Grades of Truth-functionality

Truth-functionality is a notion at the heart of classic propositional logic. A standard explication of the notion runs roughly as follows:

An n-ary connective C is truth-functional iff it corresponds to a truth-function.

More precisely, C is truth-functional iff there is a mapping from the truth-values of any n sentences S1 … Sn to the truth-value of the concatenation of C with S1 … Sn.

According to that explication, the following operators are equally truth-functional:

T It is true that …

TF It is either true or false that …

EX It is expressible in English that …

The two latter operators are only special in corresponding to a constant truth-function, namely the function that maps both True and False unto True.

But one may sense an important difference between TF and EX: the former connective actually operates on the truth-values of the embedded sentence such that the truth-value of the complex sentence is a result of that operation. But the latter connective does not operate on the truth-value of the embedded sentence at all. That it corresponds to a constant truth-function is not the result of its being sensitive to the truth-value of the embedded sentence; in some sense, it does not depend on that truth-value.

A definition of truth-functionality that captures the described difference between TF and EX can be given in terms of the explanatory connective ‘because of’:

An n-ary connective C is truth-functional iff the concatenation of C with any sentences S1 … Sn has the truth-value it has because of the truth-values of S1 … Sn.

This definition will classify the standard connectices of propositional logic as truth-functional (a true negation, for instance, is true because the negated sentence is false, and a false negation is false because the negated sentence is true). And it also classifies TF as truth-functional (‘It is either true or false that snow is white’ is true because ‘Snow is white’ is true; had the latter been false, then the former had been true because of that.)

But EX does not count as truth-functional in the defined sense (‘It is expressible in English that snow is white’ is true, but not because ‘Snow is white’ is true).

While I do not think that these considerations show that the standard definition of truth-functionality is in any way flawed (it is a technical notion after all), the alternative definition captures differences between operators that might as well be associated with the term ‘truth-functional’.

(See the papers-section for a more detailed exposition of my proposal.)

Posted by Benjamin.

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