In philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent truth value. If "This statement is false" is true, then what it says is the case; but what it says is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.
The Epimenides paradox is often considered equivalent or interchangeable with the "liar paradox", but they are not the same. The liar paradox is a statement that cannot consistently be true or false, while Epimenides' statement is simply false, as long as there exists at least one Cretan who sometimes tells the truth.
It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox, and they were probably only understood as such much later in history. The oldest known version of the liar paradox is instead attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. Eubulides reportedly said:
The simplest version of the paradox is the sentence:
If the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is false. Yet the sentence cannot be false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either hypothesis, the statement is both true and false.
However, that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is neither true nor false. This response to the paradox is, in effect, to reject the common beliefs about truth and falsity: the claim that every statement has to abide by the principle of bivalence, a concept related to the law of the excluded middle.
The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:
If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true and so one is led to another paradox.
Another reaction to the paradox of (A) is to posit, as Graham Priest has, that the statement follows paraconsistent logic and is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:
If (C) is both true and false then it must be true. This means that (C) is only false, since that is what it says, but then it cannot be true, creating another paradox.
Thus the following two statements are equivalent:
The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present similar answers.
and Jones says only these three things about Smith:
and Smith really is a big spender but is not soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.
Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.
If some statement, B, is assumed to be false, one writes B = false. The statement (C) that the statement B is false would be written as C = "B = false". Now, the liar paradox can be expressed as the statement A, that A is false:
A = "A = false"
This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to not A and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is a A being both "true" and "false". Other resolutions mostly include some modifications of the equation e.g. A. N. Prior claims that the equation should be A = "A = false" and "A = true" and therefore A is false.
In philosophy and logic, the liar paradox, known to the ancients as the pseudomenon, encompasses paradoxical statements such as "This sentence is false." or "The next sentence is false. The previous sentence is true." These statements are paradoxical because there is no way to assign them a consistent truth value. If "This statement is false" is true, then what it says is the case; but what it says is that it is false, hence it is false. On the other hand, if it is false, then what it says is not the case; thus, since it says that it is false, it must be true.
The Epimenides paradox is often considered equivalent or interchangeable with the "liar paradox", but they are not the same. The liar paradox is a statement that cannot consistently be true or false, while Epimenides' statement is simply false, as long as there exists at least one Cretan who sometimes tells the truth.
It is unlikely that Epimenides intended his words to be understood as a kind of liar paradox, and they were probably only understood as such much later in history. The oldest known version of the liar paradox is instead attributed to the Greek philosopher Eubulides of Miletus who lived in the fourth century BC. It is very unlikely that he knew of Epimenides's words, even if they were intended as a paradox. Eubulides reportedly said:
The simplest version of the paradox is the sentence:
If the statement is true, everything asserted in it must be true. However, because the statement asserts that it is itself false, it must be false. So the hypothesis that it is true leads to the contradiction that it is false. Yet the sentence cannot be false for that hypothesis also leads to contradiction. If the statement is false, then what it says about itself is not true. It says that it is false, so that must not be true. Hence, it is true. Under either hypothesis, the statement is both true and false.
However, that the liar sentence can be shown to be true if it is false and false if it is true has led some to conclude that it is neither true nor false. This response to the paradox is, in effect, to reject the common beliefs about truth and falsity: the claim that every statement has to abide by the principle of bivalence, a concept related to the law of the excluded middle.
The proposal that the statement is neither true nor false has given rise to the following, strengthened version of the paradox:
If (B) is neither true nor false, then it must be not true. Since this is what (B) itself states, it means that (B) must be true and so one is led to another paradox.
Another reaction to the paradox of (A) is to posit, as Graham Priest has, that the statement follows paraconsistent logic and is both true and false. Nevertheless, even Priest's analysis is susceptible to the following version of the liar:
If (C) is both true and false then it must be true. This means that (C) is only false, since that is what it says, but then it cannot be true, creating another paradox.
Thus the following two statements are equivalent:
The latter is a simple contradiction of the form "A and not A", and hence is false. There is therefore no paradox because the claim that this two-conjunct Liar is false does not lead to a contradiction. Eugene Mills and Neil Lefebvre and Melissa Schelein present similar answers.
and Jones says only these three things about Smith:
and Smith really is a big spender but is not soft on crime, then both Smith's remark about Jones and Jones's last remark about Smith are paradoxical.
Kripke proposes a solution in the following manner. If a statement's truth value is ultimately tied up in some evaluable fact about the world, that statement is "grounded". If not, that statement is "ungrounded". Ungrounded statements do not have a truth value. Liar statements and liar-like statements are ungrounded, and therefore have no truth value.
If some statement, B, is assumed to be false, one writes B = false. The statement (C) that the statement B is false would be written as C = "B = false". Now, the liar paradox can be expressed as the statement A, that A is false:
A = "A = false"
This is an equation from which the truth value of A = "this statement is false" could hopefully be obtained. In the boolean domain "A = false" is equivalent to not A and therefore the equation is not solvable. This is the motivation for reinterpretation of A. The simplest logical approach to make the equation solvable is the dialetheistic approach, in which case the solution is a A being both "true" and "false". Other resolutions mostly include some modifications of the equation e.g. A. N. Prior claims that the equation should be A = "A = false" and "A = true" and therefore A is false.
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