Non-extensionality in Free Logic

The language of science contains many singular terms not referring to an existent, i.e., terms that are empty (e.g., `Vulcan`, `1/0`). In contrast to classical logic, free logic allows for empty singular terms. However, this raises the question whether in simple statements with empty singular terms - like `Vulcan rotates` - linguistic expressions which have the same extensional meaning (in short: extension) are always substitutable for each other without change of the extension (i.e., whether such statements are extensional). Lambert, one of the founders of free logic, has argued that in such statements general terms which have the same extension are not always substitutable for each other without change of the truth-value as extension (i.e., he has argued that they are non-extensional). In our project this argument is analyzed. In doing so two new semantical systems for a language L of free logic are developed. Their basic assumption is that the extension of a statement is not a truth-value, but composed of the extensions of its singular as well as general terms and sub-statements, and that the composition depends on how its truth-value is determined (we call such complexes abstract states of affairs). In the first state-of-affairs semantics empty singular terms are assigned non-existents. Interpreting L according to this semantics allows for a justification of a positive answer to our question. By contrast, in the second state-of-affairs semantics empty singular terms are not assigned anything at all. Interpreting L according to this semantics allows for a justification of the thesis that in most statements of L expressions which have the same extension are always substitutable for each other without change of the state of affairs as extension. The few exceptions are sharply identifiable. However, and that is our point, Lambert`s argument does not justify the non-extensionality of these exceptions. His argument depends, of course, on the critical assumption that a statement like `Vulcan is a thing such that it rotates and exists` has the same truth-value as extension as the statement `Vulcan is a thing such that it rotates and Vulcan is a thing such that it exists`. Hence, both statements have in Lambert`s approach the same extension. However, in our second state-of-affairs semantics both statements have entirely different states of affairs as extensions. Hence, they have in our approach not the same extension. Therefore, Lambert`s argument is subverted if L is interpreted by means of the second state-of-affairs semantics. In this case this argument neither justifies his thesis that Quine`s theory of predication in Word and Object is non-extensional, nor his thesis that classical logic gets non-extensional as soon as its existence assumptions in regard to singular terms are explicated.