4. The Horned Man 悖论 英文版 逻辑学教材

The Paradoxes of Eubulides of Megara (4th century B.C.)

4.The Horned Man

I.The Paradox

Assuming you don’t suffer from calvarial homoplastic osteomata, consider:

You still have what you have not lost.

You have not lost horns.

Therefore, you still have horns.

Again, the paradox is a seemingly sound argument with a false conclusion.

The Horned Man introduces the phenomenon of presupposition, a subject that most naturally arises in considering definite descriptions, i.e., phrases of the form ‘the Φ’ (like ‘the puppy in the window’) that purport to refer to a specific individual, viz., the unique (relevant) thing that is Φ.

II.Strawson vs. Russell on the Analysis of Definite Descriptions

Consider the sentence:

1. Joe is tall.

As we learned when studying predicate logic, (1) is a singular proposition, i.e., a proposition about one specific object, for the sentence says that the individual denoted by the designating expression ‘Joe’ possesses the property expressed by the predicate expression ‘…is tall.’ Thus, we formalize (1) as:

1'. Tj j: Joe

T x: x is tall

Now, consider the following sentence, which contains a definite description rather

than a name:

2. The puppy in the window is purebred.

P.F. Strawson argued that (2) is also a singular proposition, and, so, he took definite descriptions to be designating expressions. On his view, the logical form of (2) is:

2'. P the Φthe Φ: the unique puppy that is in the (salient) window

P x: x is purebred

Call the definite description ‘the Φ’ a quasi-name.

Bertrand Russell argued against Strawson that, contrary to its appearance, a definite description is a quantifier construction rather than a designating expression. For example, Russell argued that the sentence:

3. The king of France is bald.

asserts that there exists one and only one object that has the property of being the king of France, and the unique individual with that property also has the property of baldness. That is, (3) is an existentially quantified general proposition, not a singular proposition. Thus, (3) has the logical form:

3'. (?x)(((F x∧ (?y)(F y?y = x)) ∧ B x)F x: x is a king of France

B x: x is bald

That is: There exists an x such that x is a king of France, and if any y is a king of France, t hen it is identical to x; and x is bald.

On Strawson’s theory, (3) is formalized as:

3''. B the F the F: the king of France

where ‘the F’ is a designating expression, a quasi-name, denoting the specific individual that is the king of France.

Summing up their disagreement, Russell claims that (3) asserts that there is one and only one object that is king of France, and it asserts that that object, whichever it is, is bald. Strawson claims that (3) presupposes, but does not assert, that there is one and only one object that is king of France, and it asserts that that specific individual is bald.

III.Strawsonian Presupposition

Consider some more of Strawson’s examples:

4. Wellington’s victory at Waterloo was his greatest triumph.

5. All John’s children are asleep.

According to Strawson, (4) asserts a singular proposition: a specific event—the one picked out by the quasi-name ‘Wellington’s victory at Waterloo’—has the property of b eing Wellington’s greatest triumph. But in order to assert that that event possesses that property, (4) must presuppose (though not assert) that that event occurred, i.e., that Wellington was victorious at Waterloo.

Sentence (5) asserts the general pro position that each member of the class of John’s children is asleep. And in order to assert this about each member of the class, (5) must presuppose (though not assert) that the class has members, i.e., that John has children.

OK, so precisely what is Strawsonian presupposition?

A. Sentences vs. Statements

In order to provide a precise account, we must consider the distinction that Strawson draws between a sentence and a statement. A sentence is a linguistic type; it is meaningful, but it doesn’t ac tually assert anything. It is speakers, not sentences, that make assertions, which they do by making statements. A statement is a token of a sentence type; it is a speaker’s particular use of a sentence at a particular time and in specific circumstances in order to make an assertion. For example, sentence (3) above means that the king of France is bald, but it doesn’t assert anything. However, if it is 1650, then Pierre can assert that Louis XIV is bald by using (3) to make the statement that Louis XIV is bald.

And if it is 1789, then Fran?oise can make the different assertion that Louis XVI is bald by using the same sentence to make the different statement that Louis XVI is bald.

So, a speaker’s use of a sentence produces a statement.

Now, a use of an expression type is an utterance tokening of it with certain intentions. For example, in the course of uttering sentence (3), Pierre uttered ‘the king of France’ with the intention of referring to Louis XIV, and the fact that Fran?oise’s use of t he same sentence was different from Pierre’s use consists in the fact that she uttered ‘the king of France’ with the different intention of referring to Louis XVI.

Since it is only when speakers use sentences to make statements that anything at all gets asserted, only statements have truth-values. And, a speaker’s use of a sentence type determines the truth conditions for his statement. Thus, Pierre’s statement is true if and only if Louis XIV is bald, and it is false if and only if Louis XIV is not bald, and Fran?oise’s statement is true if and only if Louis XVI is bald, and it is false if and only if Louis XVI is not bald.

B. Presupposition

Call Fran?oise’s statement that the king of France is bald ‘S.’ Now, if S is true if and only if Louis XVI is bald and false if and only if Louis XVI is not bald, then it follows that her statement can be either true or false only if the king of France (at the time of her utterance) is Louis XVI.

Now, let ‘S'’ be the (possible) statement (in 1789) that t he king of France is Louis XVI.

The truth of ‘S'’ is a necessary condition for ‘S’ to be either true or false, i.e., to have a truth-value at all. In this sense, ‘S’ presupposes‘S'.’

C. Presupposition Failure

Since ‘S'’ must be true for ‘S’ to even have a truth-value, it follows that if ‘S'’ is false, then ‘S’ is neither true nor false. S is a truth-value gap.

This conclusion about truth-value gaps makes sense in the light of Strawson’s view about statements. Consider Bertrand’s use in 1905 of sentence (3) to state that the king of France is bald. France had no king in 1905 at the time of Bertrand’s utterance. So, his intention to refer to a specific individual failed, and his use of the quasi-name ‘the king of France’ was vacuous.

Sinc e he didn’t succeed in referring to anything at all, it follows that he just wasn’t talking about anything. And, if he just wasn’t talking about anything, then it follows that he failed to make any genuine assertion at all. So, Bertrand failed to make a statement.

Since it is only statements that have truth-values, it stands to reason that Bertrand’s utterance of (3) had no truth-value.

Similarly, a use of sentence (4) to make the statement that Wellington’s victory at Waterloo was his greatest triumph presupposes the truth of the (possible) statement that Wellington was victorious at Waterloo. If Wellington was not victorious at Waterloo, then a speaker’s utterance of ‘Wellington’s victory at Waterloo’ would fail to refer, and, consequently, her use of (4) would fail to make a statement. Consequent ly, the speaker’s utterance would fail to be either true or false.1

IV.Return to the Horned Man

Presupposition resolves our trouble with the Horned Man paradox. For, the statement of the argument’s first premise:

i. You still have what you have not lost

presupposes the statement:

i′. You do possess the relevant thing.

Now, (i′) is false in the case of your horns. Since its presupposition fails, the statement of (i) is a truth-value gap. Since the statement of (i) has no truth-value, it follows that that statement is not true. Therefore, the Horned Man argument is not sound. And, the paradox is resolved.

1A problem for Strawson: My statement ‘The king of France shot my cat last night’ i s clearly false; however, on Strawson’s view it gets no truth-value.