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Predicate Calculus
Published in Janet Woodcock, Software Engineering Mathematics, 1988
This is really just a statement of the fact that we cannot use the same name to denote two different objects. We will allow ourselves the luxury of using this rule for particular terms without always having to introduce the quantified form and then removing the quantifier, thus we will write down n = n directly if we need it. This is sometimes referred to as an axiom schema since we can view the quantified form as a schema, or template, from which we can generate an infinite number of other axioms by instantiation with particular names.
Metric dynamic equilibrium logic
Published in Journal of Applied Non-Classical Logics, 2023
Arvid Becker, Pedro Cabalar, Martín Diéguez, Luis Farinas del Cerro, Torsten Schaub, Anna Schuhmann
Moreover, the first item (along with Proposition 2.2) implies that any MDHT tautology is also an MDL tautology, so the former constitutes a weaker logic. To show that, in fact, MDHT is strictly weaker, note that it does not satisfy some classical tautologies like the excluded middle, while MDL is a proper extension of classical logic. In fact, the addition of the axiom schema forces total models and so, makes MDHT collapse to MDL. Propositions 2.2 and 4.1 imply that φ is MDHT satisfiable iff it is MDL satisfiable.
Truth without standard models: some conceptual problems reloaded
Published in Journal of Applied Non-Classical Logics, 2018
The resulting theory is known as PAᴌ. However, the set of theorems of PA (with the axiom schema) over classical logic and the set of theorems of PAᴌ, replacing the axiom schema of induction by the induction rule, over ᴌukasiewicz logic are exactly the same (cf. Hàjek et al., 2000; Restall, 1992). Therefore, for simplicity, we will refer to both theories as PA.13
A family of genuine and non-algebraisable C-systems
Published in Journal of Applied Non-Classical Logics, 2021
Mauricio Osorio, Aldo Figallo-Orellano, Miguel Pérez-Gaspar
We denote to the Hilbert calculus of plus the following axiom schema: .