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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
7.8
Truth-value Gaps: Denotation Failure
Brief summary:
(7.8.1) One motivation for arguing for truth-value gaps are intuitionistic situations where neither A nor ¬A can be verified. We discussed intuitionism previously, so we turn instead to two other arguments for gaps. (7.8.2) The first sort of argument for truth-value gaps are “sentences that contain noun phrases that do not appear to refer to anything, like names such as ‘Sherlock Holmes’, and descriptions such as ‘the largest integer’ (there is no largest)” (130). (7.8.3) Frege claimed that “all sentences containing such terms are neither true nor false;” but this is too strong of a claim, because we would want for example for the following sentence to be true: “Sherlock Homes does not really exist” even though it has a non-denoting term. (7.8.4) But there are sorts of sentences with non-denoting terms, called “truths of fiction,” that would seem to really be true, false, or neither on account of the fictional world they are statements about. For example, “Holmes lived in Baker Street” would be true, because that is where the author Conan Doyle says Homes lives; “Holmes’ friend, Watson, was a lawyer,” would be false, because Doyle says that Watson was a doctor, and “Holmes had three maiden aunts” would be neither true nor false, because Doyle never says anything about Holmes’ aunts or uncles. (7.8.5) But some say that fictional truth sentences are really shorthand for sentences beginning with “In the play/novel/film (etc.), it is the case that...”. So, “in Doyle’s stories, it is the case that Holmes lived in Baker Street;” “in Doyle’s stories, it is not the case that Watson was a lawyer;” and “in Doyle’s stories, it is not the case that Holmes had three maiden aunts, and it is not the case that he did not” (thereby making all such sentences true.) (7.8.6) “Another sort of example of a sentence that can plausibly be seen as neither true nor false is a subject/predicate sentence containing a non-denoting description, like ‘the greatest integer is even’” (131). (7.8.7) But it is not necessary to say that non-denoting descriptions are neither true nor false, because they fulfill this function when being just false. (7.8.8) And in fact, in many cases non-denoting descriptions would work better being simply false. For example, let “Father Christmas” be “the old man with a white beard who comes down the chimney at Christmas bringing presents,” and thus the following is simply false: “The Greeks worshipped Father Christmas.” (7.8.9) Nonetheless, even Russell’s view that non-denoting descriptions are false does not help for cases when we would say they should be true; “For example, it appears to be true that the Greeks worshipped the gods who lived on Mount Olympus” (131-132). (7.8.10) So although we have reason to pursue non-denotation as a motivation for truth-value gaps, we see that it is problematic.
[Turning Now to Arguments for Truth-Value Gaps]
[Gap Argument Sort 1: Non-Referring Nouns and Descriptions]
[This Claim as Being Too Strong]
[“Truths of Fiction” as Sentences with Non-Denoting Terms as Gaps]
[Fictional Truths as Shorthand for True Claims about Story Facts]
[Sentences with Non-Denoting Description]
[Non-Denoting Descriptions as not Needing to Be False]
[Non-Denoting Descriptions as Often Better as Just False]
[The Failure of Russel’s View that Non-Denoting Descriptions Are False]
[The Problematic Nature of Non-Denotation as a Motivation for Truth-Value Gaps]
Summary
[Turning Now to Arguments for Truth-Value Gaps]
[One motivation for arguing for truth-value gaps are intuitionistic situations where neither A nor ¬A can be verified. We discussed intuitionism previously, so we turn instead to two other arguments for gaps.]
[(ditto)]
Let us now turn to the question of why one might suppose there to be truth-value gaps. One reason for this, we saw in the last chapter. If one identifies truth with verification then, since there may well be sentences, A, such that neither A nor ¬A can be verified, there may well be truth-value gaps. Intuitionism can be thought of as a particular case of this.4 Since we discussed intuitionism in the last chapter, we will say no more about this argument here. Instead, we will look at two different arguments.5
(130)
4. Though, note, in the Kripke semantics for intuitionist logic, every formula takes the value of either 1 or 0 at every world.
5. Other examples of truth-value gaps that are sometimes given include category mistakes. Such as ‘The number 3 is thinking about Sydney’, and other ‘nonsense’ statements; statements in the border-area of some vague predicate; and cases of presupposition failure.
(130)
[Gap Argument Sort 1: Non-Referring Nouns and Descriptions]
[The first sort of argument for truth-value gaps are “sentences that contain noun phrases that do not appear to refer to anything, like names such as ‘Sherlock Holmes’, and descriptions such as ‘the largest integer’ (there is no largest)” (130).]
[(ditto)]
The first concerns sentences that contain noun phrases that do not appear to refer to anything, like names such as ‘Sherlock Holmes’, and descriptions such as ‘the largest integer’ (there is no largest).
(130)
[This Claim as Being Too Strong]
[Frege claimed that “all sentences containing such terms are neither true nor false;” but this is too strong of a claim, because we would want for example for the following sentence to be true: “Sherlock Homes does not really exist” even though it has a non-denoting term.]
[(ditto)]
It was suggested by Frege that all sentences containing such terms are neither true nor false.6 This seems unduly strong. Think, for example, of ‘Sherlock Holmes does not really exist’, or ‘either 2 is even or the greatest prime number is’.
(130)
6. Though he also thought that denotation failure ought not to arise in a properly constructed language. Non-denoting terms should be assigned an arbitrary reference.
(130)
[“Truths of Fiction” as Sentences with Non-Denoting Terms as Gaps]
[But there are sorts of sentences with non-denoting terms, called “truths of fiction,” that would seem to really be true, false, or neither on account of the fictional world they are statements about. For example, “Holmes lived in Baker Street” would be true, because that is where the author Conan Doyle says Homes lives; “Holmes’ friend, Watson, was a lawyer,” would be false, because Doyle says that Watson was a doctor, and “Holmes had three maiden aunts” would be neither true nor false, because Doyle never says anything about Holmes’ aunts or uncles.]
[(ditto)]
Still, there are some sentences containing non-denoting terms that can plausibly be taken as neither true nor false. One sort of example | concerns ‘truths of fiction’. It is natural to suppose that ‘Holmes lived in Baker Street’ is true, because Conan Doyle says so; ‘Holmes’ friend, Watson, was a lawyer’ is false, since Doyle tells us that Watson was a doctor; and ‘Holmes had three maiden aunts’ is neither true nor false, since Doyle tells us nothing about Holmes’ aunts or uncles.
(130-131)
[Fictional Truths as Shorthand for True Claims about Story Facts]
[But some say that fictional truth sentences are really shorthand for sentences beginning with “In the play/novel/film (etc.), it is the case that...”. So, “in Doyle’s stories, it is the case that Holmes lived in Baker Street;” “in Doyle’s stories, it is not the case that Watson was a lawyer;” and “in Doyle’s stories, it is not the case that Holmes had three maiden aunts, and it is not the case that he did not” (thereby making all such sentences true.)]
[(ditto)]
This reason is not conclusive, though. An alternative view is that all such sentences are simply false. A fictional truth is really a shorthand for the truth of a sentence prefixed by ‘In the play/novel/film (etc.), it is the case that’. Thus, in Doyle’s stories (it is the case that) Holmes lived in Baker Street. Fictional falsities are similar. Thus, in Doyle’s stories it is not the case that Watson was a lawyer. And a fictional truth-value gap, A, is just something where neither A nor ¬A holds in the fiction. Thus, it is not the case in Doyle’s stories that Holmes had three maiden aunts; and it is not the case that he did not.
(131)
[Sentences with Non-Denoting Description]
[“Another sort of example of a sentence that can plausibly be seen as neither true nor false is a subject/predicate sentence containing a non-denoting description, like ‘the greatest integer is even’” (131).]
[In Logic: A Very Short Introduction, ch.4 Priest discusses Russell’s descriptions.
While we are on the topic of subjects and predicates, there is a certain kind of phrase that can be the subject of sentences, which we haven’t talked about yet. logicians usually call them definite descriptions, or sometimes just descriptions - though be warned that this is a technical term. Descriptions are phrases like ‘the man who first landed on the Moon’ and ‘the only man-made object on the Earth that is visible from space’. In general, descriptions have the form: the thing satisfying such and such a condition. Following the English philosopher/mathematician, Bertrand Russell, one of the founders of modern logic, we can write them as follows. Rewrite ‘the man who first landed on the Moon’ as ‘the object, x, such that x is a man and x landed first on the Moon’. Now write ιx for ‘the object, x, such that’, and this becomes ‘ιx(x is a man and x landed first on the Moon)’. If we write M for ‘is a man’ and F for ‘landed first on the Moon’, we then get: ιx(xM & xF). In general, a description is something of the form ιxcx, where cx is some condition containing occurrences of x. (That’s what the little subscript x is there to remind you of.)
(Priest, Logic: A Very Short Introduction, p.24, ch.4)
Now Priest is using “the greatest integer is even.” I would think that the denoting description is “the greatest integer,” as there is none.]
Another sort of example of a sentence that can plausibly be seen as neither true nor false is a subject/predicate sentence containing a non-denoting description, like ‘the greatest integer is even’. (Maybe not every predicate, though: ‘The greatest integer exists’ would seem to be false. But existence is a contentious notion anyway.)7
(131)
7. A related suggestion concerns names that may denote objects, but not objects that exist in the world or situation at which truth is being evaluated. Thus, Aristotle exists in this world, but consider some world at which he does not exist. It may be suggested that ‘Aristotle is a philosopher’ is neither true nor false at that world.
(131)
[Non-Denoting Descriptions as not Needing to Be False]
[But it is not necessary to say that non-denoting descriptions are neither true nor false, because they fulfill this function when being just false.]
[Priest’s next point seems to be that the view that certain non-denoting descriptions are neither true nor false is unnecessary, because, as Russell saw it, such sentences can just be false and still work fine.]
But again, this view is not mandatory. One may simply take such sentences to be false (so that their negations are true, etc.). This was, essentially, Russell’s view.
(131)
[Non-Denoting Descriptions as Often Better as Just False]
[And in fact, in many cases non-denoting descriptions would work better being simply false. For example, let “Father Christmas” be “the old man with a white beard who comes down the chimney at Christmas bringing presents,” and thus the following is simply false: “The Greeks worshipped Father Christmas.”]
[In section 7.8.7 above we said that non-denoting descriptions need not be neither true nor false because they function also when being simply false. Now Priest says that in fact they would work better anyway in many cases were they simply false, as for example “Father Christmas” being “the old man with a white beard who comes down the chimney at Christmas bringing presents,” and thus we would say the following is simply false: “The Greeks worshipped Father Christmas.”]
And Russell’s view would seem to work better than a truth-value gap view in many cases. Thus, let ‘Father Christmas’ be short for the description ‘the old man with a white beard who comes down the chimney at Christmas bringing presents’. Then the following would certainly appear to be false: ‘The Greeks worshipped Father Christmas’ and ‘Julius Caesar thought about Father Christmas.’
(131)
[The Failure of Russel’s View that Non-Denoting Descriptions Are False]
[Nonetheless, even Russell’s view that non-denoting descriptions are false does not help for cases when we would say they should be true; “For example, it appears to be true that the Greeks worshipped the gods who lived on Mount Olympus” (131-132).]
[(ditto)]
Note, though, that even Russell’s view appears to be in trouble with some similar examples. For example, it appears to be true that the Greeks | worshipped the gods who lived on Mount Olympus, and that little Johnny does think about Father Christmas on 24 December.
(131-132)
[The Problematic Nature of Non-Denotation as a Motivation for Truth-Value Gaps]
[So although we have reason to pursue non-denotation as a motivation for truth-value gaps, we see that it is problematic.]
[(ditto)]
Thus, though non-denotation does give some reason for supposing there to be truth-value gaps, the view has its problems, as do most views concerning non-denotation.8
(132)
8. We will meet the topic of denotation-failure again in chapter 21 (Part II).
(132)
From:
Priest, Graham. 2008 [2001]. An Introduction to Non-Classical Logic: From If to Is, 2nd edn. Cambridge: Cambridge University.
Or if otherwise noted:
Priest, Graham. 2000. Logic: A Very Short Introduction. Oxford: Oxford University.
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