Many intuitively valid arguments involving intensionality cannot be captured by first-order logic, even when extended by modal and epistemic operators. Indeed, previous attempts at providing an adequate treatment of the phenomenon of intensionality in logic and language, such as those of Frege, Church, Russell, Carnap, Quine, Montague and others are fraught with numerous philosophical and technical difficulties and shortcomings. We present Bealer's solution to this problem which\n hinges on an ontological commitment to theory of Properties, Propositions and Relations (PRP). At the most basic level we can distinguish two conceptions in the theory of PRPs. An objective one tied to modality and necessary equivalence, and a mental (intentional) one tied to concepts and the requirement of non-circularity in definitions. Building on the work of Russell, Church and Quine, Bealer proposes two distinct intensional logics T1 and T2 (presented in Hilbert form)\n corresponding to these two conceptions, both based on the language of first-order logic extended with an intensional abstraction operator. In T1 necessitation can be directly defined and the axioms entail that we obtain standard S5 modal logic. These logics have a series of striking features and desirable aspects which set them apart from higher-order approaches. Bealer constructs a non-Tarskian algebraic semantic framework, distinct from possible worlds semantics and its problematic\n ontological commitments, yielding two classes of models for which T1 and T2 are both sound and complete. Other features include being able to deal with quantifying-in, and the various substitution puzzles, being free from artificial type restrictions, having a Russellian semantics, satisfying Davidson's\n learnability requirement, etc. Bealer proposes his logic as the basis of a larger philosophical project in the tradition of logicism (or logical realism) concerning which we refer to his book Quality and Concept (1982). This includes a neo-Fregean logicist foundation of arithmetic and set-theory in which various (according to him) purely logical predication axioms ( and intensional analogues of ZF, NGB, or Kelley-Morse axioms) are adjoined to T2, thereby explaining incompleteness as a property of pure logic rather than of mathematics. Surprisingly, and rather ironically, Bealer's logic also fulfills Carnap's thesis of extensionality due precisely to its ontological commitment to the reality of PRPs. The proof of these results consists either in lemmas which are merely stated or which are given but brief sketches of a proof. We aim to give detailed proofs of all the mathematical logical results that appear in Bealer's \\emph{Quality and Concept} and in \\cite{C} and to clarify and simplify some of the concepts and techniques so as to bring Bealer's work to a larger audience of philosophers, logicians, linguists and mathematicians and to be better equipped to address some of the unsolved problems and challenges. We also include a brief introduction to other approaches to intensionality in natural language and discuss how Bealer's approach compares favourably to some of them and is likely to benefit from the insights offered by others.\n