On The Liar Paradox

PHIL 600, Presentation on Kirkham’s 9.1 – 9.2 by Vered Arnon

In the first section of chapter 9, Kirkham introduces what is known as the Liar Paradox. There are two variations, one self-referential, the other not. The self-referential one is the sentence ‘This sentence is false.’ The nonself-referential one is the sentence pair ‘The next sentence is false. The previous sentence is true.’ Kirkham lays out the ‘liar derivation’ which is a series of premises showing the contradictions in the paradox:

1. The liar sentence says of itself that it is false and says nothing more.
2. If a sentence s is true and it says that p is the case and says nothing more, then p is the case.
3. If p is the case and a sentence s says that p is the case and says nothing more, then s is true.
4. Every sentence is true or false. (Principle of bivalence.)
5. The liar sentence is true. (Conditional premise.)
6. Ergo, the liar sentence is false. (By 1,2,5.)
7. Ergo, the liar sentence is true and false. (By 5,6)
8. Ergo, if the liar sentence is true, then it is true and false. (By conditional proof.)
9. The liar sentence is false. (Conditional premise.)
10. Ergo, the liar sentence is true. (By 1,3,9.)
11. Ergo, the liar sentence is true and false. (By 9,10.)
12. Ergo, if the liar sentence is false, then it is true and false. (By conditional proof.)
13. Ergo, the liar sentence is true and false. (By 4,8,12.)

Kirkham goes on to say that there must be some problem with some of the premises, because 13 is not acceptable to him. He claims that either a reason must be found to reject the first premise, or all theories holding 2, 3, and 4 as theorems must be abandoned.

Solutions to the liar paradox are generally subject to five criteria of adequacy: specificity, no ad hoc postulations, no overkill, completeness, and compatible with intuitions. Kirkham feels that the criteria of no ad hoc postulations, completeness, and compatibility with intuitions, are too demanding. I feel that a criteria of completeness is not too demanding at all, and while an incomplete solution may indeed still be useful, it should nevertheless not be considered a valid solution. I agree with his assessment of the other two criteria though, I think ad hoc elements are necessary to resolve paradoxes, and I also think that a ban on counterintuitive solutions makes a fallacious assumption that all peoples’ intuitions come to the same conclusions.

In section 9.2, Kirkham examines Russell’s Theory of Types as a solution to the Liar Paradox. Russell claims that ‘Whatever involves all of a collection must not be one of the collection’. He calls this the vicious-circle principle. According to Russell, the liar proposition is meaningless because it is talking about all propositions but is a proposition itself. Russell postulated a hierarchy of types, with individuals lowest in the hierarchy, and all first-order propositions being just above that, continuing upwards infinitely. Propositions concerning one type must be at a higher hierarchical position than what they are concerned with. Kirkham points out that Russell seems unresolved as to whether the liar sentence is by virtue of the Theory of Types false or meaningless, but either way, Russell’s Theory of Types does help to resolve semantic and set-theoretic paradoxes. Kirkham didn’t criticise Russell much, and I think he gave an accurate portrayal of Russell’s Theory of Types.

Getting back to section 9.1, however, I have disagreement with some statements that Kirkham makes. When outlining the criteria that solutions to the Liar Paradox are supposed to meet, Kirkham mentions that Susan Haack interprets the ban against ad hoc postulations in an absolute way, and goes too far. He also claims that Willard Van Orman Quine goes too far in acceptance of counterintuitive solutions. I disagree with Kirkham’s opinion that ‘a solution is better to the extent that it minimizes ad hoc elements and minimizes damage to our existing intuitions’. He seems to feel that counterintuitive solutions are threatening and ad hoc postulations don’t really deal with the problem.

Susan Haack wrote a book titled Deviant Logic in which, among other things, she examines four different theses that are alternative reactions to problems or questions that standard logic seems to have difficulty with.

The No-Item Thesis: Despite appearances, the items in question are not the kind with which logic is, or should be, concerned. Haack argues that this thesis is trivialising and avoidant, and I agree with her opposition to it. I don’t think the Liar Paradox can be adequately resolved simply by saying that logic isn’t actually concerned with it.

The ‘Misleading Form’ Thesis: The items in question, though within the scope of logic, do not really have the form they appear to have. Russell’s Theory of Types supports the ‘misleading form’ thesis. Haack argues that while it does solve the problem, the cost is a loss of simplicity and requires an acceptance of unnatural counter-intuitive translations of sentences into formal language.

The Truth-Value Gap Thesis: The items in question, though within the scope of logic, are neither true nor false, but truth-valueless.

The ‘New Truth-Value’ Thesis: The items in question, though within the scope of logic, are neither true nor false, but have some other truth-values.

The theses concerning truth-value are are explained in terms of very complicated formal logic, and I can’t understand all the formal logic, so rather than try and fail to explain the theses, I will summarise Haack’s observations on their consequences for the theory of bivalence, since as I mentioned in the beginning, Kirkham in section 9.1 pointed out that premise 4 plays a significant role in the ‘liar derivation’. In a many-valued system of logic, where all values are designated or antidesignated, then the principle of bivalence continues to hold true, but if there is a middle value that is neither designated nor antidesignated, then the principle of bivalence can be abandoned and formal logic still used to explain things.

HANDOUT 9.1 – 9.2

1. The liar sentence says of itself that it is false and says nothing more.
2. If a sentence s is true and it says that p is the case and says nothing more, then p is the case.
3. If p is the case and a sentence s says that p is the case and says nothing more, then s is true.
4. Every sentence is true or false. (Principle of bivalence.)
5. The liar sentence is true. (Conditional premise.)
6. Ergo, the liar sentence is false. (By 1,2,5.)
7. Ergo, the liar sentence is true and false. (By 5,6)
8. Ergo, if the liar sentence is true, then it is true and false. (By conditional proof.)
9. The liar sentence is false. (Conditional premise.)
10. Ergo, the liar sentence is true. (By 1,3,9.)
11. Ergo, the liar sentence is true and false. (By 9,10.)
12. Ergo, if the liar sentence is false, then it is true and false. (By conditional proof.)
13. Ergo, the liar sentence is true and false. (By 4,8,12.)

Russell’s Theory of Types, Vicious Circle Principle: ‘Whatever involves all of a collection must not be one of the collection’. He calls this the vicious-circle principle.

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