In the first chapter I have introduced Carnapian intensional logic again st the background of Frege s and Quine s puzzles. The main body of the d issertation consists of two parts. In the first part I discussed Carnapi an modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. F irst, they are formulated in languages containing description terms. Sec ond, they contain a syst…
Read moreIn the first chapter I have introduced Carnapian intensional logic again st the background of Frege s and Quine s puzzles. The main body of the d issertation consists of two parts. In the first part I discussed Carnapi an modal logic and arithmetic with descriptions. In the second chapter, I have described three Carnapian theories, CCL, CFL, and CNL. All three theories have three things in common. F irst, they are formulated in languages containing description terms. Sec ond, they contain a system of modal logic. Third, they do not contain th e unrestricted classical substitution principle, but they do contain the classical substitution principle restricted to non-modal formulas and t he Carnapian substitution principle, which says that two terms can be s ubstituted salva veritate if they are necessarily coreferential. There a re two major differences between the three theories. First, CCL and CFL allow universal instantiation with description ter ms, whereas CNL does not. Moreover, the quantificational theo ry of the CCL is classical, whereas the quantificational theo ry of CFL is a free logic. Another difference is t hat CCL and CFL contain different description principles. Most import antly, the description principle of CCL ensures that even imp roper descriptions have a denotation, whereas the description principle of CFL does not guarantee this. CNL does not have a description prin ciple. In the third chapter, I have studied collapse arguments for CCL, CFL, and CNL. A collapse argument is an argum ent for the following statement: if p is true, then it is nec essarily true. A crucial role in the proofs of these collapse results wa s played by so-called self-predication principles, which say that unde r certain conditions the predicate that expresses the descriptive condition can be combined by the description term formed ou t of that predicate with the result being a true sentence. In this chapt er I have discussed a collapse argument for the extension of CCL with a self-predication principle, I have given a collapse argument for a similarly extended CFL, and most importantly, I have gi ven a collapse argument for the extension of CNL with a self- predication principle. Finally, I have argued that the relevant self-pre dication principles are unsound under a Carnapian interpretation. In the fourth chapter, I have studied the extension of Peano Arithmetic with a Carnapian modal logic C, which is a dummy l etter standing for either CCL or CFL. One can prov e that the principle of the necessity of identity is a theorem of CPA. This implies that one gets a collapse result for CPA. The standard principle of weak induction was crucial for the proof. O ne can also prove that, if one assumes a particular self-predication pri nciple, and if one assumes the principle of strong induction or, equivalently, the least-number principle, then one gets a partial collap se of de re modal truths in de dicto modal tr uths. I have argued that, if the box operator is interpreted as a metaph ysical necessity operator, then Platonists would not be inimical to the collapse result. But if CPA is extended with a physical theor y, then there is a threat that physical truths become physical necessiti es. It was shown that, under a Carnapian interpretation, the standard pr inciple of weak induction is unsound, and that it can be replaced by a C arnapian principle of weak induction that is sound. The probl em of logical and mathematical omniscience prevents ordinary Carnapian i ntensional logic from being taken seriously as a logic adequate for desc ribing the principles of demonstrability. Yet many of the proof-theoreti c results of the first part carry over to the part on Carnapian epistemi c arithmetic with descriptions, since proof-theoretic results are indepe ndent of the informal reading of the operators. In the fifth chapter, I looked at extensions of arithmetic with a modal logic in which the box operator is interpreted as a demonstrability oper ator. A first extension in that sense is Shapiro s Epistemic Arithmetic. Shapiro himself offered the problem of mathematic al omniscience as a reason why it is difficult to find a model theory fo r EA.Horsten attempted to provide a model theory via the deto ur of Modal-Epistemic Arithmetic. The attention of the reade r was drawn to an incoherence in the model theory of. Two al ternative solutions were presented and, after a short discussion of the problem of de re demonstrability one of those alternatives wa s chosen. The discussion of the problem of de re demonstrabil ity made it clear that it would be interesting to study the epistemic pr operties of notation systems. Horsten himself provided a framework for t his, viz. Carnapian Epistemic Arithmetic, and he started a systematic study of the epistemic properti es of notation systems within that framework. However, he did not provid e non-trivial but adequate models. To make a start with solving the prob lem of finding good models for CEA, I introduced Carnapian Mo dal-Epistemic Arithmetic In constructing CMEA I incorporated the lesson about the principle of weak induction learnt in the fourth chapter. In the sixth chapter, I gave a critical assessment of an argument concerning the limits of de re demonst rability about the natural numbers. The conclusion of the Description Ar gument is that it is undemonstrable that there is a natural number that has a certain property but of which it is undemonstrable that it has tha t property. A crucial step in the Description Argument involved a self-p redication principle. Making good use of one of the results obtained in the third chapter, I proved a collapse result for the background theory against which the Description Argument was formulated. I concluded that either the either the Description Argument is sound but its conclusion i s trivial, o r the Description Argument is unsound, or it is a cheapshot. As an appendix I included an article co-authored by prof. dr. Leon Horst en and me. The topic of the article is indirectly related to some other topics investigated in my dissertation. Also, it backs up one of the addition al theses I might be asked to publicly defend during my doctoral exam. T he topic of the appendix is the set of the so-called paradoxes of stric t implication. Jonathan Lowe has argued that a particular variation on C.I. Lewis notion of strict implication avoids the paradoxes of strict implication. Pace Lowe, it is argued that Lowe s notion of implication d oes not achieve this aim. Moreover, a general argument is offered to the effect that no other variation on Lewis notion of constantly strict imp lication describes the logical behaviour of natural language conditional s in a satisfactory way.