An Algebraic Introduction to Mathematical Logic

Boolos‐style proofs of limitative theorems

“…Consequently, T ⊢ ¬ϕ(n, t). This, however, leads to a contradiction since, by (2) in the proof of the Theorem, ¬ϕ(n, t) is true, i.e. it is a true Σ sentence.…”

“…2 In order to proceed, let us observe that, though Boolos's proof is essentially a formalization of the Berry paradox, it is not the most straightforward one. As a matter of fact, the theorem that can be considered as the most faithful formal version of the Berry paradox is Tarski's theorem on the undefinability of truth.…”

The validity degree vectors of formulae in two-valued predicate logic

“…The set consisting of all valuations of F(S) will be denoted by Ω. It is well known that every valuation υ of F(S) is completely determined by its restriction υ|S : S → {0, 1} because F(S) is the free algebra generated by S [42] . Suppose that (X n , A n , µ n ) is a probabilistic measure space, where µ n is a probability measure on X n , and A n is the family consisting of all µ n -measurable sub-…”

PART I. General Considerations

“…Our guiding theme will be to treat logic as algebra, following a well-established (and profitable) tradition; [1], [9] and [12] are good introductions. We will phrase the discussion in the language of universal algebra, to make it broad enough to include not only traditional logics, but also fuzzy logic and the logic of quantum computing; see [15], [16], [20], [8], [13] and [14] for introductions to these.…”