Shores. Jc Beall’s Current and Potential Impact on the Continental Philosophy of Non-Classical Logics (Author Manuscript)

by Corry Shores

 

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In accordance with the archiving and open access policies of Springer Nature, I am making a PDF of the Author Manuscript (AM) available here on my personal website.

This version of the article has been accepted for publication, after peer review (when applicable) and is subject to Springer Nature’s AM terms of use, but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at:

http://dx.doi.org/10.1007/s44204-023-00071-5

To access the final published version, please reach the publisher using that link or contact the author at corryshores@gmail.com or through Research Gate.

 

 

Corry Shores

Jc Beall’s Current and Potential Impact on the Continental Philosophy of Non-Classical Logics

 

PDF LINK

 

 

 

 

Shores, Corry. “Jc Beall’s Current and Potential Impact on the Continental Philosophy of Non-Classical Logics.” Asian Journal of Philosophy 2, no. 1 (2023): 1–12. doi:10.1007/s44204-023-00071-5.

https://link.springer.com/article/10.1007/s44204-023-00071-5

http://dx.doi.org/10.1007/s44204-023-00071-5

 

Research Gate link:

https://www.researchgate.net/publication/370444805_Jc_Beall%27s_current_and_potential_impact_on_the_continental_philosophy_of_non-classical_logics_in_the_Asian_Journal_of_Philosophy

Shores. Logic of Gilles Deleuze: Basic Principles. Announcement and Preview

 

by Corry Shores

 

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Announcement and Preview of

 

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The Logic of Gilles Deleuze:

Basic Principles

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My book on Deleuze’s logic is now in press. A preview of the table of contents, acknowledgments, and introduction is available here:

https://www.academia.edu/44372079/The_Logic_of_Gilles_Deleuze_Basic_Principles

 

The publisher offers a preview of the first chapter here:

https://bloomsburycp3.codemantra.com/viewer/5f4e5a6fdc0e82000176fab1

 

 

Here is the publisher’s webpage for the book:

https://www.bloomsbury.com/us/the-logic-of-gilles-deleuze-9781350062252/

 

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The Logic of Gilles Deleuze: Basic Principles (Bloomsbury Studies in Continental Philosophy)

 

 

 

I thank a number of people in the acknowledgements (also see below). But here on this blog post I want to especially thank readers of this blog who have helped me on the book and supported me throughout the process, including Clifford Duffy, Terrance Blake, and Scott Wollschleger.

 

 

 

Full Acknowledgements:

This book was first made possible by Roland Breeur, who recommended me to the person who became one of my main editors, Liza Thompson. Much of what I know about philosophy and how it should be conducted, I learned from Prof. Breeur. And Liza, along with my other editors, Frankie Mace and Lucy Russell, have extended to me an incredible amount of generosity with the scheduling for the book. It never would have made it without their help, so I thank you all very much.

The basic content of the book was first made possible by the participants and organizers of the 2014 Paraconsistent Reasoning in Science and Mathematics conference at Ludwig Maximilian University: Peter Verdée, Holger Andreas, David Ripley, Graham Priest, Diderik Batens, Fenner Tanswell, Marcos Silva, Bryson Brown, Hitoshi Omori, Heinrich Wansing, Andreas Kapsner, Cian Chartier, Franz Berto, Itala Maria Loffredo D’Ottaviano, Zach Weber, João Marcos, Luis Estrada-González, Nick Thomas, Maarten McKubre-Jordens, Maria Martinez, Diego Tajer, and Otávio Bueno. They graciously allowed me to present, despite being quite incapable with logic, and they afterward did much to help me begin my project. Peter Verdée and Holger Andreas edited an edition of the proceedings for Springer, and they were kind enough to include my paper in it, the text of which is partly used here. I thank everyone for getting me started in non-classical logics, which still I love to this day.

I could not have written this book without the enduring, loving support of my wife, Gülben Salman. Her sacrifices and efforts are the reason I was able to do all the work necessary here. As a philosopher herself, she also made substantial contributions throughout the whole compositional process, and I cannot thank her enough. Gülben, I dedicate this book to you. I also thank Yasin Ceylan, Aziz Fevzi Zambak, Deniz Yılmaz Zambak, Aret Karademir, Hikmet Ünlü, Bolkar Özkan, Scott Wollschleger, Kurt Ozment, Samet Bağçe, Karen Vanhercke, Vykintas Baltakas, along with my family, Patricia, Ebbie Victor, Fatma, Hasan, Ebbie Paul, Brandon, Aimee, Mandy, Austin, and Joseph for the companionship, support, and advice they gave me all throughout.

Certain parts specifically benefited from help I received from other scholars. Oğuz Akçelik reviewed the logic parts (and any mistakes are mine). Many of the cinema parts (Chapters 4, 7, 8) were made possible by the guidance and teaching of Ahmet Gürata. The section on Plato in Chapter 8 was improved with Hikmet Ünlü’s expert assistance, and his instruction in Ancient Greek proved indispensable for working through the Stoic material in Chapter 5. Dorothea Olkowski taught me about intuitionism and its importance in Deleuze’s philosophy, so all of Chapter 6 was made possible by her writings and comments, and also she reviewed and made suggestions on most of Chapter 5. Roland Breeur’s work on imposture influenced much of what I write on the Falsifier in Chapter 8, and he reviewed and made suggestions for both Chapters 7 and 8. Along the way, I also received help with interpretation, sourcing, and translation from Antoine Dolcerocca, Terence Blake, Clifford Duffy, Roger Vergauwen, Julie Van der Wielen, Griet Galle, Iain McKenzie, Guillaume Collet, and Steven Spileers. Meriç Aytekin contributed much to the sourcing in Chapter 2, and Çi̇si̇l Vardar, to Chapter 1. At the beginning stages, my project benefitted from the comments provided by anonymous referees and from Ronald Bogue. I am very grateful to them. And I have taken great inspiration from the work of Jeffrey Bell, who has pioneered this particular field of study and whose advice I deeply appreciate. I am also heavily indebted to the archivists, transcribers, and translators (listed in the bibliography, but let me here mention Richard Pinhas) who have made Deleuze’s courses accessible. I thank everyone mentioned here so very much.

And many of the logic parts were improved through my correspondences and conversations with Graham Priest. His philosophy is the original inspiration for this book, and he has been nothing but the most generous and supportive toward this project. I thank him for patiently and thoroughly answering all of my questions about his writings and ideas. The philosophical world is so much better because of him, and I will always be deeply grateful.

I also could not have completed this book without the support and understanding of my colleagues at the Middle East Technical University: Halil Turan, Barış Parkan, Murat Baç, David Grünberg, Ayhan Sol, Samet Bağçe, Elif Çırakman, Mehmet Hilmi Demir, Aziz Fevzi Zambak, Fulden İbrahimhakkıoğlu, Yasin Ceylan, Teo Grünberg, Ahmet İnam, Ertuğrul Rufayi Turan, Refik Güremen, James Griffith, Selma Aydın Bayram, Dilek Başar Başkaya, Ercan Erkul, Gülizar Karahan Balya, Hikmet Ünlü, Erdinç Sayan, and Tahir Kocayiğit. (Ayhan Sol helped me especially with freeing up my scheduling for more time to write.)

Many students in my classes and seminars have contributed ideas and insights to this book, including: Bolkar Özkan, Gürkan Kılınç, Ilgın Aksoy, Yıldırım Bayazit, Faik Tekin Asal, Ekin Demirors, Hazal Babur, Tanayça Ünlütürk, Aybüke Aşkar, Meli̇ke Başak Yalçın, Ulaş Murat Altay, Sedef Beşkardeşler, Toprak Seda Karaosmanoğlu, İlkyaz Taşdemir, Çınar Uysal, Handan Ağirman, Tunahan Akbulut, Yasemin Karabaş, Aybüke Aşkar, Mahsasadat Shojaei, Umut Kesi̇kkulak, Ayşe Pekdiker, Seyran Sam Kookiaei, Atakan Botasun, Esra Saçlı, Firuza Rahimova, Sona Mustafayeva, İrem Kayra Özdemir, Erkan Özmacun, Ezel Ortaç, Rada Nur Ergen, and Yiğit Baysal. I thank all of you for your interest in these topics, for your original philosophical thinking, and for helping me interpret the texts.

And finally, I thank the following publishers and journals who granted me permission to reprint texts and figures (and additionally, I thank their blind referees, who helped me improve the articles):

Tijdschrift voor Filosofie / Peeters Publishers. (“The Primacy of Falsity: Deviant Origins in Deleuze.” TijdschriftVoorFilosofie 81 (2019): 81–130).

Routledge. (“Affirmations of the False and Bifurcations of the True: Deleuze’s Dialetheic and Stoic Fatalism.” In Deleuze and Guattari’s Philosophy of Freedom: Freedom’s Refrains, edited by Dorothea Olkowski and EftichisPirovolakis, 178–223. New York: Routledge, 2019.)

Springer. (“Dialetheism in the Structure of Phenomenal Time.” In Logical Studies of Paraconsistent Reasoning in Science and Mathematics, edited by Holger Andreas and Peter Verdée, 145-157. Cham, Switzerland: Springer, 2016.)

Deleuze and Guattari Studies / Edinburgh University Press. (“In the Still of the Moment: Deleuze’s Phenomena of Motionless Time.” Deleuze Studies 8, no. 2 (2014): 199–229.)

 

 

Shores, Corry. The Logic of Gilles Deleuze: Basic Principles. London: Bloomsbury, 2020.

[Publisher’s book-webpage]

 

 

 

 

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Shores. Logic of Gilles Deleuze, 1: Basic Principles, entry directory

 

by Corry Shores

 

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Entry Directory for

 

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The Logic of Gilles Deleuze:

Basic Principles

[Publisher’s book-webpage]

 

 

 

 

 

Announcement and Preview

 

 

 

 

 

 

 

 

 

 

 

 

Shores, Corry. The Logic of Gilles Deleuze: Basic Principles. London: Bloomsbury, 2020.

[Publisher’s book-webpage]

 

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Priest (CBS) “Dialectic and Dialetheic,” collected brief summaries

 

by Corry Shores

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Collected Brief Summaries for

 

Graham Priest

 

“Dialectic and Dialetheic”

 

 

Introduction:

Dialectics Requires Dialetheism

 

Priest will argue that Hegel’s and Marx’s dialectics were based on dialetheia, that is, on true contradiction.

 

 

1

Why It Is Necessary to Argue This

 

Many scholars argue that Marx’s and Hegel’s dialectics involve a non-logical notion of contradiction or that contradiction is conceptual and does not obtain in reality. Priest, however, will argue that the logical sense of contradiction is fundamental to their philosophies of dialectic.

 

 

 

2

The Argument Against this Interpretation

 

The main argument against reading Hegel and Marx as dialetheists is that it goes against the basic restriction of classical logic that you cannot have contradictions. But this restriction is based on an assumption and is thus not a necessary one.

 

 

 

3

Dialetheic Logic

 

Dialetheic logic is just like orthodox logic except that it allows for true contradictions, and when there are true contradictions, we cannot infer from them any other proposition we want.

 

 

 

4

Motion: An Illustration

 

One way we can illustrate how dialetheic logic can apply to dialectics is by accounting for motion in a Hegelian way. An object in motion is at a certain point at a certain instant, but since it is in motion, in that instant it is already leaving that point. Thus it is both true and false that the object is at that point in that instant.

 

 

 

5

The History of Hegel’s Dialectic

 

If we look at three of Hegel’s influences – Neo-Platonists, Kant, and Fichte – we see that Hegel borrowed self-contradictory ideas from each of them. Thus Hegel is a dialetheist, that is, he believes that true contradictions exist.

 

 

 

6

Contradiction in Hegel’s Dialectic

 

In Hegel’s dialectical movement, contradictory categories result from one another and are conjoined. It is in this ways that Hegel is a dialetheist [someone who thinks that there exist true contradictions].

 

 

 

7

Contradiction in Marx’s Dialectic

 

 

 

8

Identity in Difference

 

Hegel’s dialectic takes the form of identity in difference, formulable as (a=b)&(a≠b). This is a variation on the dialetheic formulation A&~A.

 

 

 

9

Dialectics and Epistemology

 

 

 

10

Conclusion

 

 

 

 

 

 

Priest, Graham. “Dialectic and Dialetheic.” Science & Society 53, no. 4 (1990): 388–415.

 

 

 

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Priest (CBS) Logic: A Very Short Introduction, collected brief summaries

 

by Corry Shores

 

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[The following collects the brief summaries for Priest’s book. The directory of entries without the summaries is found here:

http://piratesandrevolutionaries.blogspot.com/2015/07/entry-directory-priest-logic-very-short.html

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Collected Brief Summaries for:

 

Graham Priest

 

Logic: A Very Short Introduction

 

 

Preface

 

Logic is an ancient discipline that was revolutionized in the 20th century with mathematical techniques and is currently very useful in information and computational sciences. This book will give a brief, broad, and non-technical overview.

 

 

Ch.1

Validity: What Follows from What?

 

“Logic is the study of what counts as a good reason for what, and why” (Priest, 1). An inference draws a conclusion from premisses (or from a premiss). It is valid if the conclusion follows from those premisses. It is deductively valid if it necessarily follows, that is, if no other conclusion could possibly follow, and it can be determined as such when “there is no situation in which all the premisses are true, but the conclusion is not.” An inductively valid inference is based on reasoning given in the premisses, yet other conclusions could also follow instead.

 

Ch.2

Truth Functions – Or Not?

 

Our intuitions about the validity of inferences are often correct, but sometimes they are misleading. One such case is the inference: q, ¬q / p, for example, “The Queen is rich,” “The Queen is not rich,” therefore “Pigs can fly”. Since the conclusion seems logically unrelated, we might erroneously think it is an invalid inference. By rendering these sentences into symbols and computing their truth values, we can see that there is no instance when the premisses are true and the conclusion not-true (false), and thus indeed it is valid. But since there is no situation where both the premisses can be true anyway, it is called vacuously valid. We also learn the truth tables for negation, disjunction, and conjunction, which are based on the truth conditions for these operations. If a sentence is true, then its negation is false, and vice versa. A disjunction is true only if at least one disjunct is true. And a conjunction is true only if both conjuncts are true. But conjunctions and disjunctions in English do not always map perfectly onto these truth tables.

 

 

 

Ch.3

Names and Quantifiers: Is Nothing Something?

 

When we speak of things, we might refer to some specific thing by name, like if we say, “Marcus came to the party”. In this case, what we are saying refers just to this one named person or thing. Or we might speak broadly and universally of all of a group of things, like if we said, “everyone came to the party”. In this case, what we say of the people or things applies to all of them. Or, we might refer to some thing, but without designating it specifically with a name, like when we say, “Someone came to the party”. Here we are saying something about a person or thing, but we are not specifying which one. When we want to speak of some thing or another, as in, “someone is happy,” we could use the existential quantifier and formulate this as, ∃x xH, meaning, there is some x such that x is happy. Or if we wanted to say, “Everyone is happy,” we could write ∀x xH, meaning, for all x, x is happy. Note that from just one quantified sentence an inference can be drawn. For example, if all people are happy, then there is some person who is happy. By using quantification, we can settle debates in mathematics and philosophy.

 

 

Ch.4

Descriptions and Existence: Did the Greeks Worship Zeus?

 

A definite description specifies a thing satisfying certain conditions, for example, “the man who first landed on the Moon”. Descriptions can be formulated symbolically by the use of variables that are predicated. The overall formulation takes the form ιxc_x. Here, the ιx means, “the object x, such that…”, and the c_x gives the conditions specifying the object. In our example we could write ιx(xM & xF) to mean, “the object x such that x is a man and x first landed on the Moon”. Furthermore, we may treat the whole description as something that can take predicates, and we can use Greek letters to stand for the whole description, thus possibly making the above formulation simply μ. This abbreviation will help us examine the validity of the Characterization Principle (CP), which is used in the Ontological Argument for God. We describe God as having a variety of properties that specify God, with the final one being “exists”: ιx(xP₁ & … & xP_n). The CP says that a thing characterized by certain properties in fact has those properties, and thus the whole described thing is predicated by the properties given in the description. Symbolically this involves substituting all cases of x in the description with that description itself. In this formulation we would get: ιx((xP₁ & … & xP_n)P₁ & … & (xP₁ & … & xP_n)P_n), which in part says that the object that is omniscient etc., and exists, is in fact omniscient, etc., and does really exist. Using the Greek letters we can render the above substitution as: γP₁ & … & γP_n. But there is an important rule this argument breaks, namely that any predication to a non-existing entity is false. If there is a God, then the predication that God exists is true; but if there is no God in reality, then this predication is false. This means that for the argument to work, it must assume the truth of its conclusion at the outset, and is thus invalid. Yet there are cases where this rule does not apply, for example in instances of fictional entities like Greek gods whose properties can rightly be predicated to their description even though the thing described does not exist.

 

 

Ch.5

Self Reference: What is this Chapter About?

 

Paradoxical and otherwise problematic instances of self-reference lead us to suspect that we have more options than the following two: 1) a sentence can be just true, or 2) a sentence can be just false. Consider the “liar” sentence, ‘This sentence is false.’ If it is true, then it is false; but if it is false, then it is true. Either way, it’s truth-value will contradict what it says its truth-value is. So we have option 3) a sentence can be both true and false. Or consider the “liar cousin” sentence, ‘This sentence is true.’ Normally the terms in such a declarative sentence refer to things or situations by which we may determine the truth or falsity of the statement, that is to say, whether or not the indicated situation holds in reality or not. So if we say, “this chair is red,” we look to the indicated chair and its color, and we determine if the sentence is true or not. However, the terms in “this sentence is true” does not point us to such a determining situation, since we are only able to make two equally viable assumptions about its truth value, namely, that it is either true or that it is false; but, we have no way to make the determination one way or another, since it will always be consistent with what it says of itself under both assumptions. It would seem that we have no grounds that would allow us to determine whether it is true or false, and thus we have option 4) a sentence may be neither true nor false. The classical assumptions 1 and 2 lead us to conclude certain inferences are valid when our intuitions say otherwise. For example, “The Queen is rich,” “The Queen isn’t rich,” therefore, “Pigs can fly” (q, ¬q/p). Our intuitions tell us this seems invalid. But by just using assumptions 1 and 2, it is valid, since structurally speaking there is no situation where the premises are true and the conclusion is false. For, the premises can never all be true anyway. However, under the new assumptions, particularly that sentences can be both true and false, q, ¬q/p can be valid, if q is both true and false and p just false. For, q is at least true and ¬q is also at least true. However, our intuitions tell us that q∨p, ¬q/p is valid, but the new assumptions deem it invalid. Yet, perhaps it only seems intuitively valid if we forget that there are exceptional situations where sentences can be both true and false. There are other problems with the assumptions. When we assume that the liar cousin, “This sentence is true,” is neither true nor false, that means it cannot be true, but it says of itself that it is true. And while we might go along with saying that “This sentence is false” is both true and false, we might not feel the same way about “This sentence is not-true”. Here, we might conclude that it is both true and not-true (and not just true and false), which is a stronger contradiction that we may not want to accept.

 

 

Ch.6

Necessity and Possibility: What Will be Must be?

 

We can modify a statement of fact to indicate whether or not the referenced state of affairs is possibly the case or necessarily so. Modal logic allows us to deal with these modifications formally. Suppose “it will rain” is p. We write, “Possibly it will rain” as ⋄p, and we write “necessarily it will rain” as ◻p. Unlike truth-functional operators (like negation and conjunction), these modal operators do not alter the truth values of statements in a mechanically consistent way. To formally examine modally modified sentences, we think of there being other possible worlds about which we may make the same statements of fact, and these statements may be true or false depending on which alternate possible world it is in. In one possible world, it does rain tomorrow. But in another, it will not. We say something is possible when in at least one other world this state of affairs is false. However, no matter what possible world we conceive of, in all of them, if it rains, then fluid is falling. Such things which cannot be otherwise, when for example they are governed by fixed laws of physics, are considered necessarily true; for, in every other possible world they are true. We can diagram these possible world situations using boxes. In one box we give the statements of fact and their truth values for one situation or world (this world for example), and in other boxes we give the statements and their values for the other possible worlds. This helps us see which statements are necessarily true or false in one world and which are possibly so. This manner of formulation helps with certain debates, for example, it allows us to see that Aristotle’s argument for fatalism is fallacious. The argument makes us think that there is nothing we can do now to change the future, and also, that there is nothing in the past that we can regret or feel responsible for. The reasoning is as follows. If it is true that something will happen, then it will happen no matter what. But if it is false that something will happen, it will fail to happen no matter what. Either way, whatever happens occurs no matter what. By formulating this using modal logic, we see that it infers something incorrectly. There is a difference between the following two claims: 1) it is necessarily the case that if it is true that tomorrow I will get in an accident, then I will get in an accident, and 2) if it is true that if I will get in an accident, then I will necessarily get in an accident. If we just look at the semantic references, both formulations seem to have the same meaning. But on the level of their logical structure they are making different claims, and also structurally the second claim cannot be derived from the first, which is what is needed for the argument to hold. Aristotle’s fatalist argument would want you to believe that in every possible world you will get in an accident tomorrow, which is not so. It even acknowledges that the opposite could happen. However, there is a way to twist this fatalist argument a bit to remove that fallacy, and we may wonder whether or not this modification provides a valid argument for fatalism. We first say that there is nothing we can do now to change the past. This implies that states of affairs in the past are irrevocably true and statements about those situations are necessarily true. Now, suppose we do get in an accident tomorrow. This means it is true now if we say that we will. Suppose further that we said it yesterday also. We can say now that in the past it was true that we will get in an accident tomorrow. This means that it is irrevocably true that in the past we will get into an accident, and thus it is necessarily true that we will.

 

 

Ch.7

Conditionals: What’s in an If?

 

Conditionals are of the form, “if a then c,” or a→c. The first term is the antecedent, and the second, the consequent. Conditionals are false only if the antecedent is true and the consequent false, and they are true for all other value assignments. But there are many difficulties regarding conditionals, and some of which call into question the universal applicability of these value-assignments. For example, according to the truth table for conditionals, when the antecedent is false, then the whole conditional is true, regardless of whether or not the consequent is true. This means that the following two conditionals should both be true: “If Italy is part of France, Rome is in France” and “If Italy is part of France, Beijing is in France”. But intuitively, the second one seems false. So conditionals are not truth-functional, since a lot depends on the meanings of the terms. In order to evaluate them, we can use possible worlds, like with modal operators: “the conditional a→c is true in some situation, s, just if c is true in every one of the possible situations associated with s in which a is true; and it is false in s if c is false in some possible situation associated with s in which a is true.” Since Rome is by definition in Italy, that means in no possible world would it not be in France, were Italy to be in France. So that is why the first sentence is true. However, since Beijing is by definition a city in China and not a city of Italy, then in some possible worlds Beijing will not be in France, were Italy to be in France. And that is why the second sentence is false. Another problem with conditionals has to do with ¬(a&¬c), which has the same truth table as a→c, and in fact is called the material conditional and is symbolized as a⊃c. But although we might think that we can infer a→c from ¬(a&¬c), this is not in fact a valid inference, and we can show this using the possible worlds analysis. The important difference between a→c and ¬(a&¬c) is that a→c involves the relevance of a to c, where there is no such relevance implied in ¬(a&¬c). For this reason we can think of situations where a→c will be false but ¬(a&¬c) will technically true, thereby invalidating the inference. There are other cases too of inferences using conditionals that seem valid, and yet there are troubling counter-examples that call their validity into question.

 

 

Ch.8

The Future and the Past: Is Time Real?

 

We can use tense logic to analyze the validity of inferences that are based on statements referring to different moments in time. We first think of a one-dimensional series of situations arranged in their proper chronological sequence. We then think of statements of fact. They may or may not be true for one temporalized situation or another. Suppose a statement h is true only for the temporally situated moment s₀. This statement refers to an instantaneous state of affairs, like the moment the first bullet entered Czar Nicolas’ heart. It will be false for all situations coming before and after that temporalized situation, since the event did not happen at those other moments. However, at a succeeding moment in the future, we can say truly that the event happened in the past. And likewise for a preceding moment in the past, we can say it will be happening in the future. We use the modifier P for past (“it was the case that”) and F for future (“it will be the case that”). So in moment s₁, Ph is true, and for moment s_-1, Fh is true. We can further designate temporal relations by compounding the modifiers. PPh would apply h to a situation coming before some other situation that is already in the past. FPh would apply h then to some situation coming after some other situation that is already in the past. Now, P and F refer to some determinate situation in the past or future. We can instead refer to all future situations with the modifier G (“it is always Going to be the case that”) and all past ones with the modifier H (“it Has always been the case that”). We can also make a model  for this tense logic by arranging in sequence a number of s’s, placing s₀ in the middle, and counting up and down the subscripts on both sides. This allows us to evaluate inferences based on tense modifiers. One example is McTaggart’s argument against the reality of time. If time is real, then the past and future are real, and thus they do not present logical contradictions. We then consider a sentence that is true just for the situation at one time-point. This means it did not happen in two temporally distinct time-points, and thus it did not happen both in the past and in the future: ¬(Ph&Fh). However, time flows, and so before it happened, it was in the future, and after it happened, it was in the past: Ph&Fh. The concepts of past and future present a contradiction, and thus time is unreal. One may object to the second formulation and say that it pretends that, for one situation that is located at one time point, the event can be both in the past and in the future. So to clarify the problem, we might then compound the modifiers and write ¬(PPh&FFh) to mean that the event did not happen at some determinate point coming before another in the past and at the same time happen at some determinate point coming after another in the future. Those following McTaggart’s reasoning can then say that still, because of the flow of time, PPh will be true and FFh was true, and thus, in contradiction with the prior, negated conjunction, PPh&FFh. But, by using the tense-logic model, we can display visually that the McTaggart argument is mistaken. There is never a singular temporalized situation where both terms in the past&future parings are true. Nonetheless, as this is a model that spatializes the flow of time, it might not be adequate for dealing with this argument about time’s non-spatial flow.

 

 

Ch.9

Identity and Change. Is Anything Ever the Same?

 

Over time, something’s properties might change. But it might either keep its identity or it might take on another one altogether. This presents a difficulty for philosophy and logic, especially since identity is a foundational concept in our thinking. We first distinguish objects and their properties, and we note that the properties may be variable while the objects remain constant. The ‘is’ of predication (x is red, or Rx) is different from the ‘is’ of identity (x is y, or x=y). However, Leibniz’s Law [of indiscernibles] uses properties to define identity. If two things share the same properties, then they are identical, and vice versa. This is a useful law in most applications, as for example when we use it for substituting terms in algebra. There are some other instances that at first seem to cast doubts on the applicability of the law, but these cases can be shown in the end to be mistaken for other reasons. However, there is one case that presents a big problem for the Law. We assume that identical things always were and always will be identical. When an amoeba A splits into amoebae B and C, then A has transformed into two other things in the sense of it having taken on new guises. This means that before the split, B and C  were identical to A and thus were identical to each other. However, after the split they are non-identical. This contradicts the assumption that things that are identical always are so.

 

 

Ch.10

Vagueness: How Do You Stop Sliding Down a Slippery Slope?

 

A thing can change gradually over time. A true statement about that thing’s status at the beginning can later be false at the end of the development. But in many cases, it is not clear when exactly during that development the status changes without ambiguity. “Jack is a child” is true when Jack is very young and not true when Jack is old; but, when precisely in his young adult years does it cease being entirely true and instead “Jack is an adult” becomes entirely true? This issue is related to sorites paradoxes. Consider that “Jack is a child” is true at the beginning, and “If Jack is a child at the beginning, then he is still a child one second later” also is probably also true. That means by modus ponens, “Jack is a child one second later” is true. Using this same sort of reasoning, we can then conclude that Jack is a child two seconds later, and so on, meaning that he never ceases being a child. (We reiterate the structure, taking the affirmed prior conclusion that Jack is still a child in the  succeeding second, and use it as a premise in an argument of the same structure, allowing us to conclude he is a child in yet the next succeeding second, and so on infinitely).  One solution to these issues is to use fuzzy truth values. We can say for example that when he is 3 years old, the statement “Jack is a child” has a full truth value of 1. At 9 years “Jack is  child” has a truth value of 0.75. At 14 years, 0.5. At 19 years, 0.25. And at 24 years, 0. And when we apply truth functional operators to statements with  values between 1 and 0, we can determine the different resulting fuzzy values. Also, we can say that an inference is valid when both the conclusion and the premises meet a certain minimum level of truth value, which is determined by the actual context to which the statements apply. What we find then is that the sorites paradox does not hold when we use this fuzzy system. [For, in order for the modus ponens inference to work in all steps, we will need the minimum value to be 0 (in order to accommodate the final transitional step), which is too low to be meaningful.] Also, fuzzy values do not clear up the situation entirely, because we have the same problem when we need to determine precisely at what point the values change from 1 to something less than 1.

 

 

 

 

 

 

 

 

 

Priest, Graham. Logic: A Very Short Introduction. 1st ed. Oxford: Oxford University, 2000.

Priest (6.4) An Introduction to Non-Classical Logic, ‘Tableaux for Intuitionistic Logic,’ summary

 

by Corry Shores

 

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[The following is summary of Priest’s text, which is already written with maximum efficiency. Bracketed commentary and boldface are my own, unless otherwise noted. I do not have specialized training in this field, so please trust the original text over my summarization. I apologize for my typos and other unfortunate mistakes, because I have not finished proofreading, and I also have not finished learning all the basics of these logics.]

 

 

 

 

Summary of

 

Graham Priest

 

An Introduction to Non-Classical Logic: From If to Is

 

Part I:

Propositional Logic

 

6.

Intuitionistic Logic

 

6.4

Tableaux for Intuitionistic Logic

 

 

 

 

Brief summary:

(6.4.1) Our tableaux for intuitionistic logic will build from those for modal logic, but with some modifications. In modal logic, the nodes take one of two forms: {1} A,i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r). For our intuitionistic tableaux, “The first modification is that a node on the tableau is now of the form A,+i or A,−i. The first means, intuitively, that A is true at world i; the second means that A is false at i” (107) Previously we did not need this information in the tableaux about truth and falsity, because A’s being false in a world was equivalent to its negation being true, and so we would represent that with ¬A,i. (It seems then that being “false” or at least lacking a proof, here in intuitionistic systems, means either that {1} within some world, there is a disproof (a proof that there is no proof)  for a formula, in other words, that for instance ⇁A in world 0 (maybe written as v_wo (⇁A) = 1 and) symbolized as ⇁A,+0 in the tableaux, which means that it is the case that there is a disproof for A in world 0, or that {2} there is currently neither a proof for a formula nor a disproof for that formula (I am not sure how that is written normally, maybe for instance as v_wo (A) = 0, but it is) written as A,−0 in the tableaux, meaning that it is not the case that there is a proof for A in world 0. (If there were a disproof, and thus if  v_wo (⇁A) = 1, then I think still you would thereby have v_wo (A) = 0).) (6.4.2) We form the initial list of our tableaux by setting all premises to true in world 0, thus as: B,+0. And the conclusion is set to false in world 0, thus as: A,−0. (6.4.3) We close a branch on our tableau when we obtain a contradiction, that is, “just when we have nodes of the form A,+i and A,−i. (6.4.4) Priest then gives the tableaux rules (see below. The list includes the accessibility rules from the next section also).

 

Conjunction Development, True (∧D,+)

A ∧ B,+i ↓ A,+i B,+i

 

Conjunction Development, False (∧D,−)

A ∧ B,−i ↙   ↘ A,−i      B,−i

 

Disjunction Development, True (∨D,+)

A ∨ B,+i ↙   ↘ A,+i      B,+i

 

Disjunction Development, False (∨D,−)

A ∨ B,−i ↓ ¬A,−i ↓ ¬B,−i

 

Conditional Development, True (⊐D,+)

A ⊐ B,+i irj ↙   ↘ A,−j        B,+j applied for every j on the branch

 

Conditional Development, False (⊐D,−)

A ⊐ B,−i ↓ irj A,+j ↓ B,−j the j is new

 

Negation Development, True (⇁D,+)

⇁A,+i irj ↓ A,−j applied for every j on the branch

 

Negation Development, False (⇁D,−)

⇁A,−i ↓ irj A,+j the j is new

 

Heredity, True  (hD,+)

p,+i irj ↓ p,+j . p is any propositional parameter, applied to every j (distinct from i)

(modified from p.108, section 6.4.4)

 

ρ, Reflexivity (ρrD)

ρ . ↓ iri

 

τ, Transitivity (τrD)

τ irj jrk ↓ irk

 

Priest has us “Note that, in particular, we can never ‘tick off’ any node of the form A ⊐ B,+i or ⇁A,+i, since we may have to come back and reapply the rule if anything of the form irj turns up” (108-109). (6.4.5) We also have the ρ reflexivity and τ transitivity accessibility rules (shown in the listing above). (6.4.6) Priest next gives an example tableau to show that ⊢_I p ⊐ ⇁⇁p. (6.4.7) Priest next gives another example tableau that shows that p ⊐ q ⊬_I ⇁p ∨ q. (6.4.8) “Counter-models are read off from an open branch of a tableau in a natural way. The worlds and accessibility relation are as the branch of the tableau specifies. If a node of the form p,+i occurs on the branch, p is set to true at w_i ; otherwise, p is false at w_i . (In particular, if a node of the form p,−i occurs on the branch, p is set to false at w_i )” (110). (6.4.9) Priest then gives a more visual portrayal of the counter-model from above section 6.4.8. Here “We indicate the fact that p is true (at a world) by +p, and the fact that it is false by −p” (110). (6.4.10) “The tableaux are sound and complete with respect to the semantics” (111). (6.4.11) Priest then gives an example of an infinite open tableau. (6.4.12) Priest then shows how it is easier to directly make a counter-model in cases of infinite tableaux.

 

 

 

 

 

 

 

 

 

Contents

 

6.4.1

[The Node Formulation in Intuitionistic Tableaux]

 

6.4.2

[The Initial List]

 

6.4.3

[Branch Closure]

 

6.4.4

[The Tableaux Rules]

 

6.4.5

[The ρ Reflexivity and τ Transitivity Accessibility Rules]

 

6.4.6

[Tableau Example 1: Valid]

 

6.4.7

[Tableau Example 2: Invalid]

 

6.4.8

[Counter-Models Explained]

 

6.4.9

[Counter-Model Illustrations]

 

6.4.10

[The Soundness and Completeness of the Tableaux]

 

6.4.11

[Infinite Open Tableaux]

 

6.4.12

[Counter-Models for Infinite Tableaux]

 

 

 

 

 

 

 

 

Summary

 

6.4.1

[The Node Formulation in Intuitionistic Tableaux]

 

[Our tableaux for intuitionistic logic will build from those for modal logic, but with some modifications. In modal logic, the nodes take one of two forms: {1} A,i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r). For our intuitionistic tableaux, “The first modification is that a node on the tableau is now of the form A,+i or A,−i. The first means, intuitively, that A is true at world i; the second means that A is false at i” (107) Previously we did not need this information in the tableaux about truth and falsity, because A’s being false in a world was equivalent to its negation being true, and so we would represent that with ¬A,i. (It seems then that being “false” or at least lacking a proof, here in intuitionistic systems, means either that {1} within some world, there is a disproof (a proof that there is no proof)  for a formula, in other words, that for instance ⇁A in world 0 (maybe written as v_wo (⇁A) = 1 and) symbolized as ⇁A,+0 in the tableaux, which means that it is the case that there is a disproof for A in world 0, or that {2} there is currently neither a proof for a formula nor a disproof for that formula (I am not sure how that is written normally, maybe for instance as v_wo (A) = 0, but it is) written as A,−0 in the tableaux, meaning that it is not the case that there is a proof for A in world 0. (If there were a disproof, and thus if  v_wo (⇁A) = 1, then I think still you would thereby have v_wo (A) = 0).)]

 

[We will now learn how to construct tableaux for intuitionistic logic. Let us first review some relevant matters. In section 6.2, Priest discussed the basic motivations for intuitionistic logic. He first noted that we can understand the meaning of sentences we never heard of before (6.2.1). One explanation for this is compositionality, which says that “the meaning of a sentence is determined by the meanings of its parts, and of the grammatical construction which composes these” (p.103, section 6.2.2). If meaning is given by truth-conditions, then by compositionality the meaning of sentences built-up using connectives is based on the truth-functionality of the connectives (6.2.3). Truth itself according to a correspondence theory is the correspondence of what a formula says and the facts of an extra-linguistic reality. But what about mathematical formulas? Are there real, extra-linguistic mathematical objects? (6.2.4) Mathematical realists hold that there is an extra-linguistic reality corresponding to the truths of mathematical formulations like “2 + 3 = 5;” they think for example that there are “objectively existing mathematical objects, like 3 and 5.” Intuitionists however think rather that we should not apply the correspondence theory of truth to mathematical formulations (6.2.5). Intuitionism expresses a statement’s meaning on the basis of its proof conditions, which are the conditions under which the sentence is proved (6.2.6). The proof condition of a simple sentence is whatever we would take to be a sufficient proof, and those for complex sentences that are built up using connectives will be similar to the normal conditions only now using the notion of proof (note that ⇁ and ⊐ symbolize negation and the conditional):

A proof of A ∧ B is a pair comprising a proof of A and a proof of B.

A proof of A ∨ B is a proof of A or a proof of B.

A proof of ⇁A is a proof that there is no proof of A.

A proof of A ⊐ B is A construction that, given any proof of A, can be applied to give a proof of B.

(104, section 6.2.7)

Lastly, Priest noted that these proof conditions cannot validate excluded middle, because there are formulas that cannot be proved nor can it be proven that there is no proof for them (6.2.8). Then in section 6.3, Priest outlined the semantics for intuitionistic logic. He first notes that there is a possible worlds semantics that “arguably captures the above ideas” (6.3.1). The only connectives in our intuitionist logic are ∧, ∨, ⇁ and ⊐ (with the last two being negation and the conditional, respectively) (6.3.2). Our intuitionistic possible worlds semantics takes the structure ⟨W, R, v⟩. It is mostly the same as logic K_ρτ, meaning that it is a normal modal logic in which the R accessibility relation is reflexive (all worlds have access to themselves) and transitive (whenever a first world has access to a second and that second to a third, then the first has access to that third as well.) There is one additional constraint, called the heredity condition, which means that when a proposition is true in one world, it it is true in all other worlds that are accessible from it (6.3.3). By means of certain rules we evaluate molecular formulas. Negation and the conditional involve accessible worlds (6.3.4). The heredity condition holds not just for propositional parameters but for all formulas (6.3.5).To see how the above interpretation captures intuitionist ideas, we first conceive of the way that information accumulates over time as being like one world (like our world at one moment) as being a set of proven things and another world accessible from the first having the same proven things and maybe more (like our world progressing later into a world perhaps with more information) (6.3.6). The possible world semantics for intuitionism captures the ideas in the proof conditions (6.3.7). We define validity in intuitionistic logic as truth preservation over all worlds of all interpretations, and we write intuitionistic logical consequence as ⊨_I (6.3.8). If there is only one world, the intuitionistic interpretation is equivalent to a classical one. And intuitionistic logic is a sub-logic of classical logic, because everything that is intuitionistically valid is classical valid, but not everything classical valid is intuitionistically valid (6.3.9). (We turn now to the current section).  Now, our intuitionistic tableaux will be a modification of normal modal logic tableaux (see section 2.4). Recall in particular from section 2.4.1 that tableaux in modal logic take the same branching node structure as those for propositional logic. However, the nodes themselves have a different structure, and there are two possible ones. {1} A, i, where A is a formula and i is a natural number indicating the world in which the formula holds, or {2} irj, where i is a natural number for a world that accesses world j, also given as a natural number (the r stays as r). Priest says now that for our intuitionistic tableaux, “The first modification is that a node on the tableau is now of the form A,+i or A,−i. The first means, intuitively, that A is true at world i; the second means that A is false at i.” Previously we did not need this information about truth and falsity, because A’s being false in a world was equivalent to its negation being true, and so we would represent that with ¬A. (I am not certain about the situation yet, but it seems to me now to be the following. Being “false” or at least lacking a proof, here means either that {1} within some world, there is a disproof (a proof that there is no proof)  for a formula, in other words, that for instance ⇁A in world 0 (maybe written as v_wo (⇁A) = 1 and) symbolized as ⇁A,+0 in the tableaux, which means that it is the case that there is a disproof for A in world 0, or that {2} there is currently neither a proof for a formula nor a disproof for that formula (I am not sure how that is written normally, maybe for instance as v_wo (A) = 0, but it is) written as A,−0 in the tableaux, meaning that it is not the case that there is a proof for A in world 0. (If there were a disproof, and thus if  v_wo (⇁A) = 1, then I think still you would thereby have v_wo (A) = 0).)]

To obtain tableaux for intuitionist logic, we modify those for normal modal logics. The first modification is that a node on the tableau is now of the form A,+i or A,−i. The first means, intuitively, that A is true at world i; the second means that A is false at i. For previous modal logics, the fact that A was false at a world was indicated by ¬A, i. But now, A may be false at a world without ⇁A being true there.

(107)

[contents]

 

 

 

 

 

 

6.4.2

[The Initial List]

 

[We form the initial list of our tableaux by setting all premises to true in world 0, thus as: B,+0. And the conclusion is set to false in world 0, thus as: A,−0.]

 

[ditto]

The initial list of a tableau for a given inference now comprises B,+0, for every premise, B, and A,−0, where A is the conclusion.

(107)

[contents]

 

 

 

 

 

 

6.4.3

[Branch Closure]

 

[We close a branch on our tableau when we obtain a contradiction, that is, “just when we have nodes of the form A,+i and A,−i.]

 

[ditto]

Closure of a branch occurs just when we have nodes of the form A,+i and A,−i.

(108)

[contents]

 

 

 

 

 

 

6.4.4

[The Tableaux Rules]

 

[Priest then gives the tableaux rules (see below). Priest has us “Note that, in particular, we can never ‘tick off’ any node of the form A ⊐ B,+i or ⇁A,+i, since we may have to come back and reapply the rule if anything of the form irj turns up” (108-109).]

 

[ditto]

The rules of the tableau for the connectives are as follows:

 

Conjunction Development, True (∧D,+)

A ∧ B,+i ↓ A,+i B,+i

 

Conjunction Development, False (∧D,−)

A ∧ B,−i ↙   ↘ A,−i      B,−i

 

Disjunction Development, True (∨D,+)

A ∨ B,+i ↙   ↘ A,+i      B,+i

 

Disjunction Development, False (∨D,−)

A ∨ B,−i ↓ ¬A,−i ↓ ¬B,−i

 

Conditional Development, True (⊐D,+)

A ⊐ B,+i irj ↙   ↘ A,−j        B,+j applied for every j on the branch

 

Conditional Development, False (⊐D,−)

A ⊐ B,−i ↓ irj A,+j ↓ B,−j the j is new

 

Negation Development, True (⇁D,+)

⇁A,+i irj ↓ A,−j applied for every j on the branch

 

Negation Development, False (⇁D,−)

⇁A,−i ↓ irj A,+j the j is new

 

Heredity, True  (hD,+)

p,+i irj ↓ p,+j . p is any propositional parameter, applied to every j (distinct from i)

(modified from p.108, section 6.4.4)

 

The rules for ∧ and ∨ are self-explanatory. The first rule for each of ⊐ and ⇁ is applied for every j on the branch. In the second, for each, the j is new. The rules are easier to remember if one recalls that A ⊐ B means, in effect, □(A ⊃ B), and ⇁A means, in effect, □¬A. Note that, in particular, we can never ‘tick off’ any node of the form A ⊐ B,+i or ⇁A,+i, since we may have to come back and reapply the rule if anything of the form irj turns up. The final rule is applied only to propositional parameters, and, again, to every j (distinct from i). The rule is required by the heredity condition, and we will refer to it as the heredity rule. Note that there is no corresponding rule for p, −i.

(108-109)

[contents]

 

 

 

 

 

 

6.4.5

[The ρ Reflexivity and τ Transitivity Accessibility Rules]

 

[We also have the ρ reflexivity and τ transitivity accessibility rules.]

 

ρ, Reflexivity (ρrD)

ρ . ↓ iri

 

τ, Transitivity (τrD)

τ irj jrk ↓ irk

]

 

[Recall from section 3.2.3 the constraints on the accessibility relation that generate variations of a modal logic:

ρ (rho), reflexivity: for all w, wRw.

σ (sigma), symmetry: for all w₁, w₂, if w₁Rw₂, then w₂Rw₁.

τ (tau), transitivity: for all w₁, w₂, w₃, if w₁Rw₂ and w₂Rw₃, then w₁Rw₃.

η (eta), extendability: for all w₁, there is a w₂ such that w₁Rw₂.

(p.36, section 3.2.3)

And recall the ρ, σ, and τ tableau rules from section 3.3.2.

Tableaux Rules for Kρ, Kσ, and Kτ
ρ . ↓ iri . . “ρrD”	σ irj ↓ jri . . “σrD”	τ irj jrk ↓ irk . “τrD”

(p.38, section 3.3.2 , with my naming additions)

Now recall from section 6.3.6 the idea that intuitionism can be understood in the following way. Successive times are worlds, but their relativities are such that there is heredity for affirmed information (so what is affirmed now is affirmed at all future worlds). But because of the “arrow of time,” something found true later does not mean it must be found true in the past. So the worlds do not have symmetrical access. And also note from the section above that the heredity rule is only for true statements. So something’s being false now does not mean it must be false in the future, event though if it is true now it must be true in the future.

Think of a world as a state of information at a certain time; intuitively, the things that hold at it are those things which are proved at this time. uRv is thought of as meaning that v is a possible extension of u, obtained by finding some number (possibly zero) of further proofs. Given this understanding, R is clearly reflexive and transitive. (For τ: any extension of an extension is an extension.) And the heredity condition is also intuitively correct. If something is proved, it stays proved, whatever else we prove.

(p.106, section 6.3.6)

Thus we do not have the symmetry σ rule, but just the ones for ρ (reflexivity) and τ  (transitivity).]

We also have the rules ρ and τ (of 3.3.2), as required for the reflexivity and transitivity of R.

(109)

[contents]

 

 

 

 

 

 

6.4.6

[Tableau Example 1: Valid]

 

[Priest next gives an example tableau to show that ⊢_I p ⊐ ⇁⇁p.]

 

[ditto]

As an example, here is a tableau to show that ⊢_I p ⊐ ⇁⇁p

 

⊢I p ⊐ ⇁⇁p
1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . 11. . 12.	p ⊐ ⇁⇁p,−0 ↓ 0r0 ↓ 0r1 ↓ p,+1 ↓ ⇁⇁p,−1 ↓ 1r1 ↓ 1r2 ↓ ⇁p,+2 ↓ 2r2 ↓ 0r2 ↓ p,−2 ↓ p,+2 ×	P . 1ρ . 1⊐− . 1⊐− . 1⊐− . 4ρ . 5⇁− . 5⇁− . 8ρ . 3,7τ . 8,9⇁+ .. 4,7h (12×11) valid

(enumeration and step accounting are my own and are probably mistaken)

 

(2)–(4) are obtained from (1) by the rule for false ⊐. (5) and (6) are obtained from (4) by the rule for false ⇁. (7) is obtained from (6) by the rule for true ⇁ (and the fact that 2r2). Finally, (8) is obtained from (3) by the heredity rule (and the fact that 1r2).⁴

(109)

4. Note a distinctive feature of intuitionist tableaux. Suppose that we had constructed the tableau using, not a propositional parameter, p, but an arbitrary formula, A. Then we could not apply the heredity rule to close off the tableau in the same way. But since anything of the form A ⊐⇁⇁A is logically true, and the tableau system is complete, tableaux for all such formulas will close, though not in a uniform way. (That is, for each sentence that A represents, the tableau will continue to closure in a different way.) This could be changed by making the heredity rule apply to all formulas, not just propositional parameters. And since heredity does hold for arbitrary formulas (6.3.5), this rule is sound. But this complicates tableaux enormously, and, by completeness, is unnecessary anyway.

(109)

[contents]

 

 

 

 

 

 

6.4.7

[Tableau Example 2: Invalid]

 

[Priest next gives another example tableau that shows that p ⊐ q ⊬_I ⇁p ∨ q.]

 

[ditto]

Here is another example to demonstrate that p ⊐ q ⊬_I ⇁p ∨ q. (Since the inference is classically valid – when ⊐ and ⇁ are replaced by ⊃ and |  ¬ – this shows that intuitionist logic is a proper sub-logic of classical logic.)

 

p ⊐ q ⊬I ⇁p ∨ q
1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10.	p ⊐ q,+0 ↓ ⇁p ∨ q,−0 ↓ 0r0 ↓ ⇁p,−0 ↓ q,−0 ↓ 0r1 ↓ p,+1 ↓ 1r1 ↙        ↘ p,−0         q,+0   ↙       ↘        ×      p,−1       q,+1           ×	P . P . 1ρ . 2∨− . 2∨− . 4⇁− . 4⇁− . 7ρ . 1,3⊐+ (9b×5) 2⊐+ (10a×7) open invalid

(enumeration and step accounting are my own and are probably mistaken)

 

The sixth and seventh lines are given by the rule for false ⇁, applied to the fourth line. Both splits are caused by an application of the rule for true ⊐ to the first line, to worlds 0 and 1, respectively. Note that there are no possible applications of the heredity rule.

(109-110)

 

[contents]

 

 

 

 

 

 

6.4.8

[Counter-Models Explained]

 

[“Counter-models are read off from an open branch of a tableau in a natural way. The worlds and accessibility relation are as the branch of the tableau specifies. If a node of the form p,+i occurs on the branch, p is set to true at w_i ; otherwise, p is false at w_i . (In particular, if a node of the form p,−i occurs on the branch, p is set to false at w_i )” (110).]

 

[ditto]

Counter-models are read off from an open branch of a tableau in a natural way. The worlds and accessibility relation are as the branch of the tableau specifies. If a node of the form p,+i occurs on the branch, p is set to true at w_i ; otherwise, p is false at w_i . (In particular, if a node of the form p,−i occurs on the branch, p is set to false at w_i .) Thus, reading from the open branch of the tableau of 6.4.7, W = {w₀ , w₁ }; w₀ Rw₀ , w₀Rw₁ and w₁Rw₁ ; v_wo (p) = v_wo (q) = 0 and v_w1 (p) = v_w1 (q) = 1.

(110)

[contents]

 

 

 

 

 

 

 

6.4.9

[Counter-Model Illustrations]

 

[Priest then gives a more visual portrayal of the counter-model from above section 6.4.8. Here “We indicate the fact that p is true (at a world) by +p, and the fact that it is false by −p” (110).]

 

[ditto]

In pictures:

xx↷xxxxx↷

xxw₀xx→xxw₁

xx−pxxxx +p

xx−q xxxx+q

 

We indicate the fact that p is true (at a world) by +p, and the fact that it is false by −p. It is a simple matter to check directly that the interpretation is a counter-model. At every world accessible from w₀ , p is false or q is true. | Hence, p ⊐ q is true at w₀ . p is true at w₁ ; hence ⇁p is false at w₀ . But q is also false there. Hence, ⇁p ∨ q is false there.

(110-111)

[contents]

 

 

 

 

 

 

6.4.10

[The Soundness and Completeness of the Tableaux]

 

[“The tableaux are sound and complete with respect to the semantics” (111).]

 

[ditto]

The tableaux are sound and complete with respect to the semantics. This is demonstrated in 6.7.

(111)

[contents]

 

 

 

 

 

 

6.4.11

[Infinite Open Tableaux]

 

[Priest then gives an example of an infinite open tableau.]

 

[Recall from section 6.4.4 above that “we can never ‘tick off’ any node of the form A ⊐ B,+i or ⇁A,+i, since we may have to come back and reapply the rule if anything of the form irj turns up” (p.108, section 6.4.4). Here we will see how by obtaining ⇁⇁p,+i in one line, that means we must derive ⇁p,–j in another line by the true negation rule and transitivity. But then by the false negation rule, we will get a new world related to the previous one, causing us to redo the ⇁⇁p,+i line in that new world, which then means the pattern continues without termination.]

Note that, as for Kρτ , open tableaux for intuitionist logic may be infinite. Here, for example, is the start of a tableau which establishes that ⊬_I ⇁⇁p ⊐ p :

 

⊬I ⇁⇁p ⊐ p
1. . 2. . 3. . 4. . 5. . 6. . 7. . 8. . 9. . 10. . 11. . 12.	⇁⇁p ⊐ p,−0 ↓ 0r0 ↓ 0r1 ↓ ⇁⇁p,+1 ↓ p,−1 ↓ 1r1 ↓ ⇁p,−1 ↓ 1r2 ↓ p,+2 ↓ 2r2 ↓ 0r2 ↓ ⇁p,−2 ↓ 2r3 ⫶	P . 1ρ . 1⊐− . 1⊐− . 1⊐− . 4ρ . 4⇁+ . 7⇁− . 7⇁− . 9ρ . 3,8τ . 4,8⇁+ .. 12⇁− open invalid

(enumeration and step accounting are my own and are probably mistaken)

 

Every time we open a new world, i, the fourth line (and transitivity) requires us to write ⇁p, −i there; but this requires us to open a new world, j, such that irj and p, +j, and so on.

(111)

[contents]

 

 

 

 

 

 

6.4.12

[Counter-Models for Infinite Tableaux]

 

[Priest then shows how it is easier to directly make a counter-model in cases of infinite tableaux.]

 

[ditto]

Again, as with K_ρτ , in such cases it is usually easier to con- struct counter-models directly. Thus, for ⇁⇁p ⊐ p, the following will work:

 

xx↷xxxxx↷

xxw₀xx→xxw₁

xx−pxxxx +p

 

Since p is true at w₁ , ⇁p is false at w₀ and w₁. Hence, ⇁⇁p is true at w₀ . Since p is false there, ⇁⇁p ⊐ p is false at w₀.

(111)

[contents]

 

 

 

 

 

 

From:

 

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