Farzad Didehvar

For a subset K ⊆ [0, 1], the notion of K-satisfiability is a generalization of the usual satisfiability in first order fuzzy logics. A set Γ of closed formulas in a first order language τ is K-satisfiable, if there exists a τ-structure such that ∥ σ ∥ ∈ K, for any σ ∈ Γ. As a consequence, the usual compactness property can be replaced by the K-compactness property. In this paper, the K-compactness property for Łukasiewicz first order logic is investigated. Using the ultraproduct construction, it is proved that for any closed subset K and set Γ of closed formulas, Γ is K-satisfiable if and only if it is finitely K-satisfiable

In this paper, we introduce a foundation for computable model theory of rational Pavelka logic and continuous logic, and prove effective versions of some related theorems in model theory. We show how to reduce continuous logic to rational Pavelka logic. We also define notions of computability and decidability of a model for logics with computable, but uncountable, set of truth values; we show that provability degree of a formula with respect to a linear theory is computable, and use this to carry out an effective Henkin construction. Therefore, for any effectively given consistent linear theory in continuous logic, we effectively produce its decidable model. This is the best possible, since we show that the computable model theory of continuous logic is an extension of computable model theory of classical logic. We conclude with noting that the unique separable model of a separably categorical and computably axiomatizable theory is decidable.

Throughout this paper, by representing some paradoxes and their associated proofs and arguments, we try to show the cases which proving some assertions doesn’t conclude the truth of them. In the next step, we try to find out Which proofs could be considered as reliable in a way that it shows the Truth of their related assertion, specially We claim that math- metical proofs could be considered as reliable ones in this sense. Nevertheless, we claim that the validation of the previous assertion is comparable to validation of Church Thesis in Computability Theory. So, we could use it and we consider these proofs as reliable ones but we should be doubtful about their truth. At last we give a possible explanation for this subject (based on local arguments and global arguments, the concepts which we introduce them throughout this paper), and we investigate the validity of this explanation. What we should bold it here is: In this approach we know the proofs in these paradoxes as correct proofs, but the point which we want to stress on that would be: the proof doesn’t show the truth..