Showing posts with label possible worlds. Show all posts
Showing posts with label possible worlds. Show all posts

21 May 2021

Dumoncel (0) “Les modalités deleuziennes”, Section 0, “[introductory material]”, summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Dumoncel, entry directory]

[Dumoncel, “Les modalités deleuziennes,” entry directory]

 

[The following is a paragraph by paragraph summary of Dumoncel’s text. Unless otherwise indicated, boldface, underlining, and bracketed commentary are my own. Proofreading is incomplete, so please forgive my mistakes. ]

 

 

 

 

Summary of

 

Jean-Claude Dumoncel

[Dumoncel’s academia.edu page and wikipedia page]

 

“Les modalités deleuziennes”

[link to the article]

 

Part 0

[introductory material]

 

 

 

 

 

 

Brief summary (collecting those below):

 

(0.1) Analytic philosophers and Deleuze share a common interest: Leibniz. (0.2) Deleuze’s fundamental intuition regarding Leibniz is that for him, monads are possible worlds, and possible worlds are monads. (0.3) First we will have a brief introduction to Leibniz, secondly an examination of mathematized modal logic, and finally we uncover Deleuzian modalities. (0.4) Dumoncel’s study also aims to explicate a page of Deleuze & Guattari’s What Is Philosophy?

 

 

 

Contents

 

0.1

[The Common Interest of Deleuze and Analytic Philosophers as Being Leibniz]

 

0.2

[Deleuze’s Claim That for Leibniz, Monads are Possible Worlds and Vice Versa]

 

0.3

[Preview]

 

0.4

[The Relevant Quote from What Is Philosophy?]

 

[Dumoncel’s study also aims to explicate a page of Deleuze & Guattari’s What Is Philosophy?]

 

Bibliography

 

 

 

 

 

 

 

Summary

 

0.1

[The Common Interest of Deleuze and Analytic Philosophers as Being Leibniz]

 

[Analytic philosophers and Deleuze share a common interest: Leibniz.]

 

[contents]

 

 

 

 

 

 

0.2

[Deleuze’s Claim That for Leibniz, Monads are Possible Worlds and Vice Versa]

 

[Deleuze’s fundamental intuition regarding Leibniz is that for him, monads are possible worlds, and possible worlds are monads.]

 

[contents]

 

 

 

 

 

 

0.3

[Preview]

 

[First we will have a brief introduction to Leibniz, secondly an examination of mathematized modal logic, and finally we uncover Deleuzian modalities.]

 

[contents]

 

 

 

0.4

[The Relevant Quote from What Is Philosophy?]

 

[Dumoncel’s study also aims to explicate a page of Deleuze & Guattari’s What Is Philosophy?]

 

[Here are the passages in What Is Philosophy?]

Let us proceed in a summary fashion: we will consider a field of experience taken as a real world no longer in relation to a self but to a simple “there is.” There is, at some moment, a calm and restful world. Suddenly a frightened face looms up that looks at something out of the field. The other person appears here as neither subject nor object but as something that is very different: a possible world, the possibility of a frightening world. This possible world is not real, or not yet, but it exists nonetheless: it is an expressed that exists only in its expression—the face, or an equivalent of the face. To begin with, the other person is this existence of a possible world. And this possible world also has a specific reality in itself, as possible: when the expressing speaks and says, “I am frightened,” even if its words are untruthful, this is enough for a reality to be given to the possible as such. This is the only meaning of the “I” as linguistic index. But it is not indispensable: China is a possible world, but it takes on a reality as soon as Chinese is spoken or China is spoken about within a given field of experience. This is very different from the situation in which China is realized by becoming the field of experience itself. Here, then, is a concept of the other that presupposes no more than the determination of a sensory world as condition. On this condition the other appears as the expression of a possible. The other is a possible world as it exists in a face that expresses it and takes shape in a language that gives it a reality. In this sense it is a concept with three inseparable components: possible world, existing face, and real language or speech.

Obviously, every concept has a history. This concept of the other person goes back to Leibniz, to his possible worlds and to the monad as expression of the world. But it is not the same problem, because in Leibniz possibles do not exist in the real world. It is also found in the modal logic of propositions. But these do not confer on possible worlds the reality that corresponds to their truth conditions (even when Wittgenstein envisages propositions of fear or pain, he does not see them as modalities that can be expressed in a position of the other person because he leaves the other person oscillating between another subject and a special object).

(Deleuze & Guattari, What Is Philosophy?, 17-18)

 

 

 

 

 

 

 

Bibliography:

.

Dumoncel, Jean-Claude. “Les modalités deleuziennes.” Web. Accessed 2021.05.21.

https://www.academia.edu/23505679/LES_MODALITES_DELEUZIENNES

 

 

Deleuze, Gilles, and Guattari, Félix. What Is Philosophy? Translated by Hugh Tomlinson and Graham Burchell. New York: Columbia University, 1994.

 

 

.

18 Dec 2018

Leibniz (3.405-3.417) Theodicy (The Sextus Pyramid of Incompossibles story), summary

 

by Corry Shores

 

[Search Blog Here. Index tabs are found at the bottom of the left column.]

 

[Central Entry Directory]

[Leibniz, entry directory]

[Leibniz, Theodicy, entry directory]

 

[The following is summary. Boldface is my own. Proofreading is incomplete, so please forgive my mistakes.]

 

 

 

 

Summary of

 

Gottfried Leibniz

 

Theodicy:

Essays on the Goodness of God, the Freedom of Man and the Origin of Evil

 

Essays on the Justice of God and the Freedom of Man in the Origin of Evil”

 

Part 3

 

Sections 405-417

(The Sextus Pyramid of Incompossibles Story)

 

 

 

 

Brief summary:

(3.405) Leibniz begins with Laurentius Valla’s Dialogue on Free Will (between Antonio Glarea and Laurentius), to set up a discussion about divining and freedom. (3.406) Laurentius will explain how divine foreknowledge can be compatible with human freedom. But he requires that Antonio be satisfied with his answer. (3.407) Antonio argues that since God foresaw Judas betray, that means Judas could not have done otherwise and thus is blameless. Laurentius replies that mere knowledge itself cannot cause things past or present to exist; therefore, knowledge cannot cause future things to exist. (3.408) Laurentius makes a couple distinctions. {1} Foretelling vs. foreknowing. Antonio gives an example where he asks a heathen god which foot he will put forward, then he puts the opposite foot forward (Laurentius explains here that either the god would not foretell, or if he would, then we should expect Antonio to obey the foretelling). {2} Regarding foreknowledge: the necessary vs. the certain. Events that are foreseen could have been otherwise (as their alternatives do not create logical contradictions) and thus they are not necessary, but it is certain they will happen (as they have been chosen by God in advance). “It is not impossible for what is foreseen not to happen; but it is infallably certain that it will happen. I can become a Soldier or Priest, but I shall not become one” (p.371). (3.409) Laurentius says that “Two contradictories are often both possible” (p.372).” He next illustrates why foreknowledge is not causal with a story about Sextus Tarquinius. Sextus goes to the Oracle of Apollo for a prediction. Apollo says that he will die poor and in exile. Sextus objects, asking why Jupiter would be so cruel in choosing for him such a bad fate, when he is innocent and devout. Apollo replies that Sextus will later be proud, adulterous, and a traitor to his country. Antonio agrees that Sextus cannot accuse Apollo of causing these things by foreseeing them. Similarly, Laurentius concludes, Judas cannot complain about God’s foreknowledge of his crimes. (3.410) Laurentius continues the story. Sextus tells Apollo he will not commit the foretold crimes, and he will appeal to the Gods to give him a better heart. Nonetheless, Apollo will notify Sextus that the Gods make creatures with their given traits. The ass is stupid, the lion is courageous, etc. They gave Sextus a wicked soul, and he will act in conformity with his natural disposition; and, Jupiter will accordingly punish his wicked actions. (3.411) Antonio objects that the real crime is caused by Jupiter for having created someone with a wicked nature who will certainly do wrong. Thus Sextus should not be blamed for Jupiter’s crime, as he cannot resist Jupiter’s will. Laurentius says he was able to show that foreknowledge is different from providence and that foreknowledge does not have any causal power. But he also says that he cannot know explain if and how providence restricts freedom. (3.412) Laurentius explains that we will not know why God is hard against one person but compassionate towards another. We simply have to accept that God is very good and wise, and that is enough reason for us to deem his reasons as being good. (3.413) Leibniz now speaks for himself rather than quoting the Laurentius dialogue. Leibniz notes that the dialogue’s defect is it condemns providence, as it makes Jupiter the author of sin. Leibniz now continues the story his own way. Sextus next goes to Jupiter and asks him why he condemned him to be wicked and unhappy, and Sextus asks Jupiter to fix his error and give him a fate where he becomes wise and happy. Jupiter says that Sextus can be wise and happy if he does not go to Rome and thus renounces his kingship. Sextus cannot give that up, so he accepts his doomed fate. Leibniz now adds a new character to the story, Theodorus, a high priest who has been listening to this story. Theodorus tells Jupiter that Jupiter’s worshippers are upset, because Jupiter is the one who decided to give Sextus his evil will (which resulted in him going to Rome even though he now knows it dooms him). Jupiter tells Theodorus that he should visit Jupiter’s daughter Pallas, because she will explain what Jupiter was bound to do when deciding Sextus’ will). (3.414) Theodorus goes to Jupiter’s daughter Pallas. She watches and wards over the palace of fates. It contains not just the actual world, but every possible alternative fate. Jupiter picked the best one, and he  sometimes comes to admire his selection. So at the palace, we can see all the alternative twists in fate. And for any one person, there can be a variety of different lives, and thus there are many Sextuses. Pallas then makes a geometrical illustration for this structure of bifurcating worlds. A locus is a set of points that fulfill certain conditions, but these conditions do not often determine a singular point but rather a set of points forming a line. (For instance, the center of a circle is a focus, and all points equidistant from it is both the circumference of the circle and also the locus of points for that criterion.) Pallas is saying that when you ask for a different condition in the world, like Sextus not going to Rome, then you have defined a whole range of alternative possibilities. She will now show this ranging of various Sextuses.

You see here the palace of the fates, where I keep watch and ward. Here are representations not only of that which happens but also of all that which is possible. Jupiter, having surveyed them before the beginning of the existing world, classified the possibilities into worlds, and chose the best of all. He comes sometimes to visit these places, to enjoy the pleasure of recapitulating things and of renewing his own choice, which cannot fail to please him. I have only to speak, and we shall see a whole world that my father might have produced, wherein will be represented anything that can be asked of him; and in this way one may know also what would happen if any particular possibility should attain unto existence. And whenever the conditions are not determinate enough, there will be as many such worlds differing from one another as one shall wish, which will answer differently the same question, in as many ways as possible. You learnt geometry in your youth, like all well-instructed Greeks. You know therefore that when the conditions of a required point do not sufficiently determine it, and there is an infinite number of them, they all fall into what the geometricians call a locus, and this locus at least (which is often a line) will be determinate. Thus you can picture | to yourself an ordered succession of worlds, which shall contain each and every one the case that is in question, and shall vary its circumstances and its consequences. But if you put a case that differs from the actual world only in one single definite thing and in its results, a certain one of those determinate worlds will answer you. These worlds are all here, that is, in ideas. I will show you some, wherein shall be found, not absolutely the same Sextus as you have seen (that is not possible, he carries with him always that which he shall be) but several Sextuses resembling him, possessing all that you know already of the true Sextus, but not all that is already in him imperceptibly, nor in consequence all that shall yet happen to him. You will find in one world a very happy and noble Sextus, in another a Sextus content with a mediocre state, a Sextus, indeed, of every kind and endless diversity of forms.

(475-476)

(3.415) Pallas takes Theodorus to one of the hall’s in the palace of fates, which displays a whole complete world. In this one, Sextus obeys Jupiter, goes off to a place like Corinth, cultivates a garden, finds treasure, is loved by the city, and dies old. In the hall is this particular world’s book of fate. Sextus has a number on his head. Theodorus follows Pallas’ instruction to find that number in the book, where he finds a more extensive account of this Sextus’ life. Pallas explains that he only needs to point to a line of the text, and its story will come to life. This way, he could see all the characteristics of some part of Sextus’ life. Next they go to another hall, which displays a different possible world. Here, there is another Sextus who also obeys Jupiter, but this time he goes instead to Thrace where he marries the king’s daughter, succeeds the king, and is adored by the subjects. Theodorus and Pallas go to a number of rooms, each time with another possible world and life for Sextus. (3.416) The halls are stacked in the form of a pyramid, with them being better and better worlds as one gets higher and with the one at the apex being the best possible world that Jupiter/God therefore chose as the existing one. But even though there is a certain apex world, the pyramid has no bottom, because the worlds degrade infinitely going down. In the best possible world, Sextus is displayed as he actually is. He goes to Rome, creates confusion and violates his friend’s wife and is then driven out of Rome, beaten and unhappy. But even though choosing that world would have made a happy Sextus, God could not choose it, because then God would be renouncing God’s divine wisdom (which calls him to choose the best of all possible worlds.) So in order to make Sextus actual, he had to choose a world where he was wicked. (For, suppose he did not choose that world; he would either choose no world or a less perfect one. But God is wise and cannot choose a less perfect world. So God must choose the world where Sextus is wicked.) This means that “The crime of Sextus serves for great things: it renders Rome free; thence will arise a great empire, which will show noble examples to mankind.” (So it comes back to God being responsible for Sextus’ wickedness, but God had no better alternative options in the matter.)

The halls rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful worlds. Finally they reached the highest one which completed the pyramid, and which was the most beautiful of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. That is (as the Goddess explained) because amongst an endless number of possible worlds there is the best of all, else would God not have determined to create any; but there is not any one which has not also less perfect worlds below it: that is why the pyramid goes on descending to infinity. Theodorus, entering this highest hall, became entranced in ecstasy; he had to receive succour from the Goddess, a drop of a divine liquid placed on his tongue restored him; he was beside himself for joy. We are in the real true world (said the Goddess) and you are at the source of happiness. Behold what Jupiter makes ready for you, if you continue to serve him faithfully. Here is Sextus as he is, and as he will be in reality. He issues from the temple in a rage, he scorns the counsel of the Gods. You see him going to Rome, bringing confusion everywhere, violating the wife of his friend. There he is driven out with his father, beaten, unhappy. If Jupiter had placed here a Sextus happy at Corinth or King in Thrace, it would be no longer this world. And nevertheless he could not have failed to choose this world, which surpasses in perfection all the others, and which forms the apex of the pyramid. Else would Jupiter have renounced his wisdom, he would have banished me, me his daughter. You see that my father did not make Sextus wicked; he was so from all eternity, he was so always and freely. My father only granted him the existence which his wisdom could not refuse to the world where he is included: he made him pass from the region of the possible to that of the actual beings. The crime of Sextus serves for great things: it renders Rome free; thence will arise a great empire, which will show noble examples to mankind. But that | is nothing in comparison with the worth of this whole world, at whose beauty you will marvel, when, after a happy passage from this mortal state to another and better one, the Gods shall have fitted you to know it.

(377-378)

(3.417) Theodorus awakes from this dream being even more convinced of his beliefs, and he carries out his priestly duties zealously. Leibniz explains that this Sextus story fills out what is missing in Valla’s dialogue. Valla discussed God’s knowledge of vision (of the future) which concerns beings that exist. But Pallas shows knowledge of simple intelligence, which embraces all that is possible (that is, knowledge not just of the existing world but of all other possibilities). Leibniz ends by saying that the source of things should be sought in this second kind of knowledge.

 

 

 

 

 

Contents

 

3.405

[Using the Laurentius Valla Dialogue on Freedom]

 

3.406

[Laurentius’ Agreement to Answer]

 

3.407

[Knowledge as Having No Power to Cause Existence, Particularly Knowledge of the Future]

 

3.408

[The Foreknown as Certain but Not Necessary]

 

3.409

[Apollo’s Foreknowledge of Sextus Tarquinius’s Crimes as Not Causing Them]

 

3.410

[Sextus as Having the Hope or Intention to Change His Ways but Still Failing to Do So]

 

3.411

[The Crime as Seeming to Lie with Jupiter and Not Sextus]

 

3.412

[Our Inability to Know Why God Treats Some People Badly and the Need to Simply Trust that God Is Being Good Nonetheless]

 

3.413

[Theodorus as Being the One Who Will Learn Why God Must Make Some People Wicked]

 

3.414

[Jupiter’s Daughter Pallas as Guarding the Palace of Fates, Which Holds Every Alternative Fate for the World]

 

3.415

[The Different Rooms with Different Worlds, Fates, and Sextuses]

 

3.416

[The Best of All Possible Worlds is One Where Some People Are Made as Wicked and Suffer]

 

3.417

[God’s Foreknowledge as Being More Than Mere Foreseeing the Events of This World but Instead as Knowing all Events in All Worlds]

 

 

 

 

 

 

 

Summary

 

3.405

[Using the Laurentius Valla Dialogue on Freedom]

 

[Leibniz begins with Laurentius Valla’s Dialogue on Free Will (between Antonio Glarea and Laurentius), to set up a discussion about divining and freedom.]

 

[Leibniz will give comments on free will contra Boethius. He will do it by quoting in the abstract a dialogue, Laurentius Valla’s Dialogue on Free Will. In it, a Spaniard named Antonio Glarea asks Laurentius to elaborate on the difficulty of free will, because justice and injustice, punishment and reward, in this life and the next, depend upon it.  Laurentius’ point is that we must accept that we have a certain ignorance with regard to free will, and we must come to terms with it like how we must accept that we do not wave the wings of birds. (Note: In the following quotations, “Ant.” is for Antonio and “Laur.” is for Laurentius.)]

405. I intended to finish here after having met (as it seems to me) all the objections of M. Bayle on this matter that I could find in his works. But remembering Laurentius Valla’s Dialogue on Free Will, in opposition to Boethius, which I have already mentioned, I thought it would be opportune to quote it in abstract, retaining the dialogue form, and then to continue from where it ends, keeping up the fiction it initiated; and that less with the purpose of enlivening the subject, than in order to explain myself towards the end of my dissertation as clearly as I can, and in a way most likely to be generally understood. This Dialogue of Valla and his books on Pleasure and the True Good make it plain that he was no less a philosopher than a humanist. These four books were opposed to the four books on the Consolation of Philosophy by Boethius, and the Dialogue to the fifth book. A certain Spaniard named Antonio Glarea requests of him elucidation on the difficulty of free will, whereof little is known as it is worthy to be known, for upon it depend justice and injustice, punishment and reward in this life and in the life to come. Laurentius Valla answers him that we must console ourselves for an ignorance which we share with the whole world, just as one consoles oneself for not having the wings of birds.

(370)

[contents]

 

 

 

 

 

 

3.406

[Laurentius’ Agreement to Answer]

 

[Laurentius will explain how divine foreknowledge can be compatible with human freedom. But he requires that Antonio be satisfied with his answer.]

 

[Antonio references Boethius and says we can still obtain wings like Daedalus, and so please enlighten us a little about free will, Laurentius. Specifically, we want to know how to reconcile foreknowledge with freedom. Laurentius agrees to give an answer, so long as Antonio makes his curiosities satisfied with it and will not ask another question after it.]

406. Antonio—I know that you can give me those wings, like another Daedalus, so that I may emerge from the prison of ignorance, and rise to the very region of truth, which is the homeland of souls. The books that I have seen have not satisfied me, not even the famous Boethius, who meets with general approval. I know not whether he fully understood himself what he says of God’s understanding, and of eternity superior to time; and I ask for your opinion on his way of reconciling foreknowledge with freedom. Laurent—I am fearful of giving offence to many people, | if I confute this great man; yet I will give preference over this fear to the consideration I have for the entreaties of a friend, provided that you make me a promise. ant.—What? Laur.—It is, that when you have dined with me you do not ask me to give you supper, that is to say, I desire that you be content with the answer to the question you have put to me, and do not put a further question.

(370-371)

[contents]

 

 

 

 

 

 

3.407

[Knowledge as Having No Power to Cause Existence, Particularly Knowledge of the Future]

 

[Antonio argues that since God foresaw Judas betray, that means Judas could not have done otherwise and thus is blameless. Laurentius replies that mere knowledge itself cannot cause things past or present to exist; therefore, knowledge cannot cause future things to exist.]

 

[Antonio states the crux of the problem with the example of Judas. Suppose God foresaw Judas committing treason. If God foresaw it, then it is necessary that Judas betray (because God cannot be wrong, including about the future, so it cannot be otherwise), and thus it is impossible for Judas not to betray. Now, since there is no obligation to do the impossible, Judas did not sin (having not broken any obligation even in having betrayed), and thus he did not deserve to be punished. So generally speaking, if God foresees such things, then there is no justice or religion and no fear of God. Laurentius replies that even though God foresaw the sin, he did not compel people to commit the sin. For, sin is voluntary. Antonio restates that since the will to sin was foreseen, it was necessary. Laurentius replies that knowledge cannot cause things past or present to exist, therefore, knowledge cannot cause future things to exist. ]

Ant.—I promise you. Here is the heart of the difficulty. If God foresaw the treason of Judas, it was necessary that he should betray, it was impossible for him not to betray. There is no obligation to do the impossible. He therefore did not sin, he did not deserve to be punished. That destroys justice and religion, and the fear of God. Laur.—God foresaw sin; but he did not compel man to commit it; sin is voluntary. Ant.—That will was necessary, since it was foreseen. Laur.—If my knowledge does not cause things past or present to exist, neither will my foreknowledge cause future things to exist.

(371)

[contents]

 

 

 

 

 

 

3.408

[The Foreknown as Certain but Not Necessary]

 

[Laurentius makes a couple distinctions. {1} Foretelling vs. foreknowing. Antonio gives an example where he asks a heathen god which foot he will put forward, then he puts the opposite foot forward (Laurentius explains here that either the god would not foretell, or if he would, then we should expect Antonio to obey the foretelling). {2} Regarding foreknowledge: the necessary vs. the certain. Events that are foreseen could have been otherwise (as their alternatives do not create logical contradictions) and thus they are not necessary, but it is certain they will happen (as they have been chosen by God in advance).]

 

[Antonio tries to show that were the future foreknown, it is fixed and necessary. (But his illustration is odd, because it seems to show a different point, namely, that the future cannot be foreknown.) We think of a heathen god who claims to know the future. We ask, which foot will we put forward? Whichever he foretells, we do the opposite. Laurentius says that there is a false supposition here, namely, that God would answer Antonio’s question in the first place; and if He did, then Antonio would obey Him. But at any rate, Laurentius says, we have changed the issue here, because now we are speaking about foretelling rather than foreknowing. For foreknowledge, we must distinguish the necessary from the certain. “It is not impossible for what is foreseen not to happen; but it is infallably certain that it will happen. I can become a Soldier or Priest, but I shall not become one.”]

408. Ant.—That comparison is deceptive: neither the present nor the past can be changed, they are already necessary; but the future, movable in itself, becomes fixed and necessary through foreknowledge. Let us pretend that a god of the heathen boasts of knowing the future: I will ask him if he knows which foot I shall put foremost, then I will do the opposite of that which he shall have foretold. Laur.—This God knows what you are about to do. Ant.—How does he know it, since I will do the opposite of what he shall have said, and I suppose that he will say what he thinks? Laur.—Your supposition is false: God will not answer you; or again, if he were to answer you, the veneration you would have for him would make you hasten to do what he had said; his prediction would be to you an order. But we have changed the question. We are not concerned with what God will foretell but with what he foresees. Let us therefore return to foreknowledge, and distinguish between the necessary and the certain. It is not impossible for what is foreseen not to happen; but it is infallibly sure that it will happen. I can become a Soldier or Priest, but I shall not become one.

(371)

[contents]

 

 

 

 

 

 

3.409

[Apollo’s Foreknowledge of Sextus Tarquinius’s Crimes as Not Causing Them]

 

[Laurentius says that “Two contradictories are often both possible” (p.372).” He next illustrates why foreknowledge is not causal with a story about Sextus Tarquinius. Sextus goes to the Oracle of Apollo for a prediction. Apollo says that he will die poor and in exile. Sextus objects, asking why Jupiter would be so cruel in choosing for him such a bad fate, when he is innocent and devout. Apollo replies that Sextus will later be proud, adulterous, and a traitor to his country. Antonio agrees that Sextus cannot accuse Apollo of causing these things by foreseeing them. Similarly, Laurentius concludes, Judas cannot complain about God’s foreknowledge of his crimes.]

 

[Antonio says that “The philosophers’ rule maintains that all that which is possible can be considered as existing.” [(Antonio’s idea might be the following, but I am not certain. He might be appealing to Diodorus’ Master Argument. Quoting Epictetus (and note the final sentence):

The ‘Master’ argument appears to have been proposed on the basis of some such principles as these. Of the following propositions, any two imply a contradiction to the third. They are these. That everything that has happened is necessarily true; that the impossible cannot be a consequence of the possible; and that something is a possibility which neither is nor ever will be true. Diodorus, perceiving this contradiction, made use of the probability of the two first to demonstrate this conclusion: nothing is possible which neither is nor ever will be true.

(Epictetus 1995: 122, boldface mine. See Long trans at Perseus)

) Antonio next says: “But if that which you affirm to be possible, namely an event different from what has been foreseen, actually happened, God would have been mistaken.” (Maybe the idea here is the following. Assume God cannot be mistaken. So whatever God foresees must happen. We want to know if there can be more than one possible future and thus if there can be freedom. We established (the Diodorus Master Argument idea) that whatever is possible must exist (and so the only possible outcomes are the ones that in fact will come about, and their alternatives were thus never possible). But if we affirm that one of two alternatives is possible but it does not happen, then that means something which is possible (and thus that will happen) in fact did not happen, and (supposing God foresaw it) than God was mistaken. (So, since God cannot be mistaken, it cannot be that anything deemed possible does not happen. So only one thing can happen, with the alternative options being impossible, and thus we have no freedom, because we have no options or choices to make. I am guessing this is a Master sort of argument like this, but I am not sure. Please see the quotation below.) Laurentius replies that “The rules of the philosophers are not oracles for me. This one in particular is not correct.” (Maybe he is saying that he does not accept Diodorus’ assumptions, because they are wrong.) For, he continues: “Two contradictories are often both possible.” Laurentius then wonders if two contradictories can both exist. (As we will see, it seems they cannot both exist, but they can coincide as possible worlds.) He then begins a story of Sextus Tarquinius (wiki), who comes to Delphi to consult the Oracle of Apollo. The oracle says, “Exul inopsque cades irata pulsus ab urbe (You will die poor and in exile, driven in anger from your fatherland). A beggared outcast of the city’s rage, Beside a foreign shore cut short thy age.” Sextus is disappointed. Apollo says he is merely foretelling what will happen; he is not causing it to come about. Apollo says that if Sextus has any complaints, he should take them to Jupiter and the Parcae (the fates). So, Laurentius says, Sextus would be ridiculous if he continued to complain to Apollo. Antonio says that Sextus will ask why Jupiter would be so cruel, when he is innocent and devout. Laurentius explains that Apollo would reply that Sextus will later be proud, adulterous, and a traitor to his country. Antonio agrees that Sextus cannot accuse Apollo of causing these things by foreseeing them. Similarly, Laurentius concludes, Judas cannot complain about God’s foreknowledge of his crimes.]

Ant.—Here I have you firmly held. The philosophers’ rule maintains that all that which is possible can be considered as existing. But if that which you affirm to be possible, namely an event | different from what has been foreseen, actually happened, God would have been mistaken. Laur.—The rules of the philosophers are not oracles for me. This one in particular is not correct. Two contradictories are often both possible. Can they also both exist? But, for your further enlightenment, let us pretend that Sextus Tarquinius, coming to Delphi to consult the Oracle of Apollo, receives the answer:

Exul inopsque cades irata pulsus ab urbe.

A beggared outcast of the city’s rage,

Beside a foreign shore cut short thy age.

The young man will complain: I have brought you a royal gift, O Apollo, and you proclaim for me a lot so unhappy? Apollo will say to him: Your gift is pleasing to me, and I will do that which you ask of me, I will tell you what will happen. I know the future, but I do not bring it about. Go make your complaint to Jupiter and the Parcae. Sextus would be ridiculous if he continued thereafter to complain about Apollo. Is not that true? Ant.—He will say: I thank you, O holy Apollo, for not having repaid me with silence, for having revealed to me the Truth. But whence comes it that Jupiter is so cruel towards me, that he prepares so hard a fate for an innocent man, for a devout worshipper of the Gods? Laur.—You innocent? Apollo will say. Know that you will be proud, that you will commit adulteries, that you will be a traitor to your country. Could Sextus reply: It is you who are the cause, O Apollo; you compel me to do it, by foreseeing it? Ant.—I admit that he would have taken leave of his senses if he were to make this reply. Laur.—Therefore neither can the traitor Judas complain of God’s foreknowledge. And there is the answer to your question.

(371-372)

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3.410

[Sextus as Having the Hope or Intention to Change His Ways but Still Failing to Do So]

 

[Laurentius continues the story. Sextus tells Apollo he will not commit the foretold crimes, and he will appeal to the Gods to give him a better heart. Nonetheless, Apollo will notify Sextus that the Gods make creatures with their given traits. The ass is stupid, the lion is courageous, etc. They gave Sextus a wicked soul, and he will act in conformity with his natural disposition; and, Jupiter will accordingly punish his wicked actions.]

 

[Antonio is now satisfied with Laurentius’ reply and story. But Laurentius says they should take the story a little further. We now suppose that Sextus will tell Apollo that he will not commit the foretold crimes. Antonio says that God (Apollo) would then proclaim that he is not a liar and that Sextus will in fact do the foretold things. Laurentius then claims that Sextus may try to alter his fate by praying to the Gods and asking them to give him a better heart. Antonio says that the answer the Gods would give him is that they cannot make God’s foreknowledge be a lie (Desine fata Deum flecti sperare precando / Hope not that the fates of the gods may be turned aside by prayers. (Virgil, Aeneid, VI, 376)), and so there is nothing they can do to change his fate. And thus Sextus would complain that therefore he is not free, and it is not in his power to follow virtue. Laurentius says that Apollo would tell Sextus that the Gods make creatures with their given traits. The ass is stupid, the lion is courageous, etc. They gave Sextus a wicked soul, and he will act in conformity with his natural disposition; and, Jupiter will accordingly punish his wicked actions.]

Ant.—You have satisfied me beyond my hopes, you have done what Boethius was not able to do: I shall be beholden to you all my life long. Laur.—Yet let us carry our tale a little further. Sextus will say: No, Apollo, I will not do what you say. Ant.—What! the God will say, do you mean then that I am a liar? I repeat to you once more, you will do all that I have just said. Laur.—Sextus, mayhap, would pray the Gods to alter fate, to give him a better heart. Ant.—He would receive the answer: |

Desine fata Deum flecti sperare precando.

He cannot cause divine foreknowledge to lie. But what then will Sextus say? Will he not break forth into complaints against the Gods? Will he not say? What? I am then not free? It is not in my power to follow virtue? Laur.—Apollo will say to him perhaps: Know, my poor Sextus, that the Gods make each one as he is. Jupiter made the wolf ravening, the hare timid, the ass stupid, and the lion courageous. He gave you a soul that is wicked and irreclaimable; you will act in conformity with your natural disposition, and Jupiter will treat you as your actions shall deserve; he has sworn it by the Styx.

(472-473)

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3.411

[The Crime as Seeming to Lie with Jupiter and Not Sextus]

 

[Antonio objects that the real crime is caused by Jupiter for having created someone with a wicked nature who will certainly do wrong. Thus Sextus should not be blamed for Jupiter’s crime, as he cannot resist Jupiter’s will. Laurentius says he was able to show that foreknowledge is different from providence and that foreknowledge does not have any causal power. But he also says that he cannot know explain if and how providence restricts freedom.]

 

[Antonio then notes that the real crime lies in Jupiter for having created someone who will certainly be wicked. So there again is no grounds to punish Sextus, as he is the victim of Jupiter’s crimes, and Sextus is powerless to resist Jupiter’s will. Laurentius then notes that he cannot say much more here. He distinguished Apollo and Jupiter so to distinguish foreknowledge and providence. He has shown that foreknowledge does not impair freedom, but he cannot explain if and how providence does not impair freedom. Antonio notes that even though Laurentius helped with one matter, he created an equally intractable one that still needs resolution. Laurentius reminds Antonio about the agreement: Antonio must not ask further questions (so he cannot ask about providence and free will).]

Ant.—I confess to you, it seems to me that Apollo in excusing himself accuses Jupiter more than he accuses Sextus, and Sextus would answer him: Jupiter therefore condemns in me his own crime; it is he who is the only guilty one. He could have made me altogether different: but, made as I am, I must act as he has willed. Why then does he punish me? Could I have resisted his will? Laur.—I confess that I am brought to a pause here as you are. I have made the Gods appear on the scene, Apollo and Jupiter, to make you distinguish between divine foreknowledge and providence. I have shown that Apollo and foreknowledge do not impair freedom; but I cannot satisfy you on the decrees of Jupiter’s will, that is to say, on the orders of providence. Ant.—You have dragged me out of one abyss, and you plunge me back into another and greater abyss. Laur.— Remember our contract: I have given you dinner, and you ask me to give you supper also.

(372)

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3.412

[Our Inability to Know Why God Treats Some People Badly and the Need to Simply Trust that God Is Being Good Nonetheless]

 

[Laurentius explains that we will not know why God is hard against one person but compassionate towards another. We simply have to accept that God is very good and wise, and that is enough reason for us to deem his reasons as being good.]

 

[Antonio says that Laurentius deceived him. Laurentius admits that he has resorted to something that cannot be explained. But this is something we have to accept. We will not know why God is hard against one person but compassionate towards another. We simply have to accept that God is very good and wise, and that is enough reason for us to deem his reasons as being good. And moreover, God is just, because he makes it such that his decrees and operations do not destroy our freedom. We must think of Jesus Christ and put our trust in God’s divine mercy.]

Ant.—Now I discover your cunning: You have caught me, this is not an honest contract. Laur.—What would you have me do? I have given you wine and meats from my home produce, such as my small estate can provide; as for nectar and ambrosia, you will ask the Gods for them: that divine nurture is not found among men. Let us hearken to St. Paul, that chosen vessel who was carried even to the third heaven, who heard there unutterable words: he will answer you with the comparison of the potter, with the incomprehensibility of the ways of God, and wonder at the depth of his wisdom. Nevertheless it is well to observe that one does not ask why God foresees the thing, for that is understood, | it is because it will be: but one asks why he ordains thus, why he hardens such an one, why he has compassion on another. We do not know the reasons which he may have for this; but since he is very good and very wise that is enough to make us deem that his reasons are good. As he is just also, it follows that his decrees and his operation do not destroy our freedom. Some men have sought some reason therein. They have said that we are made from a corrupt and impure mass, indeed of mud. But Adam and the Angels were made of silver and gold, and they sinned notwithstanding. One sometimes becomes hardened again after regeneration. We must therefore seek another cause for evil, and I doubt whether even the Angels are aware of it; yet they cease not to be happy and to praise God. Boethius hearkened more to the answer of philosophy than to that of St. Paul; that was the cause of his failure. Let us believe in Jesus Christ, he is the virtue and the wisdom of God: he teaches us that God willeth the salvation of all, that he willeth not the death of the sinner. Let us therefore put our trust in the divine mercy, and let us not by our vanity and our malice disqualify ourselves to receive it.

(373-374)

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3.413

[Theodorus as Being the One Who Will Learn Why God Must Make Some People Wicked]

 

[Leibniz now speaks for himself rather than quoting the Laurentius dialogue. Leibniz notes that the dialogue’s defect is it condemns providence, as it makes Jupiter the author of sin. Leibniz now continues the story his own way. Sextus next goes to Jupiter and asks him why he condemned him to be wicked and unhappy, and Sextus asks Jupiter to fix his error and give him a fate where he becomes wise and happy. Jupiter says that Sextus can be wise and happy if he does not go to Rome and thus renounces his kingship. Sextus cannot give that up, so he accepts his doomed fate. Leibniz now adds a new character to the story, Theodorus, a high priest who has been listening to this story. Theodorus tells Jupiter that Jupiter’s worshippers are upset, because Jupiter is the one who decided to give Sextus his evil will (which resulted in him going to Rome even though he now knows it dooms him). Jupiter tells Theodorus that he should visit Jupiter’s daughter Pallas, because she will explain what Jupiter was bound to do when deciding Sextus’ will).]

 

[Now Leibniz is talking. He notes that this dialogue’s defect is it condemns providence, as it makes Jupiter the author of sin. So Leibniz will take the story a little further. Sextus leaves Apollo and Delphi and seeks out Jupiter at Dodona. He asks Jupiter, why have you condemned me to be wicked and unhappy? Change my fate and my heart, and acknowledge your error. Jupiter replies that if Sextus renounces Rome, the Parcae (the Fates) will spin him a different fate where he becomes wise and happy. (“Renouncing Rome” seems to mean not going there to become King and thus not committing his subsequent crimes.) Sextus then asks Jupiter, why do I need to renounce the hope of the crown? Why can I not be a good king? Jupiter says that he “knows what is needful for you”, and so if he goes to Rome, he is lost. (The idea might be that given Sextus’ nature, if he puts himself in that situation, he will certainly act wickedly. So he needs to remove himself from those influences in order to be good. This seems similar to Stoic co-fatality. See Goldschmidt Le système stoïcien et l'idée de temps section 2.1.4.3.49) Sextus could not sacrifice his hopes of becoming a king, so he accepts his fate. Now we have a new character, Theodorus, who is a high priest and who was listening to this discussion. Theodorus says to Jupiter that Jupiter shows that the unhappiness Sextus receives will be the result of his own evil will, and he no longer has any right to complain. But now, Theodorus continues, the followers of Jupiter are upset, because it was still Jupiter’s decision to have given Sextus this evil will to begin with rather than a better one. Jupiter then tells Theodorus that he should now visit Jupiter’s daughter Pallas, because she will explain what Jupiter was bound to do (when deciding Sextus’ will).]

This dialogue of Valla’s is excellent, even though one must take exception to some points in it: but its chief defect is that it cuts the knot and that it seems to condemn providence under the name of Jupiter, making him almost the author of sin. Let us therefore carry the little fable still further. Sextus, quitting Apollo and Delphi, seeks out Jupiter at Dodona. He makes sacrifices and then he exhibits his complaints. Why have you condemned me, O great God, to be wicked and unhappy? Change my lot and my heart, or acknowledge your error. Jupiter answers him: If you will renounce Rome, the Parcae shall spin for you different fates, you shall become wise, you shall be happy. Sextus—Why must I renounce the hope of a crown? Can I not come to be a good king? Jupiter—No, Sextus; I know better what is needful for you. If you go to Rome, you are lost. Sextus, not being able to resolve upon so great a sacrifice, went forth from the temple, and abandoned himself to his fate. Theodorus, the High Priest, who had been present at the dialogue between God and Sextus, addressed these words to Jupiter: Your wisdom is to be revered, O great Ruler of the Gods. You have convinced this man of his error; he must henceforth impute his unhappiness to his | evil will; he has not a word to say. But your faithful worshippers are astonished; they would fain wonder at your goodness as well as at your greatness: it rested with you to give him a different will. Jupiter—Go to my daughter Pallas, she will inform you what I was bound to do.

(374-375)

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3.414

[Jupiter’s Daughter Pallas as Guarding the Palace of Fates, Which Holds Every Alternative Fate for the World]

 

[Theodorus goes to Jupiter’s daughter Pallas. She watches and wards over the palace of fates. It contains not just the actual world, but every possible alternative fate. Jupiter picked the best one, and he  sometimes comes to admire his selection. So at the palace, we can see all the alternative twists in fate. And for any one person, there can be a variety of different lives, and thus there are many Sextuses. Pallas then makes a geometrical illustration for this structure of bifurcating worlds. A locus is a set of points that fulfill certain conditions, but these conditions do not often determine a singular point but rather a set of points forming a line. (For instance, the center of a circle is a focus, and all points equidistant from it is both the circumference of the circle and also the locus of points for that criterion.) Pallas is saying that when you ask for a different condition in the world, like Sextus not going to Rome, then you have defined a whole range of alternative possibilities. She will now show this ranging of various Sextuses.]

 

[Theodorus then goes to Athens, sleeps in the temple of the Goddess Pallas. In his dream, Pallas appears at the gate of a great palace, with the quote, Qualisque videri, Coelicolis et quanta solet: In the guise and grandeur in which she is wont to appear before the denizens of heaven. (Virgil, Aeneid, II, 591-592). She touches Theodorus’ face with an olive branch so he can converse with her. Pallas says to Theodorus that this is the palace of fates, and she keeps watch and ward over it.  In it are the representations not only of what happens, but also of all that which is possible. Jupiter surveyed all these possibilities at the beginning of the existing world, he classified the possibilities into worlds, and chose the best of all. He sometimes comes to the palace to enjoy his selection. So because we can also see whole alternate worlds with a different fates, we can know what would happen if any particular possibility should attain into existence. “And whenever the conditions are not determinate enough, there will be as many such worlds differing from one another as one shall wish, which will answer differently the same question, in as many ways as possible.” (So if in the alternate one change somewhere along the line creates a path that itself admits of further determinations and thus many further alternative outcomes, then we can see all those ramifying possibilities too.) Pallas then makes a geometrical illustration for this structure of bifurcating worlds. A locus is a set of points that fulfill certain conditions, but these conditions do not often determine a singular point but rather a set of points forming a line. (For instance, the center of a circle is a focus, and all points equidistant from it is both the circumference of the circle and also the locus of points for that criterion.) Pallas is saying that when you ask for a different condition in the world, like Sextus not going to Rome, then you have defined a whole range of alternative possibilities. She will now show this ranging of various Sextuses.]

Theodorus journeyed to Athens: he was bidden to lie down to sleep in the temple of the Goddess. Dreaming, he found himself transported into an unknown country. There stood a palace of unimaginable splendour and prodigious size. The Goddess Pallas appeared at the gate, surrounded by rays of dazzling majesty.

Qualisque videri

Coelicolis et quanta solet.

She touched the face of Theodorus with an olive-branch, which she was holding in her hand. And lo! he had become able to confront the divine radiancy of the daughter of Jupiter, and of all that she should show him. Jupiter who loves you (she said to him) has commended you to me to be instructed. You see here the palace of the fates, where I keep watch and ward. Here are representations not only of that which happens but also of all that which is possible. Jupiter, having surveyed them before the beginning of the existing world, classified the possibilities into worlds, and chose the best of all. He comes sometimes to visit these places, to enjoy the pleasure of recapitulating things and of renewing his own choice, which cannot fail to please him. I have only to speak, and we shall see a whole world that my father might have produced, wherein will be represented anything that can be asked of him; and in this way one may know also what would happen if any particular possibility should attain unto existence. And whenever the conditions are not determinate enough, there will be as many such worlds differing from one another as one shall wish, which will answer differently the same question, in as many ways as possible. You learnt geometry in your youth, like all well-instructed Greeks. You know therefore that when the conditions of a required point do not sufficiently determine it, and there is an infinite number of them, they all fall into what the geometricians call a locus, and this locus at least (which is often a line) will be determinate. Thus you can picture | to yourself an ordered succession of worlds, which shall contain each and every one the case that is in question, and shall vary its circumstances and its consequences. But if you put a case that differs from the actual world only in one single definite thing and in its results, a certain one of those determinate worlds will answer you. These worlds are all here, that is, in ideas. I will show you some, wherein shall be found, not absolutely the same Sextus as you have seen (that is not possible, he carries with him always that which he shall be) but several Sextuses resembling him, possessing all that you know already of the true Sextus, but not all that is already in him imperceptibly, nor in consequence all that shall yet happen to him. You will find in one world a very happy and noble Sextus, in another a Sextus content with a mediocre state, a Sextus, indeed, of every kind and endless diversity of forms.

(475-476)

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3.415

[The Different Rooms with Different Worlds, Fates, and Sextuses]

 

[Pallas takes Theodorus to one of the hall’s in the palace of fates, which displays a whole complete world. In this one, Sextus obeys Jupiter, goes off to a place like Corinth, cultivates a garden, finds treasure, is loved by the city, and dies old. In the hall is this particular world’s book of fate. Sextus has a number on his head. Theodorus follows Pallas’ instruction to find that number in the book, where he finds a more extensive account of this Sextus’ life. Pallas explains that he only needs to point to a line of the text, and its story will come to life. This way, he could see all the characteristics of some part of Sextus’ life. Next they go to another hall, which displays a different possible world. Here, there is another Sextus who also obeys Jupiter, but this time he goes instead to Thrace where he marries the king’s daughter, succeeds the king, and is adored by the subjects. Theodorus and Pallas go to a number of rooms, each time with another possible world and life for Sextus.]

 

[The Goddess Pallas then leads Theodorus to a hall of the palace, which is no longer a hall, but a world, with the quotation Solemque suum, sua sidera norat: A sun of its own it knew, and stars of its own. Here, Sextus obeys Jupiter and goes to some city resembling Corinth between two seas; he cultivates a garden and discovers treasure, is esteemed and beloved by the city, dying old. Theodorus sees Sextus’ life unfold like a stage production. Theodorus then notes a great volume of writings in this hall. Pallas explains that it is the history of this world; it is its book of fates. Sextus had a number on his head. Pallas says, look for that number in the book. Theodorus does that, and he finds an even more extensive history of this Sextus’ life. Pallas then says that Theodorus need only point to a line of text, and its story will come to life, and so by pointing to one, he saw all the characteristics of one particular part of Sextus’ life. They then go to another hall, which presents another possible world. Here there is another Sextus who also obeys Jupiter, and he goes to Thrace. There he marries the king’s daughter, succeeds the king, and is adored by the subjects. Theodorus and Pallas go to a number of rooms, each time with another possible world and life for Sextus.]

Thereupon the Goddess led Theodorus into one of the halls of the palace: when he was within, it was no longer a hall, it was a world,

Solemque suum, sua sidera norat.

At the command of Pallas there came within view Dodona with the temple of Jupiter, and Sextus issuing thence; he could be heard saying that he would obey the God. And lo! he goes to a city lying between two seas, resembling Corinth. He buys there a small garden; cultivating it, he finds a treasure; he becomes a rich man, enjoying affection and esteem; he dies at a great age, beloved of the whole city. Theodorus saw the whole life of Sextus as at one glance, and as in a stage presentation. There was a great volume of writings in this hall: Theodorus could not refrain from asking what that meant. It is the history of this world which we are now visiting, the Goddess told him; it is the book of its fates. You have seen a number on the forehead of Sextus. Look in this book for the place which it indicates. Theodorus looked for it, and found there the history of Sextus in a form more ample than the outline he had seen. Put your finger on any line you please, Pallas said to him, and you will see represented actually in all its detail that which the line broadly indicates. He obeyed, and he saw coming into view all the characteristics of a portion of the life of | that Sextus. They passed into another hall, and lo! another world, another Sextus. who, issuing from the temple, and having resolved to obey Jupiter, goes to Thrace. There he marries the daughter of the king, who had no other children; he succeeds him, and he is adored by his subjects. They went into other rooms, and always they saw new scenes.

(376-377)

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3.416

[The Best of All Possible Worlds is One Where Some People Are Made as Wicked and Suffer]

 

[The halls are stacked in the form of a pyramid, with them being better and better worlds as one gets higher and with the one at the apex being the best possible world that Jupiter/God therefore chose as the existing one. But even though there is a certain apex world, the pyramid has no bottom, because the worlds degrade infinitely going down. In the best possible world, Sextus is displayed as he actually is. He goes to Rome, creates confusion and violates his friend’s wife and is then driven out of Rome, beaten and unhappy. But even though choosing that world would have made a happy Sextus, God could not choose it, because then God would be renouncing God’s divine wisdom (which calls him to choose the best of all possible worlds.) So in order to make Sextus actual, he had to choose a world where he was wicked. (For, suppose he did not choose that world; he would either choose no world or a less perfect one. But God is wise and cannot choose a less perfect world. So God must choose the world where Sextus is wicked.) This means that “The crime of Sextus serves for great things: it renders Rome free; thence will arise a great empire, which will show noble examples to mankind.” (So it comes back to God being responsible for Sextus’ wickedness, but God had no better alternative options in the matter.)]

 

[The halls are stacked up into a pyramid, and as Theodorus and Pallas proceed through them, they ascend toward the structure’s apex, with each new hall representing yet a more beautiful world. Finally they reached the top one, which was the most beautiful of all. Such a best one is needed, for otherwise God would have no basis to choose one to be the existing world. (This is an important point as we will see. If there were no best possible world, there would be no basis to choose one, namely, that best one ((and note that Jupiter cannot choose more than one! There seems to be a non-contradiction constraint in Leibniz’s metaphysics)).) However, there are an infinity of lesser and lesser perfect worlds, so the pyramid, while having a determinate apex, had no ultimate base. When Theodorus enters this highest world, he is entranced in ecstasy, and Pallas needs to give him some divine medicine to restore him. Here we see Sextus as he actually is. He leaves the temple of Jupiter (or Apollo?) in rage and scorns the council of the gods. He goes to Rome, creating confusion and violating his friend’s wife. He is then driven out of Rome with his father, beaten and unhappy. Yet if Sextus were to be happy at Corinth or king of Thrace, it would no longer be this most perfect world but another, less perfect one, altogether. But even though choosing that world would have made a happy Sextus, God could not choose it, because then God would be renouncing God’s divine wisdom (which calls for God to choose the best of all possible worlds.) So in order to make Sextus actual, God had to choose a world where he was wicked. For, suppose God did not choose that world; God would either choose no world or a less perfect one. But God is wise and cannot choose a less perfect world. So God must choose the world where Sextus is wicked. (This seems similar to providence in Stoic theology. See for instance Goldschmidt, système stoïcien, section 2.1.4.1.37.5). This means that “The crime of Sextus serves for great things: it renders Rome free; thence will arise a great empire, which will show noble examples to mankind.” (So it comes back to God being responsible for Sextus’ wickedness, but God had no better alternative options in the matter.)]

The halls rose in a pyramid, becoming even more beautiful as one mounted towards the apex, and representing more beautiful worlds. Finally they reached the highest one which completed the pyramid, and which was the most beautiful of all: for the pyramid had a beginning, but one could not see its end; it had an apex, but no base; it went on increasing to infinity. That is (as the Goddess explained) because amongst an endless number of possible worlds there is the best of all, else would God not have determined to create any; but there is not any one which has not also less perfect worlds below it: that is why the pyramid goes on descending to infinity. Theodorus, entering this highest hall, became entranced in ecstasy; he had to receive succour from the Goddess, a drop of a divine liquid placed on his tongue restored him; he was beside himself for joy. We are in the real true world (said the Goddess) and you are at the source of happiness. Behold what Jupiter makes ready for you, if you continue to serve him faithfully. Here is Sextus as he is, and as he will be in reality. He issues from the temple in a rage, he scorns the counsel of the Gods. You see him going to Rome, bringing confusion everywhere, violating the wife of his friend. There he is driven out with his father, beaten, unhappy. If Jupiter had placed here a Sextus happy at Corinth or King in Thrace, it would be no longer this world. And nevertheless he could not have failed to choose this world, which surpasses in perfection all the others, and which forms the apex of the pyramid. Else would Jupiter have renounced his wisdom, he would have banished me, me his daughter. You see that my father did not make Sextus wicked; he was so from all eternity, he was so always and freely. My father only granted him the existence which his wisdom could not refuse to the world where he is included: he made him pass from the region of the possible to that of the actual beings. The crime of Sextus serves for great things: it renders Rome free; thence will arise a great empire, which will show noble examples to mankind. But that | is nothing in comparison with the worth of this whole world, at whose beauty you will marvel, when, after a happy passage from this mortal state to another and better one, the Gods shall have fitted you to know it.

(377-378)

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3.417

[God’s Foreknowledge as Being More Than Mere Foreseeing the Events of This World but Instead as Knowing all Events in All Worlds]

 

[Theodorus awakes from this dream being even more convinced of his beliefs, and he carries out his priestly duties zealously. Leibniz explains that this Sextus story fills out what is missing in Valla’s dialogue. Valla discussed God’s knowledge of vision (of the future) which concerns beings that exist. But Pallas shows knowledge of simple intelligence, which embraces all that is possible (that is, knowledge not just of the existing world but of all other possibilities). Leibniz ends by saying that the source of things should be sought in this second kind of knowledge.]

 

[(ditto).]

At this moment Theodorus wakes up, he gives thanks to the Goddess, he owns the justice of Jupiter. His spirit pervaded by what he has seen and heard, he carries on the office of High Priest, with all the zeal of a true servant of his God, and with all the joy whereof a mortal is capable. It seems to me that this continuation of the tale may elucidate the difficulty which Valla did not wish to treat. If Apollo has represented aright God’s knowledge of vision (that which concerns beings in existence), I hope that Pallas will have not discreditably filled the role of what is called knowledge of simple intelligence (that which embraces all that is possible), wherein at last the source of things must be sought.

(378)

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From:

 

Leibniz, Gottfried. 2007. Theodicy: Essays on the Goodness of God, the Freedom of Man and the Origin of Evil. English translation by E. Huggard. Charleston, S.C.: BiblioBazaar.

 

 

For Latin translations, I consulted also:

Jorgensen, L.M. 2008. “PHIL 375: Advanced History of Philosophy”. Available at:

https://docslide.net/documents/phil-375-advanced-history-of-philosophy-ljorgensteachrestheodicy-packetpdfleibnizs.html

and

Leibniz, Gottfried. Philosophical Writings. English translation by Mary Morris. London: Dent. New York: Dutton. Available at:

https://archive.org/details/in.ernet.dli.2015.214800

 

 

 

 

 

 

4 Jul 2018

Priest (2.8) An Introduction to Non-Classical Logic, ‘Meinongianism,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

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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

 

2.

Basic Modal Logic

 

2.8

Meinongianism

 

 

 

 

Brief summary:

(2.8.1) In modal realism, possible worlds and their members are concrete objects, and in modal actualism, they are abstract objects. In both cases, they are existing objects. Now we will consider the idea that they are non-existent objects (a position called Meinongianism). (2.8.2) We are already familiar with such non-existent things as Santa Claus and phlogiston. We can think of possible worlds in the same light. (2.8.3) Meinong famously held that there are non-existent objects, and the arguments against his position are not especially cogent. (2.8.4) An example of an uncogent argument against Meinongism is that non-existent possible worlds cannot causally interact with us, and thus we can know nothing about them. Yet, this objection would also hold for modal actualism and modal realism too. (2.8.5) That same objection also fails to take into account the fact that we do know facts about certain non-existent objects on account of these facts being stipulated, for example: “Holmes lived in Baker Street – and not Oxford Street – because Conan Doyle decided it was so” (31). (2.8.6) Priest ends by noting that {1} the aforementioned ideas do not settle the matter, as there are more suggestions to consider, and {2} there are more objections to consider.

 

 

 

 

 

Contents

 

2.8.1

[Possible Worlds as Non-Existing Objects]

 

2.8.2

[Our Familiarity with Non-Existing Things]

 

2.8.3

[Meinong’s Non-Existent Objects]

 

2.8.4

[An Objection. Its Ineffectivity as Resulting from Its Additional Application to Actualism and Realism]

 

2.8.5

[The Objection’s Ineffectivity as Resulting from the Fact that Stipulations Tell Us About Non-Existent Objects]

 

2.8.6

[Our Treatment on Possible Worlds Representation as Incomplete]

 

 

 

 

 

Summary

 

 

2.8.1

[Meinongianism: Possible Worlds as Non-Existing Objects]

 

[In modal realism, possible worlds and their members are concrete objects, and in modal actualism, they are abstract objects. In both cases, they are existing objects. Now we will consider the idea that they are non-existent objects (a position called Meinongianism).]

 

[In section 2.6.1 we learned that modal realism is the view that possible worlds are real worlds that exist at different times and/or places. And in section 2.7.1 we saw that under modal actualism, possible worlds are understood not as physically real entities, like in modal realism, but rather as abstract entities, like numbers. One version of modal actualism understands a possible world as a set of propositions or other language-like entities and as being “individuated by the set of things true at it, which is just the set of propositions it contains” (p.29, section 2.7.2). There is another sort of modal actualism called combinatorialism. Here a possible world is understood as “the set of things in this world, rearranged in a different way. So in this world, my house is in Australia, and not China; but rearrange things, and it could be in China, and not Australia” (p.30, section 2.7.5). So in modal realism, possible worlds and their members are concrete objects, and in modal actualism, they are abstract objects. In both cases, they are existing objects. Now we will consider the idea that they are non-existent objects (a position called Meinongianism).]

Both realism and actualism take possible worlds and their denizens, whatever they are, to exist, either as concrete objects or as abstract objects. Another possibility is to take them to be non-existent objects. (We know, after all, that such things do not really exist!)

(30)

[contents]

 

 

 

 

2.8.2

[Our Familiarity with Non-Existing Things]

 

[We are already familiar with such non-existent things as Santa Claus and phlogiston. We can think of possible worlds in the same light.]

 

[(ditto)]

We are all, after all, familiar with the thought that there are non-existent things, like fairies, Father Christmas (sorry) and phlogiston. Possible worlds are things of this kind.

(30)

[contents]

 

 

 

 

2.8.3

[Meinong’s Non-Existent Objects]

 

[Meinong famously held that there are non-existent objects, and the arguments against his position are not especially cogent.]

 

[(ditto)]

The view that there are non-existent objects was espoused, famously, by Meinong. It had a very bad press for a long time in English-speaking philosophy, but it is fair to say that many of the old arguments against the possibility of there being non-existent objects are not especially cogent.

(30)

[contents]

 

 

 

 

2.8.4

[An Objection. Its Ineffectivity as Resulting from Its Additional Application to Actualism and Realism]

 

[An example of an uncogent argument against Meinongism is that non-existent possible worlds cannot causally interact with us, and thus we can know nothing about them. Yet, this objection would also hold for modal actualism and modal realism too.]

 

[Priest then gives an example of an uncogent argument against non-existent objects. Someone might object that since non-existent possible worlds cannot causally interact with us, that means we can know nothing about them. Yet, this objection would also hold for modal actualism and modal realism too.]

For example, one argument against such objects is that, since they cannot interact with us causally, we would have no way of knowing anything | about them. But exactly the same is true, of course, of possible worlds as both the realist and the actualist conceive them, so this can hardly count to their advantage against Meinongianism about worlds.

(30-31)

[contents]

 

 

 

 

2.8.5

[The Objection’s Ineffectivity as Resulting from the Fact that Stipulations Tell Us About Non-Existent Objects]

 

[That same objection also fails to take into account the fact that we do know facts about certain non-existent objects on account of these facts being stipulated, for example: “Holmes lived in Baker Street – and not Oxford Street – because Conan Doyle decided it was so” (31).]

 

[(ditto).]

Moreover, it is very clear how we know facts about at least some nonexistent objects: they are simply stipulated. Holmes lived in Baker Street – and not Oxford Street – because Conan Doyle decided it was so.

(31)

[contents]

 

 

 

 

2.8.6

[Our Treatment on Possible Worlds Representation as Incomplete]

 

[Priest ends by noting that {1} the aforementioned ideas do not settle the matter, as there are more suggestions to consider, and {2} there are more objections to consider.]

 

[Priest ends these four sections on the question of what possible worlds represent by noting that {1} the aforementioned ideas do not settle the matter, as there are more suggestions to consider, and {2} there are more objections to consider.]

The preceding considerations hardly settle the matter of the nature of possible worlds. There are many other suggested answers (most of which are some variation on one or other of the themes that I have mentioned); and there are many objections to the suggestions I have raised, other than the ones that I have given, as well as possible replies to the objections I have raised; philosophers can have hours of fun with possible worlds. This will do for the present, though.

(31)

[contents]

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

 

 

 

.

 

27 Jun 2018

Priest (9.7) Introduction to Non-Classical Logic, ‘Impossible Worlds and Relevant Logic,’ summary

 

by Corry Shores

 

[Search Blog Here. Index-tags are found on the bottom of the left column.]

 

[Central Entry Directory]

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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

 

9.

Logics with Gaps, Gluts and Worlds

 

9.7

Impossible Worlds and Relevant Logic

 

 

 

 

Brief summary:

(9.7.1) We will now discuss philosophical matters regarding K4 , N4 , K , and N. (9.7.2) We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different. (9.7.3) Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds. (9.7.4) We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.” (9.7.5) Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling. (9.7.6) In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to. (9.7.7) Logically impossible worlds can also in fact be ones where laws of logic indeed are broken. (9.7.8) Relevant propositional logics are ones where whenever “A B is logically valid, A and B have a propositional parameter in common” (172). (9.7.9) But N4 is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds. (9.7.10) In a similar way, N is also a relevant logic. (9.7.11) Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters. (9.7.12) There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173). (9.7.13) Relevance in our systems here however is not conditions added on top of classical validity. (9.7.14) If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.

 

 

 

 

Contents

 

9.7.1

[Moving to a Discussion on K4 , N4 , K , and N]

 

9.7.2

[Logically Impossible Worlds]

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

9.7.8

[Relevant Logics and the Conditional]

 

9.7.9

[N4 as Relevant]

 

9.7.10

[N as Relevant]

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

9.7.12

[Filter Logics]

 

9.7.13

[More Than Classical Relevance]

 

9.7.14

[Preserving Conventional Truth in This System]

 

 

 

 

 

 

 

 

Summary

 

9.7.1

[Moving to a Discussion on K4 , N4 , K , and N]

 

[We will now discuss philosophical matters regarding K4 , N4 , K , and N.]

 

[Recall the semantics and tableau constructions from previous sections: K4 (9.2 and 9.3); N4 (9.4 and 9.5) and K and N (9.6). Now Priest will discuss philosophical matters regarding these constructions.]

We are now in a position to make some comments on the import of the previous constructions.

(171)

[contents]

 

 

 

 

9.7.2

[Logically Impossible Worlds]

 

[We will now call non-normal worlds “logically impossible worlds,” because they are worlds where the laws of logic are different.]

 

[Recall from section 4.2.3 that for modal logics, non-normal worlds are ones where nothing is necessary and all is possible; for, at non-normal worlds, all necessary propositions (those starting with □) are always false, and all possible propositions (those starting with ◊) are always true. In section 9.2, we discussed a possible worlds First Degree Entailment (and thus four value-situationed) system called K4. In section 9.4, we noted that K4 has the following problematic valid formula: ⊨ p → (qq). This was problematic, because we want to be able to say things about what would follow if certain laws of logic were suspended. So the following would be valid, even though we would want it to be invalid: “if every instance of the law of identity failed, then, if cows were black, cows would be black. If every instance of the law failed, then it would precisely not be the case that if cows were black, they would be black” (p.167, section 9.4.3 ). But “we need to countenance worlds where the laws of logic are different, and so where laws of logic, like the law of identity, may fail. This is exactly what non-normal worlds are” (p.167). We thus incorporated non-normal worlds into K4 in order to get N4. But we note here that in non-normal worlds, the normal laws of logic are different, and we will call such non-normal worlds “logically impossible worlds.”]

As we saw (9.4.4 9.4.6), non-normal worlds of the kind we have employed in this chapter are worlds where the laws of logic are different. Let us call these ‘logically impossible worlds’.

(171)

[contents]

 

 

 

 

9.7.3

[Logically Impossible Worlds as Admissible]

 

[Just as there is no problem in conceiving physically impossible worlds, there should likewise be no problem in conceiving logically impossible worlds.]

 

[In section 3.6.5 (not yet summarized), Priest discussed physically impossible worlds: “Something is physically necessary if it is determined by the laws of nature, and physically possible if it is compatible with the laws of nature. Thus, it is physically impossible for me to jump thirty metres into the air (though this is not a logical impossibility)” (46). Priest says now that there is no reason why there cannot likewise be logically impossible worlds.]

There seems to be no reason why there should not be logically impossible worlds, in whatever sense there are possible worlds. Physically impossible worlds, where the laws of physics are different, are entirely routine (see 3.6.5). And just as there are worlds where the laws of physics are different, there must be worlds where the laws of logic are different.

(171)

[contents]

 

 

 

 

9.7.4

[Our Seeming Assumption of Logically Impossible Worlds]

 

[We already seem to suppose such logically impossible worlds when we note how certain laws of logic fail in particular non-classical logics, as for example when we say: “if intuitionist logic were correct, the law of double negation would fail.”]

 

[In fact, when we discuss non-classical logics, we seem to suppose such “logically impossible” worlds (even though in fact the real world might be one where the laws of classical logic do indeed fail). It is implied for example when we say that “if intuitionist logic were correct, the law of double negation would fail.”]

After all, we seem to envisage just such worlds when we evaluate conditionals such as ‘if intuitionist logic were correct, the law of double negation would fail’ (true), ‘if intuitionist logic were correct, the law of | identity would fail’ (false). Even if one is a modal realist (2.6), why should there not be such worlds?

(171-172)

[contents]

 

 

 

 

9.7.5

[Failure of Objections to Impossible Worlds]

 

[Objections to logically impossible worlds do not work. For, we cannot simply require that the laws of logic admit of no variation, when in fact that is what we are successfully and fruitfully modelling.]

 

[Priest next deals with some objections to the idea that there can be logically impossible worlds. {1} Objection: Someone might say that logical laws should always hold at possible worlds, by definition. Reply: we are not dealing with possible worlds but rather impossible ones. {2} Objection: One might say that some proposed logically law for a possible world that breaks one of our normal logical laws cannot be the case, simply because it is breaking a normal logical law. For example, suppose someone claims that there is a world where it is a logical law that A → (B ∧ ¬B) holds and so does A. Then, by modus ponens we can infer that B ∧ ¬B. The objector can say that this is a contradiction and it cannot be the case. Reply: {2a} Some might have philosophical reasons to say that the normal laws can be broken. For example, a dialetheist would say that the law of non-contradiction is breakable. {2b} The objection assumes that modus ponens holds in this world. But as an impossible world, it may not.]

One might suggest that there can be no worlds at which logical laws fail: by definition, logical laws hold at all possible worlds. Maybe so. But it is precisely impossible worlds that we are dealing with here. Or one might say: take a world in which it is a logical law that A → (B ∧ ¬B) and in which A is also true. It would follow that B ∧ ¬B is true at that world, which cannot be the case. This argument is hardly likely to persuade someone who accepts the possibility of truth-value gluts. But in any case, it is fallacious. For who says that modus ponens holds at that world? In the semantics we have looked at, it is entirely possible to have both A and A C holding at a non-normal world, without C holding there.

(172)

[contents]

 

 

 

 

9.7.6

[The Non-Necessity for Logical Laws To Be Broken in Logically Impossible Worlds]

 

[In a logically impossible world, it could still be that no normal laws of logic be broken, just like how in a physically impossible world, normally-impossible physical events can take place, but for contingent reasons happen not to.]

 

[Priest next notes that logically impossible worlds do not necessarily have cases of broken laws of logic, just like how physically impossible worlds may allow for certain alternate physical situations without them ever obtaining. However, one could specifically define logically impossible worlds as ones where the laws of logic are in fact broken.]

Note that one might take ‘logically impossible world’ to mean something other than ‘world where the laws of logic are different’. One might equally take it to mean ‘world where the logically impossible happens’. This need not be the same thing. If this is not clear, just consider physically impossible worlds. The fact that the laws of physics are different does not necessarily mean that physically impossible things happen there (though the converse is true). For example, even if the laws of physics were to permit things to accelerate past the speed of light, it does not follow that anything actually would. Things at that world might be accelerating very slowly, and the world might not last long enough for any of them to reach super-luminal speeds.

(172)

[contents]

 

 

 

 

9.7.7

[Logically Impossible Worlds Where Normal Laws of Logic Are In Indeed Broken]

 

[Logically impossible worlds can also in fact be ones where laws of logic indeed are broken.]

 

[I might be mistaken about this next point. It might be that we know there are logically impossible worlds where the laws of logic are broken, because we have already seen that there is a world where A and A C are true, but C is not. In the footnote Priest mentions some inferences that do not hold in any impossible world. However, these are instances without conditionals, and it is conditionals that express the laws of logic (but I do not myself know why that is.)]

But logically impossible worlds, in the sense that these occur in the semantics we have been looking at, may be logically impossible in the second sense as well. For example, there are, as has just been noted, worlds where A and A C are true, but C is not.6

(172)

6. There are no worlds at which AB is true, but A is not, or at which ¬¬A is true, but A is not. But it is conditionals that express the laws of logic, not conjunctions or negations. That is why it is their behaviour (and only theirs) that changes at non-normal worlds.

(172)

[contents]

 

 

 

 

9.7.8

[Relevant Logics and the Conditional]

 

[Relevant propositional logics are ones where whenever “A B is logically valid, A and B have a propositional parameter in common” (172). ]

 

[Priest now defines relevant logic: “A propositional logic is relevant iff whenever A B is logically valid, A and B have a propositional parameter in common” (172). This may seem odd, because A and B would seem to be propositional parameters, and surely we are not saying that conditions need to be of the form A A to be relevant. So recall from section 1.2.3: “I use capital Roman letters, A, B, C, ..., to represent arbitrary formulas of the object language. Lower-case Roman letters, p, q, r, ..., represent arbitrary, | but distinct, propositional parameters” (4-5). So maybe we would need to look at the lower-case sorts of formulations, when looking for relevance. We will see some examples. Or maybe A B is shorthand for more complex formulations, like we will see below, as with ⊨ A ⥽ (B ∨ ¬B). He says that conditionals that suffer from the paradoxes of implication, including the strict conditional, are not relevant. Let us look at the paradoxes of strict implication. In section 4.6.3 we saw that the following  are valid for the strict conditional.

A ⥽ (B ∨ ¬B)

⊨ (A ∧ ¬A) ⥽ B

We might fill them out with propositional parameters I am going to guess in the following way.

p ⥽ (q ∨ ¬q)

⊨ (p ∧ ¬p) ⥽ q

But I am not sure about much here yet. Yet we can see that the strict conditional is not part of a relevant logic. And these same forms are not valid in K4.

K4 pq ∨ ¬q

K4 (p ∧ ¬p) → q

Nonetheless, Priest says that neither K4 and K are relevant. To see why, we first recall from section 9.4.2 that in K4, ⊨ p → (qq) is valid, and in section 9.6.6 we saw that it if valid in K too.]

A propositional logic is relevant iff whenever A B is logically valid, A and B have a propositional parameter in common. Obviously, any conditional that suffers from paradoxes of implication (material implication, | strict implication, the intuitionist conditional) is not relevant. Neither are K4 and K relevant, as we have seen (9.4.2 and 9.6.6).

(172-173)

[contents]

 

 

 

 

9.7.9

[N4 as Relevant]

 

[But N4 is a relevant logic, on account of how conditionals are evaluated in normal worlds (they depend on the values in non-normal worlds) in combination with the arbitrarity of their value assignments in non-normal worlds.]

 

[Priest will now show that N4 is a relevant logic. It gets very technical, and I am the wrong person to summarize this, so please skip to the quotation below. I will try to say some things still, but they probably will not help you. Recall that in N4, there are non-normal worlds. And recall from section 4.2.5 and 9.4.9 that inferences are valid only if they preserve truth in all interpretations at all normal worlds. I might have this wrong, but I think that means it cannot be that the premises are at least true and the conclusion not at least true. If I am following even a little here (and probably not), Priest is going to do the following. Let me first note that I am not certain if we are dealing with an inference, like in his cited problem, or a simple conditional, like mentioned in the last line of this paragraph. I am also not sure if it makes a difference. Let us for now say that we are dealing with the conditional A B, and we want to know if it would be valid/true in N4 whenever there is no relevance of the antecedent A to the consequent B. Priest will make a model where it is false, even though in K4 presumably it would be true. It seems that the way this will work will have to do with the fact that in normal worlds we evaluate conditionals on the basis of all other worlds, whether normal or not. But as we saw in section 9.4.6, we do not evaluate the conditionals in non-normal worlds compositionally in terms of the component terms’ values but rather we assign their values arbitrarily however we please. Priest will exploit those two features of N4 in order to make a non-relevant A B be false/invalid in a normal world. So our model will have two worlds, 0 and 1, and world 1 is the non-normal one. It is still unclear to me if we are dealing with an inference from A to B or a conditional, but I am guessing wildly that Priest is covering both options. (Sorry, please read the text). For non-relevance, we suppose that we have antecedent A and consequent B (or premise(s) A and conclusion B), but A and B share no propositional parameters in common. We also consider a propositional parameter (or conditional) called D, which can be included either in antecedent A or consequent B (or in premise(s) A or conclusion B); but it cannot be in both, because as we said, A and B share no parameters in common, thus if D is in one, it cannot be in the other. We will assign our values for the conditional in non-normal world 1 arbitrarily, as that is how it works in non-normal worlds (see section 9.4.6). So we say, if D is among the antecedent (or premises), then we assign it both as true and also as false. Or, if instead it is in the consequent (or conclusion), then we assign it neither true nor false. Now, let us stick with conditionals for a second. Recall from section 9.2.4 that this is how we evaluate conditionals in N4:

A Bρw1 iff for all w′ ∈ W such that Aρw1, Bρw1

A Bρw0 iff for some w′ ∈ W, Aρw1 and Bρw0

(p.164, section 9.2.4)

We ask, A Bρw0??? In other words, we want to determine the value of A B in world 0, the normal world. Priest’s way of proving this uses the induction method, which I have not learned yet (see section 0.2). Were we to perform it, we would find that somehow, regardless of whether D is in the antecedent (or premises) or in the consequent (or conclusion), A will be both true and false (and thus at least true) and B will be neither true nor false (and thus not at least true). And hence the formula will be false/invalid in world 0, the normal world. Please read the quotation, as I am not grasping this one very well at all.]

But N4 is a relevant logic. This can be seen by modifying the argument of 8.10, problem 5. Suppose that A and B share no propositional parameters, and consider an interpretation ⟨W, N, ρ⟩, where W = {w0, w1}; N ={w0}; if D is a propositional parameter or a conditional in A, w11 and w10; if D is a propositional parameter or a conditional in B, neither w11 nor w10. (D cannot occur in both, since A and B have no parameters in common.) It is easy to check that w11 and w10, but neither w11 nor = w10.7 In particular, A is true at w1 and B is not. Hence A B is not true at w0.

(173)

7. Proof: For the first, what we show is that every formula made up from the propositional parameters occurring in A – and so, in particular, A – the result holds. Similarly for B. This is proved by induction on the construction of sentences, but an induction slightly different from the normal kind. Note that every formula can be built up from conditionals and parameters using the extensional connectives. Hence, the result may be proved by induction, with parameters and conditionals as the basis case, and induction cases for the extensional connectives. The basis case is true by definition. The induction cases are as in the notes to 8.4.6 and 8.4.9.

(173)

[contents]

 

 

 

 

9.7.10

[N as Relevant]

 

[In a similar way, N is also a relevant logic.]

 

[Priest next shows how in a similar way N is a relevant logic. Please consult the text for the details.]

A similar argument shows that Nis a relevant logic. Take a ∗ interpretation ⟨W, N, ∗, v⟩, where W = {w0, w1, w2}; N = {w0}, w*o  = w0, w*1 = w2, w*2 = w1; for every propositional parameter or conditional, D, in A, vw1(D) = 1 and vw2(D) = 0; for every propositional parameter or conditional, D, in B, vw1(D) = 0 and vw2(D) = 1. One can check that vw1(A) = 1, and vw1(B) = 0. Hence vw0(A B) = 0. Details are left as an exercise.

(172)

[contents]

 

 

 

 

9.7.11

[Relevant Logics Meet Our Intuitions About Conditionals and Relevance]

 

[Relevant logics tend to our intuitions that there should be relevance between antecedent and consequent of conditionals, and this can be done by requiring them to share parameters.]

 

[We have the intuition already that “for a conditional to be true there must be some connection between its antecedent and consequent” (172). But it is not always obvious how to do that in a formalized way. Yet we saw in section 9.7.8 that one way is to require shared parameters.]

It is a natural thought that for a conditional to be true there must be some connection between its antecedent and consequent. It was precisely this idea that led to the development of relevant logic. A sensible notion of connection is not so easy to spell out, however (as we saw, in effect, in 4.9.2). The parameter-sharing condition of 9.7.8 gives some content to the idea.

(172)

[contents]

 

 

 

 

9.7.12

[Filter Logics]

 

[There is another sort of relevant logic that is of a whole different class, called filter logics, in which “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter” (173).]

 

[Priest then notes another sort of relevant logic called filter logics. Here “a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter.” But it is a different sort of logic than the ones of this book. Priest notes that often times filter logics break the principle of transitivity.]

There are some approaches to relevant logic where a conditional is taken to be valid iff it is classically valid and satisfies some extra constraint, for example that antecedent and consequent share a parameter. (These are | sometimes called filter logics, since the extra constraint filters out ‘undesirables’.) Characteristically, such approaches give rise to relevant logics of a kind different from those considered in this book. For example, if the parameter-sharing filter is used, (p ∧ (¬pq)) → q is valid, which it is not in the relevant logics of this, and subsequent, chapters. Typically (though not invariably), a feature of filter logics is the failure of the principle of transitivity: if AB and B C then A C (thus breaking the argument of 4.9.2).

(173-174)

[contents]

 

 

 

 

9.7.13

[More Than Classical Relevance]

 

[Relevance in our systems here however is not conditions added on top of classical validity.]

 

[The way that we are dealing with relevance here “is not some extra condition imposed on top of classical validity” (174). Rather, it is something else (but I am not sure I understand what it is and what the distinction is. So see the quote below.)]

In the present approach, relevance is not some extra condition imposed on top of classical validity. Rather, relevance, in the form of parameter sharing, falls out of something more fundamental, namely the taking into account of a suitably wide range of situations.

(174)

[contents]

 

 

 

 

9.7.14

[Preserving Conventional Truth in This System]

 

[If we wanted to keep this system but reserve a real world where truth operates in a more conventional way, then we can designate an @ actual world that has certain constraints. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts in the actual real world @.]

 

[Priest then notes some additional concerns. We might want within this four value-situationed non-normal worlds logic to reserve certain conventional properties of truth for the “real” or actual world. Priest then explains how this would work. We symbolize @ as the actual world, and we say truth in this restricted sense is truth at @, and validity is truth preservation at @ for all interpretations. Then, we can add constraints in @ to model the properties we think truth should have in the actual real world. For example, we could add exhaustion and exclusion constraints to eliminate truth gaps and gluts (see section 8.4.6 and section 8.4.9).]

One final comment: one might hold that truth – real truth, not just truth in some world – has some special properties; that unlike truth in an arbitrary world, truth itself can have no gaps or gluts. To accommodate this view, one could take an interpretation to include a distinguished normal world, @ (for actuality), such that truth (simpliciter) is truth at @. Validity would then be defined as truth preservation at @ in all interpretations.8 The special properties of truth would be reflected in semantic constraints on @. Thus, if it be held that there are no truth value gluts in @, one would impose the constraint that ρ@ satisfy the condition Exclusion of 8.4.6. If it be held that there are no truth-value gaps in @, then one would impose the constraint that ρ@ satisfy the condition Exhaustion of 8.4.9.9 Or in a ∗ interpretation, one might require that @ = @∗, which rules out gaps and gluts. But from the present | perspectives, these conditions would require justification by some novel considerations.

(174-175)

8 One could, in fact, set up all the possible-world semantics that we have had till now in this way. But since these semantics contain nothing to distinguish @ from any other normal world, this would have had no effect on validity.

9 Strictly speaking, these conditions are not sufficient. To rule out truth-value gluts and gaps with formulas containing → s, we need to make another change as well. Specifically, to rule out truth-value gaps, the falsity conditions for A B at @ have to read:

A Bρ@0 iff (for some w′, Aρw1 and Bρw 0) or (it is not the case that A Bρw@1) and to rule out truth-value gluts, they have to read:

A Bρ@0 iff (for some w′, Aρw1 and Bρw 0) and (it is not the case that A Bρw@1).

(174)

[contents]

 

 

 

 

 

 

 

From:

 

Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.

 

 

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