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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
7.
Many-Valued Logics
7.6
Truth-value Gluts: Inconsistent Laws
Brief summary:
(7.6.1) We will now examine philosophical motivations for advocating for multi-valued logics with truth-value gaps or gluts. (7.6.2) In this chapter subsection, Priest will elaborate on the issue of inconsistent laws. (7.6.3) For example, consider if long ago there were the laws {1} that no aborigines have the right to vote, but {2} all property-holders have that right. At the time it was unthinkable for aborigines to own property, but later in history they do. Thus in the legal system, later on in history, aborigines both have and do not have the right to vote. (7.6.4) In cases of insistent laws, normally they are rectified to make them consistent. Nonetheless, they will remain inconsistent for some time until that change will be made. (7.6.5) Priest next considers a possible objection, namely, that such seemingly contradictory laws are actually consistent, because there is always some other law that clarifies which of the two contradicting laws takes precedent; “for example lex posterior (that a later law takes precedence over an earlier law), or that constitutional law takes precedence over statute law, which takes precedence over case law. One might insist that all contradictions are only apparent” (128). (7.6.6) Priest’s reply to this objection is that while it may be that in actual fact there are many cases where additional laws dissolve the apparent legal contradiction, in principle it is still possible, as for example were both laws made at the same rank.
[Philosophical Motivations for Multi-Valued Logics]
[The Topic: Inconsistent Laws]
[An Example of Inconsistent Laws: Aborigine Land-Owners
[The Temporary Persistence of Inconsistent Laws]
[Objection: Other Laws are Always in Place to Clarify the Law that Takes Precedent]
[Reply: This Holds in Fact but not in Principle]
Summary
[Philosophical Motivations for Multi-Valued Logics]
[We will now examine philosophical motivations for advocating for multi-valued logics with truth-value gaps or gluts.]
[We are dealing with 3-valued logics, which have the truth values 1, 0, and i (true, false, and indeterminate; see Nolt’s Logics section 15.2). We examined two pairings of 3-valued logics. The first pairing, K3 and Ł3, regards the value i as having the sense of neither true nor false (see section 7.3), and the second pairing, LP and RM3, regard i as meaning both true and false (see section 7.4). (I am not sure how that distinction of their sense comes about from the semantics. My best guess is that it has something to do with the fact that for K3 and Ł3, the designated value is 1, and for LP and RM3 it is 1 or i. So maybe if we think of validity as truth preservation, then to have i as a designated value means that it has at least some 1 in it, and when it is not a designated value means it has no 1 in it. I am just wondering aloud.) In the following sections we will consider some philosophical motivations for advocating for either truth-value gaps or gluts. The motivations for gaps will be denotation failure and future contingents. The motivations for gluts will be inconsistent laws and paradoxes of self reference. But Priest notes some other motivations for gluts that we will not examine here, namely: “the state of affairs realised at an instant of change; statements about some object in the border-area of a vague predicate; contradictory statements in the dialectical tradition of Hegel and Marx; statements with predicates whose criteria of application are over-determined; and certain statements about micro-objects in quantum mechanics” (128). See In Contradiction chapter 11 and 12 for inconsistency and motion, and see “Dialectic and Dialetheic” for contradiction in Hegel and Marx.]
Let us now turn to the issue of the philosophical motivations for many-valued logics and, in particular, the 3-valued logics we have met. Typically, the motivations for those logics that treat i as both true and false (a truth-value glut), like LP and RM3, are different from those that treat i as neither true nor false (a truth-value gap), like K3 and Ł3. Let us start | with the former. We will look at two reasons for supposing that there are truth-value gluts.2
(127-128)
2. Other examples of truth-value gluts that have been suggested include the state of affairs realised at an instant of change; statements about some object in the border-area of a vague predicate; contradictory statements in the dialectical tradition of Hegel and Marx; statements with predicates whose criteria of application are over-dertermined; and certain statements about micro-objects in quantum mechanics.
(128)
[The Topic: Inconsistent Laws]
[In this chapter subsection, Priest will elaborate on the issue of inconsistent laws.]
[In section 4.8.3, Priest gave an example where a contradiction of laws does not create a logical “explosion” entailing everything.
Another example: pieces of legislation are often inconsistent. To avoid irrelevant historical details, here is an hypothetical example. Suppose that an (absent-minded) state legislator passes the following traffic laws. At an unmarked junction, the priority regulations are:
(1) Any woman has priority over any man.
(2) Any older person has priority over any younger person.
(We may suppose that clause 2 was meant to resolve the case where two men or two women arrive together, but the legislator forgot to make it subordinate to clause 1.) The legislation will work perfectly happily in three out of four combinations of sex and age. But suppose that Ms X, of age 30, approaches the junction at the same time as Mr Y, of age 40. Ms X has priority (by 1), but has not got priority (by 2 and the meaning of ‘priority’). Hence, the situation is inconsistent. But, again, it would be stupid to infer from this that, for example, the traffic laws are consistent.
(p.75, section 4.8.3)
In this chapter subsection, Priest will elaborate on this matter of inconsistent laws.]
The first concerns inconsistent laws, and the rights and obligations that agents have in virtue of these. We have already had an example of this in 4.8.3 concerning inconsistent traffic regulations.
(128)
[An Example of Inconsistent Laws: Aborigine Land-Owners]
[For example, consider if long ago there were the laws {1} that no aborigines have the right to vote, but {2} all property-holders have that right. At the time it was unthinkable for aborigines to own property, but later in history they do. Thus in the legal system, later on in history, aborigines both have and do not have the right to vote.]
[(ditto)]
Here is another example. Suppose that in a certain (entirely hypothetical) country the constitution contains the following clauses:
(1) No aborigine shall have the right to vote.
(2) All property-holders shall have the right to vote. We may suppose that when the law was made, the possibility of an aboriginal property-holder was so inconceivable as not to be taken seriously. Despite this, as social circumstances change, aborigines do come to hold property. Let one such be John. John, it would appear, both does and does not have the right to vote.
(128)
[The Temporary Persistence of Inconsistent Laws]
[In cases of insistent laws, normally they are rectified to make them consistent. Nonetheless, they will remain inconsistent for some time until that change will be made.]
[(ditto)]
Of course, if a situation of this kind comes to light, the law is likely to be changed to resolve the contradiction. The fact remains, though, that until the law is changed the contradiction is true.
(128)
[Objection: Other Laws are Always in Place to Clarify the Law that Takes Precedent]
[Priest next considers a possible objection, namely, that such seemingly contradictory laws are actually consistent, because there is always some other law that clarifies which of the two contradicting laws takes precedent; “for example lex posterior (that a later law takes precedence over an earlier law), or that constitutional law takes precedence over statute law, which takes precedence over case law. One might insist that all contradictions are only apparent” (128).]
[(ditto)]
One way that one might object to this conclusion is as follows. The law contains a number of principles for resolving apparent contradictions, for example lex posterior (that a later law takes precedence over an earlier law), or that constitutional law takes precedence over statute law, which takes precedence over case law. One might insist that all contradictions are only apparent, and can be defused by applying one or other of these principles.
(128)
[Reply: This Holds in Fact but not in Principle]
[Priest’s reply to this objection is that while it may be that in actual fact there are many cases where additional laws dissolve the apparent legal contradiction, in principle it is still possible, as for example were both laws made at the same time or they are of the same rank.]
[(ditto)]
It is clear, however, that there could well be cases where none of these principles are applicable. Both laws are made at the same time; they are both laws of the same rank, and so on. Hence, though some legal contradictions may be only apparent, this need not always be the case.
(128)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
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