Exception paradox

Exception paradox: if every rule has an exception, then there must be an exception to the rule that every rule has an exception.

From the logical point of view, this can be taken as a proof that the sentence "every rule has an exception" is false - a simple example of a proof technique known as reductio ad absurdum. However, if the rule to which there was no exception was the rule that all rules have an exception, then the sentence 'all rules have an exception' could remain true without paradox.

The Exception paradox is quite similar to the liar paradox, in which either answer creates an infinite loop of questions which again leads back to the beginning of each initial question being asked (or answered).

A similar idea is this: If everything is possible, then it is possible for anything to be impossible. Also: The only rule is that there are no rules.

Epimenides Paradox

The Epimenides paradox is a problem in logic. It is named after the Cretan philosopher Epimenides of Knossos (alive circa 600 BC), who stated Κρῆτες ἀεί ψεύσται (Kretes aei pseystai), "Cretans, always liars". There is no single statement of the problem; a typical variation is given in the book Gödel, Escher, Bach, by Douglas R. Hofstadter:

Epimenides was a Cretan who made one immortal statement: "All Cretans are liars."

It is commonly supposed that self-referential paradox arises when one considers whether Epimenides spoke the truth. However, if Epimenides knew of one Cretan (other than himself) who is not a liar, his statement is a lie (because he asserts all) even though it correctly describes the speaker as a liar.

Contents:
1. History of the phrase
2. Logical analysis
3. References

1. History of the phrase

Epimenides was a philosopher and religious prophet who, against the general sentiment of Crete, proposed that Zeus was immortal, as in the following poem:

They fashioned a tomb for thee, O holy and high one The Cretans, always liars, evil beasts, idle bellies!
But thou art not dead: thou livest and abidest forever,
For in thee we live and move and have our being.

- Epimenides, Cretica

Denying the immortality of Zeus, then, is the lie of the Cretans. It appears that by "Cretans", Epimenides intended "Cretans other than myself". The phrase "Cretans, always liars" was quoted by the poet Callimachus in his Hymn to Zeus, with the same theological intent as Epimenides. The entire second line is quoted by the Apostle Paul in the Epistle to Titus.

One of Crete's own prophets has said it: 'Cretans are always liars, evil brutes, lazy gluttons'. He has surely told the truth.

- Epistle to Titus, 1:12

The logical inconsistency of a Cretan asserting all Cretans are always liars may not have occurred to Epimenides, nor to Callimachus. In the original context, Epimenides necessarily meant "Cretans other than myself", so there is no self-reference and thus no logical problem to speak of. The liar paradox was known in antiquity, but it was not associated with Epimenides and Saint Augustine restates the liar paradox, without mentioning Epimenides or Titus, in Against the Academicians (III.13.29). Many variations of the liar paradox (called insolubilia) were studied in the Middle Ages, but none of the extant medieval works on insolubilia refer to Epimenides, neither directly nor through the Epistle to Titus. The earliest appearance of Epimenides in the context of a logical problem dates only to the nineteenth century. Since that time, the Epimenides paradox has been commonly employed in discussions of logic.

2. Logical analysis

If one defines "liar" as someone who is never truthful, then the statement "All Cretans are liars," if uttered by a Cretan, Epimenides, leads inescapably to the conclusion that the speaker is a liar who knows that some Cretans are not liars.

Several interpretations and analyses are available, if the statement is considered false. It might be contended that the truth-value "false" can be consistently assigned to the simple proposition that "All Cretans are liars," so that this statement by itself, when deemed false, is not, strictly speaking, paradoxical. Thus, if there ever existed a Cretan (not Epimenides in this instance) who even once spoke the truth, the categorical statement "All Cretans are (always) liars," would be false, and Epimenides might be simply regarded as having made a false statement himself. But if Epimenides' statement is understood as in essence asserting its own falsehood, then the statement cannot consistently be false, either, because its falsehood would imply the truth of its self-asserted falsehood.

An interesting asymmetry is possible under one interpretation: the statement's truth clearly implies its falsehood, but, unless the statement is interpreted to refer specifically to itself (rather than referring categorically to all statements by Cretans), the statement could be contingently false without implying its own truth.

Paradoxical versions of the Epimenides problem are closely related to a class of more difficult logical problems, including the liar paradox, Russell's paradox, and the Burali-Forti paradox, all of which have self-reference in common with Epimenides. Indeed, the Epimenides paradox is usually classified as a variation on the liar paradox, and sometimes the two are not distinguished. The study of self-reference led to important developments in logic and mathematics in the twentieth century.

3. References

All of the works of Epimenides are now lost, and known only through quotations by other authors. The quotation from the Cretica of Epimenides is given by R.N. Longenecker, "Acts of the Apostles", in volume 9 of The Expositor's Bible Commentary, Frank E. Gaebelein, editor (Grand Rapids, Michigan: Zondervan Corporation, 1976-1984), page 476. Longenecker in turn cites M.D. Gibson, Horae Semiticae X (Cambridge: Cambridge University Press, 1913), page 40, "in Syriac". Longenecker states the following in a footnote:

The Syr. version of the quatrain comes to us from the Syr. church father Isho'dad of Mero (probably based on the work of Theodore of Mopsuestia), which J.R. Harris translated back into Gr. in Exp ["The Expositor"] 7 (1907), p 336.

An oblique reference to Epimenides in the context of logic appears in "The Logical Calculus" by W. E. Johnson, Mind (New Series), volume 1, number 2 (April, 1892), pages 235-250. Johnson writes in a footnote,

Compare, for example, such occasions for fallacy as are supplied by "Epimenides is a liar" or "That surface is red," which may be resolved into "All or some statements of Epimenides are false," "All or some of the surface is red."

The Epimenides paradox appears explicitly in "Mathematical Logic as Based on the Theory of Types", by Bertrand Russell, in the American Journal of Mathematics, volume 30, number 3 (July, 1908), pages 222-262, which opens with the following:

The oldest contradiction of the kind in question is the Epimenides. Epimenides the Cretan said that all Cretans were liars, and all other statements made by Cretans were certainly lies. Was this a lie?

In that article, Russell uses the Epimenides paradox as the point of departure for discussions of other problems, including the Burali-Forti paradox and the paradox now called Russell's paradox. Since Russell, the Epimenides paradox has been referenced repeatedly in logic. Typical of these references is Gödel, Escher, Bach by Douglas Hofstadter, which accords the paradox a prominent place in a discussion of self-reference.

Paradox of entailment

The paradox of entailment is an apparent paradox derived from the principle of explosion, a law of classical logic stating that inconsistent premises always make an argument valid; that is, inconsistent premises imply any conclusion at all. This seems paradoxical, as it suggests that the following is a good argument:

It is raining
It is not raining

Therefore:

George Washington was a zombie.

Contents:
1. Understanding the paradox
2. Explaining the paradox
3. References
4. See also

1. Understanding the paradox

Validity is defined in classical logic as follows: An argument (consisting of premises and a conclusion) is valid if and only if there is no possible situation in which all the premises are true and the conclusion is false.

For example an argument might run:

If it is raining, water exists (1st premise)
It is raining (2nd premise)
Water exists (Conclusion)

In this example there is no possible situation in which the premises are true while the conclusion is false. Since there is no counterexample, the argument is valid.

But one could construct an argument in which the premises are inconsistent. This would satisfy the test for a valid argument since there would be no possible situation in which all the premises are true and therefore no possible situation in which all the premises are true and the conclusion is false.

For example an argument with inconsistent premises might run:

Matter has mass (1st premise; true)
Matter does not have mass (2nd premise; false)
All numbers are equal to 42 (Conclusion)

As there is no possible situation where both premises could be true, then there is certainly no possible situation in which the premises could be true while the conclusion was false. So the argument is valid whatever the conclusion is; inconsistent premises imply all conclusions.

2. Explaining the paradox

The strangeness of the paradox of entailment comes from the fact that the definition of validity in classical logic does not always agree with the use of the term in ordinary language. In everyday use validity suggests that the premises are consistent. In classical logic, the additional notion of soundness is introduced. A sound argument is a valid argument with all true premises. Hence a valid argument with an inconsistent set of premises can never be sound. Other suggested improvements to the notion of logical validity include strict implication and relevant implication.

3. References

4. See also

* Correlation does not imply causation
* False dilemma

Excluded Middle

See False Dilemma or Black-or-White.

Exaggeration

When we overstate or overemphasize a point that is a crucial step in a piece of reasoning, then we are guilty of the fallacy of exaggeration. This is a kind of error called Lack of Proportion.

Example:

She's practically admitted that she intentionally yelled at that student while on the playground in the fourth grade. That's assault. Then she said nothing when the teacher asked, "Who did that?" That's lying, plain and simple. Do you want to elect as secretary of this society someone who is a known liar prone to assault? Doing so would be a disgrace to the Collie Society.

When we exaggerate in order to make a joke, though, we aren't guilty of the fallacy.

Every and All

The fallacy of every and all turns on errors due to the order or scope of the quantifiers "every" and "all" and "any." This is a version of the scope fallacy.

Example:

Every action of ours has some final end. So, there is some common final end to all our actions.

In proposing this fallacious argument, Aristotle believed the common end is the supreme good, so he had a rather optimistic outlook on the direction of history.

Etymological

The etymological fallacy occurs whenever someone falsely assumes that the meaning of a word can be discovered from its etymology or origins.
Example:

The word "vise" comes from the Latin "that which winds", so it means anything that winds. Since a hurricane winds around its own eye, it is a vise.

Equivocation

Equivocation is the illegitimate switching of the meaning of a term during the reasoning.

Example:

Brad is a nobody, but since nobody is perfect, Brad must be perfect, too.

The term "nobody" changes its meaning without warning in the passage. So does the term "political jokes" in this joke: I don't approve of political jokes. I've seen too many of them get elected.

Either/Or

See Black-or-White.