Second order logic Wikipedia. In logic and mathematicssecond order logic is an extension of first order logic, which itself is an extension of propositional logic. Second order logic is in turn extended by higher order logic and type theory. First order logic quantifies only variables that range over individuals elements of the domain of discourse second order logic, in addition, also quantifies over relations. For example, the second order sentence PxxPxPdisplaystyle forall P,forall xxin Plor xnotin P says that for every unary relation or set P of individuals, and every individual x, either x is in P or it is not this is the principle of bivalence. Second order logic also includes quantification over functions, and other variables as explained in the section Syntax and fragments below. Both first order and second order logic use the idea of a domain of discourse often called simply the domain or the universe. The domain is a set over which individual elements may be quantified. Boolos Computability And Logic Pdf' title='Boolos Computability And Logic Pdf' />Syntax and fragmentsedit The syntax of second order logic tells which expressions are well formed formulas. In addition to the syntax of first order logic, second order logic includes many new sorts sometimes called types of variables. These are A sort of variables that range over sets of individuals. If S is a variable of this sort and t is a first order term then the expression t S also written St, or St to save parentheses is an atomic formula. Sets of individuals can also be viewed as unary relations on the domain. For each natural number k there is a sort of variables that ranges over all k ary relations on the individuals. If R is such a k ary relation variable and t. Boolos Computability And Logic Pdf' title='Boolos Computability And Logic Pdf' />Rt. For each natural number k there is a sort of variables that ranges over all functions taking k elements of the domain and returning a single element of the domain. If f is such a k ary function variable and t. Each of the variables just defined may be universally andor existentially quantified over, to build up formulas. Thus there are many kinds of quantifiers, two for each sort of variables. A sentence in second order logic, as in first order logic, is a well formed formula with no free variables of any sort. Versions of this paper were presented at UNILOG 2013World Congress on Universal Logic, Rio de Janiero, Brazil Institut Jean Nicod, Paris, France Sociedad Argentina. Its possible to forgo the introduction of function variables in the definition given above and some authors do this because an n ary function variable can be represented by a relation variable of arity n1 and an appropriate formula for the uniqueness of the result in the n1 argument of the relation. Shapiro 2. 00. 0, p. Monadic second order logic MSO is a restriction of second order logic in which only quantification over unary relations i. Quantification over functions, owing to the equivalence to relations as described above, is thus also not allowed. The second order logic without these restrictions is sometimes called full second order logic to distinguish it from the monadic version. Monadic second order logic is particularly used in the context of Courcelles theorem, an algorithmic meta theorem in graph theory. Art Machine Software. Boolos, G. 1971. The iterative conception of set. The Journal of Philosophy, 688, 215231. Reprinted in G. Boolos 1998. Logic, logic, and logic. Just as in first order logic, second order logic may include non logical symbols in a particular second order language. These are restricted, however, in that all terms that they form must be either first order terms which can be substituted for a first order variable or second order terms which can be substituted for a second order variable of an appropriate sort. A formula in second order logic is said to be of first order and sometimes denoted 0. Sigma 01 or 0. Pi 01 if its quantifiers which may be of either type range only over variables of first order, although it may have free variables of second order. A 1. 1displaystyle Sigma 11 existential second order formula is one additionally having some existential quantifiers over second order variables, i. R0Rmdisplaystyle exists R0ldots exists Rmphi, where displaystyle phi is a first order formula. The fragment of second order logic consisting only of existential second order formulas is called existential second order logic and abbreviated as ESO, as 1. Sigma 11, or even as SO. The fragment of 1. Pi 11 formulas is defined dually, it is called universal second order logic. More expressive fragments are defined for any k 0 by mutual recursion k1. Sigma k11 has the form R0Rmdisplaystyle exists R0ldots exists Rmphi, where displaystyle phi is a k. Pi k1 formula, and similar, k1. Pi k11 has the form R0Rmdisplaystyle forall R0ldots forall Rmphi, where displaystyle phi is a k. Sigma k1 formula. See analytical hierarchy for the analogous construction of second order arithmetic. SemanticseditThe semantics of second order logic establish the meaning of each sentence. Unlike first order logic, which has only one standard semantics, there are two different semantics that are commonly used for second order logic standard semantics and Henkin semantics. In each of these semantics, the interpretations of the first order quantifiers and the logical connectives are the same as in first order logic. Only the ranges of quantifiers over second order variables differ in the two types of semantics Vnnen 2. In standard semantics, also called full semantics, the quantifiers range over all sets or functions of the appropriate sort. Thus once the domain of the first order variables is established, the meaning of the remaining quantifiers is fixed. It is these semantics that give second order logic its expressive power, and they will be assumed for the remainder of this article. In Henkin semantics, each sort of second order variable has a particular domain of its own to range over, which may be a proper subset of all sets or functions of that sort. Leon Henkin 1. 95. Gdels completeness theorem and compactness theorem, which hold for first order logic, carry over to second order logic with Henkin semantics. This is because Henkin semantics are almost identical to many sorted first order semantics, where additional sorts of variables are added to simulate the new variables of second order logic. Second order logic with Henkin semantics is not more expressive than first order logic. Henkin semantics are commonly used in the study of second order arithmetic. Vnnen 2. 00. 1 argued that the choice between Henkin models and full models for second order logic is analogous to the choice between ZFC and V as a basis for set theory As with second order logic, we cannot really choose whether we axiomatize mathematics using V or ZFC. The result is the same in both cases, as ZFC is the best attempt so far to use V as an axiomatization of mathematics. Expressive powereditSecond order logic is more expressive than first order logic. For example, if the domain is the set of all real numbers, one can assert in first order logic the existence of an additive inverse of each real number by writing x y x y 0 but one needs second order logic to assert the least upper bound property for sets of real numbers, which states that every bounded, nonempty set of real numbers has a supremum. If the domain is the set of all real numbers, the following second order sentence split over two lines expresses the least upper bound property A w w A z uu A u z. A w y x yThis formula is a direct formalization of every nonempty, bounded set A has a least upper bound. It can be shown that any ordered field that satisfies this property is isomorphic to the real number field. On the other hand, the set of first order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least upper bound property cannot be expressed by any set of sentences in first order logic.