Marc Horowitz 24.151 March 6, 1992

II

(1) ‘\underline{1} is small’. The concept takes objects as arguments, and returns truth values as values.

(2) ‘king of \underline{1}’. This function takes countries as arguments, and returns a function which decides if a person is the king of that country.

(3) ‘Something’. If the argument to ‘Something’ returns True for any object, ‘Something’ returns true.

(4) ‘the’. This is the ι (iota) function. It takes a function, and returns the object for which the function returns true.

(5) ‘not \underline{1}’. Returns the opposite truth value.

(6) ‘Marc Horowitz is the ruler of the Universe.’ This sentence has no truth value because ‘the ruler of the Universe’ has no reference. Therefore, the equivalence relation has only one argument, and the sentence has no truth value.

III

The first sentence is of the logical form

(queen(France))((ιx)(queen(England)x))

This sentence has the truth value False. ‘is queen of France’ denotes a function which takes an object as an argument and returns a truth value. The argument refers to an actual person, who may also be referred to as ‘Queen Elizabeth II’.

The second sentence is of the logical form

(ιx)(queen(England)x) = (ιx)(queen(France)x)

This sentence has no truth value at all. ‘=’ denotes a two argument function. But, the second argument (the left-hand side) has no reference (since there is no queen of France), so the = function has no value.

This is a case where the language of the two sentences masks the logical forms of the sentences. Although they are very similar grammatically, their logical forms are completely different.

IV

In the first discussion, Frege points out that the sentence expresses two basic propositions, one derived from the dependent clause, and one derived from the independent clause. In the second discussion, Frege notes that there may be a difference in cognitive significance if one sentence is replaced by a proposition of similar reference. There is a third proposition which is expressed by the logical form of the sentence. I’m probably misusing some technical term here, but this third proposition is an implication that the proposition expressed by the independent clause has a causal relationship to the proposition expressed by the dependent clause.

V

Take a sentence like ‘Dave believes that MIT is in California.’ ‘MIT is in California’ denotes False. However, this merely substituting coreferentials here leads to the nonsensical sentence ‘Dave believes that False’. Frege gets around this by stating that in the context of a propositional verb such as ‘believes’, the reference of a sentence is not its truth value, but its sense. Then, we can substitute the sense into the sentence as a coreferential, and it keeps the same truth value.