Second order logic allows various interpretations;It is often considered included in thesubsetOn, or on a function from this field to itself, not just on individual members of this field.For example, if this field is allreal numberOfaggregate, you can write inFirst-order logicThe Existence of Additive Inverses of Each Real Number in Discontinuity
But you need to use second-order logic to assert the minimum upper bound property of real numbers:
And insert a statement at the point. If A is non empty and has an upper bound in R, then A has a minimum upper bound in R.
staymathematical logicIn, second-order logic is a propositionLogical ORThe extension of first-order logic, which includes variables in the predicate position (rather than the position of items as first-order logic can only do), and quantifiers that constrain them.Therefore, we can express Jones' binary principle: for all properties, Jones either has it or does not have it.
The expressive power of various forms of second-order logic is closely connected withComputational complexity theory。especially:NPIt is a set of languages that can be expressed by existential second-order logic.Co NP is a set of languages that can be expressed by the full name of second-order logic.PH is a set of languages that can be expressed by second-order logic.PSPACEIt is added withTransitive closureThe set of languages that can be expressed by the second-order logic of operators.EXPTIMEIs a set of languages that can be expressed by second order logic with an increasing minimum fixed point operator.The connection between these language classes directly affects the relative expressive power of logic;For example, ifPH=PSPACE, the transitive closure operator added to second-order logic does not make it more expressive.
History and value
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WhenPredicate logiccoverfrege (Independent and more influential Peirce, who proposed the term second-order logic) When introduced to the mathematical community, he did use different variables to distinguish between quantification on objects and quantification on attributes and sets;But he himself did not distinguish between two different kinds of logic.Found onRussell paradoxLater, he realized that there was something wrong with his system.In the end, logicians established Frege logic to limit in various ways calledFirst-order logic- Eliminates the problem that sets and predicates cannot be quantified separately in first-order logic.The order level of standard logic began at that time.
Foundset theoryIt can be formulated into an axiomatic system in the facility of first-order logic (someCompleteness, but not as bad as Russell's paradox), and really did so (see Zermelo Fraenkel set theory), because sets are the key to mathematics.Arithmetic, mereology and various other powerful logic theories can be formulated by axiomatization without using more logic facilities than first-order quantization. With Godel and Skolem's loyalty to first-order logic, the work of second-order (or higher) logic was generally abandoned.
This abandonment is actively promoted by some logicians, the most famous of which isquine 。Quine advanced this view. In predicate language sentences such as Fx, "x" is considered to be a variable or the name of an object, so it can be quantified, such as "for all things, the case is...".But "F" is considered an abbreviation of an incomplete sentence, not the name of an object (or even an abstract object such as a property).For example, it may mean "... is a dog". It is meaningless to think that quantification can be done on such things.(This position is the same asfrege My discussion on the concept object difference is very consistent).Therefore, if you want to use a predicate as a variable, you need to make it occupy the position of a name that only individual variables can occupy.This reasoning was rejected by Boolos.
present situation
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In recent years, second-order logic has been restored to some extent. George Boolos interpreted second-order quantization as supported by complex quantization in the same domain of the same order quantization.Boolos further pointed out the non first-order expressiveness of sentences, such as "some criminals only admire each other" and "some Fianchetto people enter the warehouse without any escort".This can only be expressed in terms of the complete power of second-order quantification.(In fact, this is not true, because general quantification and partial (or branched) quantification are also sufficient to express specific types of non first order expressible sentences without using second order quantification).
However, it has been said that in some branches of mathematics, such astopologyThe ability of second-order logic is required for complete expression.This work has been completed by Stephen G. Simpson in the name of inverse mathematics.It has been proved that second-order logic is not only necessary to express some important parts of classical mathematics, but also can be used asModel theoryAnd basic mathematical tools.
Examples of mathematical propositions
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Riemann conjecture is a second order logic problem
Riemann conjectured that all "zeros" are a set, and zeros are functions on this object. According to the usual mathematical definition, an n-ary function is a mapping from the set of all n-ary groups of individuals of universe A to A.When we use "all individuals" and "existential individuals", the quantifier is added to the individual of the universe, which is calledFirst order quantifier。“[1]
”All functions "," existential functions "," all relations "and" existential relations "are second order quantifiers, that is, second order logic.Riemann's "all zeros" is the second order quantifier of "all functions".
Riemann conjecture has gone beyond the first-order logic formal system (predicate calculus) established by G Frege, and involves extremely complex logic system, which is unknown to ordinary mathematicians.
If you can't understand second-order logic, let me make a metaphor. "Acceleration" is not a basic quantity (such as length or mass), it is the second-order rate of change, that is, the rate of change.There are also three body problems (moon, earth, sun) and multi-body problems in the second-order logic of physics, which cannot be solved at one time.[2]
Riemann conjecture means that the necessary and sufficient condition for all A (zeros) to hold is that B (x=1/2 when s=x+yi) in A is held.
When all the principal terms can be established, the proposition that must depend on the predicate is a second-order logical proposition。All mathematical theorems are first-order logic problems.
Transcendental number problem is a second order logic problem
There are so-called "transcendental numbers" in mathematics, which are more irrational than irrational numbers, such as pi=3.1415926535898.... And e=2.718281828459.
Why can't people get an accurate value?
In circle cutting, Pythagorean theorem is constantly used to calculate the side length of a positive N-polygon. Every time N increases a value (first order change rate), it will cause a second order change.Because they are second-order change rates, for example, as long as you know the process of calculating pi, you will naturally know why.
The freight forwarder is in trouble
The millennium p=np problem is a second-order logic problem
Freeman Dyson wrote in "Frogs and Birds": Mathematics is becoming more and more complex as we continue to explore chaos and many new fields opened by computers.Mathematicians found the central puzzle of computability, which is expressed as P is not equal to NP.
This conjecture claims that there is such a mathematical problem that its cases can be solved quickly, but there is no fast algorithm applicable to all cases to solve all problems.
The most famous example of this problem is the travel salesman problem, that is, under the premise of knowing the distance between every two cities, find the shortest path for the salesman to travel between these series of cities.All experts believe that the conjecture is correct. The problem of travel salesman is that P is not equal to NP.But no one knows a clue to this problem.In the mathematical world of Herman Weir in the 19th century, this puzzle has not even formed.
The problem here is a second-order logic problem. Every increase in the number of cities n is a first-order change rate, and the distance between cities will have a second-order change rate.
