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

by Corry Shores

 

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

 

[Central Entry Directory]

[Corry Shores, entry directory]

 

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

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

 

by Corry Shores

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

 

[Central Entry Directory]

[Logic & Semantics, Entry Directory]

[Graham Priest, entry directory]

[Priest, “Dialectic and Dialetheic”, entry directory]

 

 

 

 

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.

 

 

 

.

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

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest’s, Logic: A Very Short Introduction, entry directory]

 

[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

]

 

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 (7.5) An Introduction to Non-Classical Logic, ‘Many-valued Logics and Conditionals,’ summary

 

by Corry Shores

 

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

 

[Central Entry Directory]

[Logic and Semantics, entry directory]

[Graham Priest, entry directory]

[Priest, Introduction to Non-Classical Logic, entry directory]

 

[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 II:

Quantification and Identity

 

7

Many-valued Logics

 

 

7.5

Many-valued Logics and Conditionals

 

 

 

 

Brief summary:

(7.5.1) We will now examine the conditional operator in many-valued logics. (7.5.2) We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table. (In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.)

 

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(7.5.3) Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional. (7.5.4) We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold. (7.5.5) Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated. (7.5.6) But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)

 

 

 

 

 

 

Contents

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

7.5.3

[Evaluating the Many-Valued Logics]

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

7.5.6

[Counter-Examples to This Claim]

 

 

 

 

 

 

Summary

 

7.5.1

[Turning to the Conditional in Many-Valued Logics]

 

[We will now examine the conditional operator in many-valued logics.]

 

[(ditto)]

Further details of the properties of ∧, ∨ and ¬ in the logics we have just met will emerge in the next chapter. For the present, let us concentrate on the conditional.

(125)

[contents]

 

 

 

 

 

 

7.5.2

[A Table of Problematic Conditional Inferences in Many-Valued Logics]

 

[We will assess whether or not some problematic inferences using conditionals are valid in K₃, Ł₃, LP₃, and RM₃, by making a table.]

 

[The issue of problematic inferences using the conditional is something we have explored quite a bit in previous sections. See sections 1.6, 1.7, 1.8, 1.9, 1.10, 4.5, 4.6, 4.8, 4.9, and 5.2. Priest now summarizes many of these problematic inferences that use the conditional in a table where we can see whether or not they are are valid in K₃, Ł₃, LP₃, and RM₃. Recall that part of this evaluation involves the designated value, which is the truth-preserving value (like 1 in classical logic and i and 1 in LP) and which is not the same in all of these systems. In the table below, a ‘✓’ means the inference or formula is valid in the given system, and an ‘×’ means it is not valid.]

In past chapters, we have met a number of problematic inferences concerning conditionals. The following table summarises whether or not they hold in the various logics we have looked at. (A tick means yes; a cross means no.)

|

		K3	Ł3	LP	RM3
1	q ⊨ p ⊃ q	✔	✔	✔	×
2	¬p ⊨ p ⊃ q	✔	✔	✔	×
3	(p ∧ q) ⊃ r ⊨ (p ⊃ r) ∨ (q ⊃ r)	✔	✔	✔	✔
4	(p ⊃ q) ∧ (r ⊃ s) ⊨ (p ⊃ s) ∨ (r ⊃ q)	✔	✔	✔	✔
5	¬(p ⊃ q) ⊨ p	✔	✔	✔	✔
6	p ⊃ r ⊨ (p ∧ q) ⊃ r	✔	✔	✔	✔
7	p ⊃ q, q ⊃ r ⊨ p ⊃ r	✔	✔	×	✔
8	p ⊃ q ⊨ ¬q ⊃ ¬p	✔	✔	✔	✔
9	⊨ p ⊃ (q ∨ ¬q)	×	×	✔	×
10	⊨ (p ∧ ¬p) ⊃ q	×	×	✔	×

 

(1) and (2) we met in 1.7, and (3)–(5) we met in 1.9, all in connection with the material conditional. (6)–(8) we met in 5.2, in connection with conditional logics. (9) and (10) we met in 4.6, in connection with the strict conditional. The checking of the details is left as a (quite lengthy) exercise. For K₃, a generally good strategy is to start by assuming that the premises take the value 1 (the only designated value), and recall that, in K₃, if a conditional takes the value 1, then either its antecedent takes the value 0 or the consequent takes the value 1. For L₃, it is similar, except that a conditional with value 1 may also have antecedent and consequent with value i. For LP, a generally good strategy is to start by assuming that the conclusion takes the value 0 (the only undesignated value), and recall that, in LP, if a conditional takes the value 0, then the antecedent takes the value 1 and the consequent takes the value 0. For RM₃, it is similar, except that if a conditional has value 0, the antecedent and consequent may also take the values 1 and i, or i and 0, respectively. And recall that classical inputs (1 or 0) always give the classical outputs.

(125-126)

[contents]

 

 

 

 

 

 

7.5.3

[Evaluating the Many-Valued Logics]

 

[Generally speaking, the many-valued logics still validate many of the problematic inferences using the conditional.]

 

[If we look at the table, we can see that there are many check-marks, which means that these problematic inferences are commonly valid in the many-valued logics. If if we ignore conditionals with an enthymematic ceteris paribus clause (see section 5.2), all these many-valued logics still “suffer from some of the same problems as the material conditional” (126). Priest then writes: “K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4) ...” (126-127). I am not sure I get that, but maybe the ideas are the following. One of the problems of the strict conditional is the explosion of contradictions, where contradictions entail everything (see especially section 4.8). But, as we can see from row 10, this is not valid in K₃ and Ł₃. Let us consider the evaluation of ‘⊨ (p ∧ ¬p) ⊃ q’ in K₃ and also ‘p ∧ ¬p ⊨ q’ in K₃, because I am guessing that is the distinction here. Recall from section 7.3.2 that the evaluation for negation, conjunction, and the conditional are the following.

 

f¬
1	0
i	i
0	1

 

f∧	1	i	0
1	1	i	0
i	i	i	0
0	0	0	0

 

f⊃	1	i	0
1	1	i	0
i	1	i	i
0	1	1	1

(122)

 

And the designated value is 1. So we need to see if there is any instance in the evaluation where the value of the whole conditional is not 1. If there is such an instance, it is invalid, and if there is no such instance, it is valid.

 

p  q	(p	∧	¬p)	⊃	q
1  1	1	0	0	1	1
1  i	1	0	0	1	i
1  0	1	0	0	1	0
i  1	i	i	i	1	1
i  i	i	i	i	i	i
i  0	i	i	i	i	0
0  1	0	0	1	1	1
0  i	0	0	1	1	i
0  0	0	0	1	1	0

 

As we can see, ‘⊨ (p ∧ ¬p) ⊃ q’ is not valid in K₃, because when p is i and q is either i or 0, then the conditional is i. Now let us evaluate p ∧ ¬p ⊨ q’ in K₃. If there are any cases where the premises are 1 and the conclusion is not 1, then it is valid. But note that if there are no cases where the premises are 1 to begin with, then it will still be valid, although “vacuously valid” (see Priest’s Logic: A Very Short Introduction, section 2.)

 

p  q	(p	∧	¬p)	⊨	q
1  1	1	0	0		1
1  i	1	0	0		i
1  0	1	0	0		0
i  1	i	i	i		1
i  i	i	i	i		i
i  0	i	i	i		0
0  1	0	0	1		1
0  i	0	0	1		i
0  0	0	0	1		0

 

As we can see, there are no cases where the premises are 1, so it is vacuously valid and thus explosion holds in K₃, and it also holds in Ł₃. But, since in LP, i is a designated value, and since when p is i and q is 0 the premises are i but the conclusion 0, that means it is not valid in LP (see section 7.4.4). Yet Priest continues, “but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows.” I am not sure why this formula shows that modus ponens is invalid. I will need to come back to this. But for now I will note that under the exact same evaluation there is the same row as the counter-example for modus ponens, as we saw in section 7.4.5:

 

p  q	p	p ⊃ q	⊨ q
1  1	1	1	1
1  i	1	i	i
1  0	1	0	0
i  1	i	1	1
i  i	i	i	i
i  0	i	i	0
0  1	0	1	1
0  i	0	1	i
0  0	0	1	0

 

In all, the system that has the fewest valid problematic inferences using the conditional is RM₃.]

As can be seen from the number of ticks, the conditionals do not fare very well. If one’s concern is with the ordinary conditional, and not with conditionals with an enthymematic ceteris paribus clause, then one may ignore lines (6)–(8). But all the logics suffer from some of the same problems as the material conditional. K₃ and Ł₃ also suffer from some of the problems that the strict conditional does. In particular, even though (10) tells us that (p ∧ ¬p) ⊃ q is not valid in these logics, contradictions still entail everything, since p ∧ ¬p can never assume a designated value. By contrast, this is not | true of LP (as we saw in 7.4.4), but this is so only because modus ponens is invalid, since (p ∧ ¬p) ⊃ q is valid, as (10) shows. (Modus ponens is valid for the other logics, as may easily be checked.) About the best of the bunch is RM₃.

(126-127)

[contents]

 

 

 

 

 

 

7.5.4

[Some Disjunction and Biconditional Rules for Many-Valued Logics to Show Why Many Problematic Conditional Inferences are Inevitable in Them]

 

[We have the intuitions that in finitely many-valued logics, the following two things should hold:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Only in K₃ does (ii) not hold.]

[Priest will now explain why the conditional in finitely many-valued logics will probably be problematic. In this section we make the first step, so the full reasoning will not here be given. But we need to note a few important things here. First consider how disjunction works. If we can affirm either just A on the one hand or alternatively just B on the other hand, then we should be able to affirm A ∨ B, because we know one of the disjuncts would be true, which is enough for the whole disjunction to be true. Thus:

(i) if A (or B) is designated, so is A ∨ B

(127)

The next observation is that A ≡ A should be a logical truth, even in a many-valued logic. Why? The reasoning is a little tricky, but it seems to be the following. Suppose A is valued at i. Let us make that concrete with an example. We have a moving object, and for A we have, “the moving object is at point 1 at time 1.” Suppose we think that should be both true and false. Regardless, would we not still think that the following should at least also be both true and false: “If the moving object is at point 1 at time 1, then the moving object is at point 1 at time 1”? So, we are claiming that it is reasonable to say that ‘if A then A’ has designated value whenever A itself has one. Now, since A is A, then that implies the inversion of the ‘if A then A’ also holds, and so A iff A should also hold as well. Now, in this case, A will have to have the same value as itself, so both sides of the biconditional will always have the same value, and regardless, the whole biconditional will have a designated value. The next idea seems to be that we suppose we have B, but we assign it the same value as A. And then we say that the same thing should hold for the biconditional here, namely, that:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

I may have the reasoning wrong there, but this is my guess. Priest ends by saying that this holds in all the finitely many-valued logics we have seen, but for K₃, (ii) fails. ]

But there are quite general reasons as to why the conditional of any finitely many-valued logic is bound to be problematic. For a start, if disjunction is to behave in a natural way, the inference from A (or B) to A ∨ B must be valid. Hence, we must have:

(i) if A (or B) is designated, so is A ∨ B

Also, A ≡ A ought to be a logical truth. (Even if A is neither true nor false, for example, it would still seem to be the case that if A then A, and so, that A iff A.) Hence:

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Note that both of these conditions hold for all the logics that we have looked at, with the exception of K₃, for which (ii) fails.

(127)

[contents]

 

 

 

 

 

 

7.5.5

[The Disjunction of all Biconditionals as a Logical Truth]

 

[Given these two rules, suppose we have a many-valued logic with one more formula than there are truth-values; that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

 

[The argumentation continues to get tricky. I may have the reasoning wrong, so see the quotation below. I am guessing we are doing the following. We are dealing with finitely many-valued logics. So that means regardless of the number of values, they will have a total number of values, that we can use the variable n to signify. So first we will consider any n-valued logic that satisfies

(i) if A (or B) is designated, so is A ∨ B

and

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Now we will consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Recall from section 1.2 that a propositional parameter is something like a formula which in our texts above we are writing as A, B, or C. The point is that we have one more formula than the total number of possible truth-values for that system. So at least two formulas will need to have the same truth-value. Now, given what (ii) says, that means for these two same-valued formulas, their biconditional must have a designated value. And, since we now have at least one designated biconditional, that would make it that the disjunction of all biconditionals will have to be designated, by rule (i). So in sum, suppose we have a many-valued logic with one more formula than there are truth-values, that means a disjunction of all of its biconditionals will need to be logically valid, because at least one of them will have to have both biconditional terms with the same value and thus be designated.]

Now, take any n-valued logic that satisfies (i) and (ii), and consider n+1 propositional parameters, p₁, p₂, . . . , p_n+1. Since there are only n truth values, in any interpretation, two of these must receive the same value. Hence, by (ii), for some j and k, p_j ≡ p_k must be designated. But then the disjunction of all biconditionals of this form must also be designated, by (i). Hence, this disjunction is logically valid.

[contents]

 

 

 

 

 

 

7.5.6

[Counter-Examples to This Claim]

 

[But there are counter-examples to this claim (and in these counter-examples, the intuitive sense of the sentences does not allow for any true biconditional combinations of two different sentences, even though technically they should evaluate as true). For instance, “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true” (127). (So suppose we have a three-valued logic, and John has 1 hair on his head. That means “John has 2 hairs on his head if and only if John has 3 hairs on his head” is true (or at least true, or ‘designated’ whatever way), on account of both sides being false, even though the intuitive sense of the formulation would make the biconditional false (or at least senseless); for, John’s having x number of hairs on his head should not be conditional on his having x ± 1 hairs on his head. Thus, finitely many-valued logics will always be potentially vulnerable to the following problem: because the disjunction of all biconditionals should be true, then at least one must be true, meaning that in the case of propositions like “John has x number of hairs”, there must be at least one true one that reads “John has x number of hairs on his head only if John has x + 1 number of hairs on his head.” But that is senseless even though it would be evaluated as true.)]

 

[So as we saw in section 7.5.5, if we accept the following two intuitive notions:

(i) if A (or B) is designated, so is A ∨ B

(ii) if A and B have the same value, A ≡ B must be designated (since A ≡ A is).

Then we should conclude that for any finitely many-valued logical system with n truth-values and n+1 propositional parameters, that the disjunction of all its biconditionals will necessarily be true. Priest then gives a counter-example. “Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’.” Priest says that any biconditional of these formulas would seem to be false, and thus the disjunction of all of them will not have any true one in it. I am not following this well, and I also do not see very how this shows us why the conditional for any finitely many-valued logic is bound to be problematic. Let me go through this as best I can. The idea originally was that in such a set of propositions, at least two would have to have the same value. So here, I would think that there would be two with the value false, and thus at least one biconditional that is true. Suppose we are using a three-valued logic with the values 0, 1, and i. And suppose John has 1 hair on his head. That means we have the following four propositions:

(1) John has 1 hair on his head.

(2) John has 2 hairs on his head.

(3) John has 3 hairs on his head.

(4) John has 4 hairs on his head.

Now, 3 and 4 are false, so their biconditional is true. So I do not see yet why “Any biconditional relating a pair of these would appear to be false.” But this is my failing. I am just not sure where I go wrong in the above explication. My best guess at the moment is that the problems come from the intuitive sense of these biconditional formulations: One of them will be, “John has 1 hair on his head if and only if John has 2 hairs on his head.” Now, as we know, this is senseless. John does in fact have 1 hair on his head, but this cannot be biconditional with him having 2 hairs on his head. By extension, even though technically “John has 3 hairs on his head if and only if John has 4 hairs on his head” is true, it is also for the same reason senseless. In other words, John’s not having some number of hairs on his head should not be biconditional on him having some other false number of hairs on his head. In other words, because every biconditional will biconditionally equate statements saying John has a different number of hairs on his head would seem on the level of its sense to necessarily always be false, even though technically they might evaluate as true whenever both sides of the biconditional are false. So on the level of sense, John’s having x number of hairs cannot be conditioned on him having x+1 number of hairs, and vice versa. That is my best guess at the moment. Still, even supposing this to be the case, I am not entirely sure why that would show there to be something problematic with conditionals in finitely many-valued logics. Is it because the biconditional is composed of conditionals, so if there is a paradox with the biconditionals there is a problem with conditionals in general? In other words, since the biconditionals imply that we will have  conditionals of the form “John has x number of hairs if John has x+1 number of hairs” where at least one biconditional combination of them will have to be true, even though we know that cannot be so in any case, given the intuitive sense of the constituent conditional sentences, that we will always have this problem with conditionals in finitely many-valued logics. Let me quote, as I am guessing very wildly here.]

But this seems entirely wrong. Consider n + 1 propositions such as ‘John has 1 hair on his head’, ‘John has 2 hairs on his head’, . . ., ‘John has n + 1 hairs on his head’. Any biconditional relating a pair of these would appear to be false. Hence, the disjunction of all such pairs would also appear to be false – certainly not logically true.

[contents]

 

 

 

 

 

 

 

 

From:

 

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