To gain an intuition, we explore various equivalent notions of the essence of classical reasoning including the Law of the Excluded Middle and Double- Negation Elimination. To understand this, one needs to see the matter in its historical context. In other words, we refute Birkhoff and von Neumann's classic thesis that the logic (the formal character) of Quantum Mechanics would be non-classical as well as Putnam's thesis that quantum logic (of his kind) would be the correct logic for propositional inference in general. (ii) Double negation elimination: ¬¬φ → φ. It has been shown to be sound and complete if one interprets the relations expressed by the categorical sentences set-theoretically as a system of non-empty classes as follows: A a B is true if and only if the class A contains the class B . Deutsch’s three-qubit generalization of the Toffoli gate 2. Either φ holds, or φ does not hold. MODERN LOGIC: SINCE GÖDEL: THE PROLIFERATION OF NONCLASSICAL LOGICS The twentieth century, and especially its second half, was marked by a fairly spectacular proliferation of what are sometimes called nonclassical logics. Fuzzy Quantum Logic is, de ned by Pykacz, partial and in nite - valued (V(P) 2[0;1]) which is the connection to a probablistic interpretation of quantum mechanics. Objectives: (1 ) To introduce the modern classification of propositions and the branches of symbolic logic , and to show the relation between the two. Classical Logic 15-317: Constructive Logic William Lovas Lecture 7 September 15, 2009 1 Introduction In this lecture, we design a judgmental formulation of classical logic. If it is not the case that φ does not hold, then φ holds. 2.3 Explosion in Modern Logic. modern logic as a development of classical logic. From a modern perspective, Aristotle’s system can be understood as a sequent logic in the style of natural deduction and as a fragment of first-order logic. Bits and Boolean functions vs. Qubits and unitary matrices 2. well before the beginning of mathematical logic, which only started in the 19th century, and was only finalised in the 1930’s. Graham Priest, in Handbook of the History of Logic, 2007. Astronomy could predict the orbits of planets using Newton’s law of gravitation already in the 18th century. Things changed dramatically with the rise of modern logic at the end of the 19th century. With Fuzzy Quantum Logic we can analyze non - tested experimental sentences. Fuzzy Quantum Logic Jaroslaw Pykacz used the fuzzy sets idea to build Quantum Logic. Lovre Grisogono Classical Logic vs. Quantum Logic. For the logical theory invented by Frege, and subsequently taken up by Russell — classical logic — is explosive. Universal Quantum Logic Gates (1989, 1995-) ----- p.12 1. Classical Logic (1980s) ----- p. 8 1. Classical reversible logic contained in quantum logic B. (iii) Peirce’s law: ((φ → ψ) → φ) → φ. Thus our classical logic is a completion of the quantum logic QL. Sufficiency of two-qubit gates for quantum computation 3. A. Quantum vs. (iv) Reductio ad absurdum: (¬φ → φ) → φ. Classical logic differs from intuitionistic logic by embracing these principles: (i) Tertium non datur (law of the excluded middle): φ∨¬φ.

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