Predication, Truth, and Extensionality in Free Logic

A free logic (= FL) is a logic free of existence assumptions with respect to its singular and general terms, but whose quantifiers are understood as in classical predicate logic (= CPL). FL admits in contrast to CPL singular terms not referring to an existent (e.g. `Vulcan`, `1/0`). However, this creates problems not arising thus in CPL at all and which are investigated in the present project - as for example the truth-value problem whether simple statements with such empty singular terms are true or false or truth-valueless, and the extensionality problem whether such statements are extensional in any substitutivity sense. The species of FL can be classified according to the different answers to the truth-value problem: Thus in positive FL some simple statements with empty singular terms are true, in negative FL all such statements are false and in neutral FL all such statements are truth- valueless (except, perhaps, statements like `Vulcan exists` which are false). Many different semantic approaches to as well as theories of predication for FL are known. However, every theory of predication requires a formal theory of general terms. Therefore, an aim of this project is the development of such a theory that considers the difference, which is important in natural science, between characterizing and classifying an object. Whereas a characterizing statement like `Vulcan is round` doesn`t imply the existence of Vulcan, a classifying statement like `Vulcan is a planet` does. Considering this difference leads to different truth conditions for such statements. Their conversion into one single formal theory of general terms allows to harmonize positive and negative FL and leads thereby to two different theories of predication. Another aim is the formal reconstruction of an argument for the thesis that in simple statements with empty singular terms co-extensive expressions aren`t always substituteable for each other without changing the truth-value as extension. This non-extensionality phenomenon is especially troublesome for certain applications of FL in computer sciences (as for example in the formalization of the logic of program specification and verification). Therefore, it is attempted in this project to subvert this argument and specifically the question is investigated whether the extensionality of not existence implying, respectively, existence implying simple statements can be guaranteed under the same conditions. A further aim is the development of varieties of a semantics with structured extensions for statements on the model of the different varieties of single-domain semantics as well as the formal theory of general terms for FL. Thus the conditions should be found under which a language for FL is extensional in the substitutivity sense, which role existence assumptions thereby play, and whether such structured extensions can be understood as abstract states of affairs.

A free logic (= FL) is a logic free of existence assumptions with respect to its singular and general terms, but whose quantifiers are understood as in classical predicate logic (= CPL). FL admits in contrast to CPL singular terms not referring to an existent (e.g. `Vulcan`, `1/0`). However, this creates problems not arising thus in CPL at all and which are investigated in the present project - as for example the truth-value problem whether simple statements with such empty singular terms are true or false or truth-valueless, and the extensionality problem whether such statements are extensional in any substitutivity sense. The species of FL can be classified according to the different answers to the truth-value problem: Thus in positive FL some simple statements with empty singular terms are true, in negative FL all such statements are false and in neutral FL all such statements are truth- valueless (except, perhaps, statements like `Vulcan exists` which are false). Many different semantic approaches to as well as theories of predication for FL are known. However, every theory of predication requires a formal theory of general terms. Therefore, an aim of this project is the development of such a theory that considers the difference, which is important in natural science, between characterizing and classifying an object. Whereas a characterizing statement like `Vulcan is round` doesn`t imply the existence of Vulcan, a classifying statement like `Vulcan is a planet` does. Considering this difference leads to different truth conditions for such statements. Their conversion into one single formal theory of general terms allows to harmonize positive and negative FL and leads thereby to two different theories of predication. Another aim is the formal reconstruction of an argument for the thesis that in simple statements with empty singular terms co-extensive expressions aren`t always substituteable for each other without changing the truth-value as extension. This non-extensionality phenomenon is especially troublesome for certain applications of FL in computer sciences (as for example in the formalization of the logic of program specification and verification). Therefore, it is attempted in this project to subvert this argument and specifically the question is investigated whether the extensionality of not existence implying, respectively, existence implying simple statements can be guaranteed under the same conditions. A further aim is the development of varieties of a semantics with structured extensions for statements on the model of the different varieties of single-domain semantics as well as the formal theory of general terms for FL. Thus the conditions should be found under which a language for FL is extensional in the substitutivity sense, which role existence assumptions thereby play, and whether such structured extensions can be understood as abstract states of affairs.