicit to mark this passage, but … there is an “epistemological break”, a rupture in Deleuze’s preferred conceptual vocabulary. After his encounter with Guattari, Deleuze ceases talking in terms of difference, and sticks to multiplicity.
Given this change, which I find to be a progress, I see no reason to confine Deleuze to the category “philosopher of difference”. Deleuze always presented himself as a pluralist and a philosopher of multiplicities, long before DIFFERENCE AND REPETITION. So I think it is a mistake to give too much prominence to the concept of difference.”
(Blake)
From Terence Blake, ‘DELEUZE: philosopher of difference or philosopher of multiplicity’
[Central Entry Directory]
[Terence Blake, entry directory]
Terence Blake has an excellent post on Zizek’s misreading of Deleuze’s “Letter to a Severe Critic”:
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