All of them are theorems about first-order model theory.
First order logic and second-order logic are in a sense two opposite extremes.
Algebraic logic then broadened its interests to first order logic and modal logic.
Nonetheless there are many results for first-order probability logic.
Hilbert concludes his discussion of first-order logic with the remark:
These technical results were of great importance for the subsequent debate over first-order logic.
One reason why first order logic has such a rich model theory is that first order logic is relatively weak.
Together, these last two points highlight just how central classical first-order logic is to Skolem's Paradox.
Any sentence of a first-order sentence comes with a certain quantificational depth —the number of its overlapping quantifiers.
Independence friendly logic (IF logic, IF first-order logic) is an extension of first-order logic.
With general models second-order logic has similar model theoretic properties as first order logic, as it can simply be thought of as many sorted first order logic (see §9.1 and Manzano 1996).
The paper Barbosa, Martins, and Carreteiro (2014) gives an axiomatization of a fragment of first-order hybrid logic called equational first-order hybrid logic.
First-order sentences lack the expressive power of first-order theories in general, and first-order languages lack the expressive resources of higher-order languages.
The translations above can be extended to first-order hybrid logic, in which case the relevant target logic is two-sorted first-order logic with equality, one sort for worlds and one sort for individuals, see Chapter 6 of Braüner (2011a).
As we will see in Section 3.6, however, not all first-order properties of temporal frames are definable by temporal formulae; and vice versa, not all properties of temporal frames that can be expressed by formulae of TL are first-order definable.
It seems wrong that second-order evidence should always swamp the first-order verdict completely, so the calibration idea has been re-formulated so as to incorporate dependence on first-order evidence explicitly, in the evidential calibration constraint (EC):
The syntax and semantics above can be extended in a number of ways, in particular, first-order machinery can be added (of course, an equivalent way to obtain first-order hybrid logic is by adding hybrid-logical machinery to first-order modal logic).
Kripke’s 1962 “The Undecidability of Monadic Modal Quantification Theory” develops a parallel between first-order logic with one dyadic predicate and first-order monadic modal logic with just two predicate letters, to prove that this fragment of first-order modal logic is already undecidable.
That is, there are sentences of first-order arithmetic that can be deduced from the second-order induction axiom (together with the other axioms of arithmetic, which are common to first-order and second-order arithmetic) but not from the instances of the first-order induction schema (see Shapiro 1991: 110).
To expound them we will assume knowledge of first-order logic (see the entries on classical logic and first-order model theory) and we will call algebraic first-order languages, or simply algebraic languages, the first-order languages with equality and without any relational symbols, so that these languages have only operation symbols (also called function symbols), if any, in the set of their non-logical symbols.
first-order
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To expound them we will assume knowledge of first-order logic see the entries on classical logic and first-order model theory and we will call algebraic first-order languages or simply algebraic languages the first-order languages with equality and without any relational symbols so that these languages have only operation symbols also called function symbols if any in the set of their non-logical symbols