Mathematical logic is a discipline within mathematics, studying formal systems in relation to the way they encode intuitive concepts of proof and computation as part of the foundations of mathematics.
Although the layperson may think that mathematical logic is the logic of mathematics, the truth is rather that it more closely resembles the mathematics of logic. It comprises those parts of logic that can be modelled mathematically. Earlier appellations were symbolic logic (as opposed to philosophical logic), and metamathematics, which is now restricted as a term to some aspects of proof theory.
- 1 History
- 2 Topics in mathematical logic
- 3 Some fundamental results
- 4 Technical reference
- 5 References
- 6 External links
- 7 See also
Mathematical logic was the name given by Giuseppe Peano to what is also known as symbolic logic. In essentials, it is still the logic of Aristotle, but from the point of view of notation it is written as a branch of abstract algebra.
Attempts to treat the operations of formal logic in a symbolic or algebraic way were made by some of the more philosophical mathematicians, such as Leibniz and Lambert; but their labors remained little known and isolated. It was George Boole and then Augustus De Morgan, in the middle of the nineteenth century, who presented a systematic mathematical (of course non-quantitative) way of regarding logic. The traditional, Aristotelian doctrine of logic was reformed and completed; and out of it developed an adequate instrument for investigating the fundamental concepts of mathematics. It would be misleading to say that the foundational controversies that were alive in the period 1900-1925 have all been settled; but philosophy of mathematics was greatly clarified by the 'new' logic.
While the traditional development of logic (see list of topics in logic) put heavy emphasis on forms of arguments, the attitude of current mathematical logic might be summed up as the combinatorial study of content. This covers both the syntactic (for example, sending a string from a formal language to a compiler program to write it as sequence of machine instructions), and the semantic (constructing specific models or whole sets of them, in model theory).
Some landmark publications were the Begriffsschrift by Gottlob Frege, Studies in Logic by Charles Peirce, Principia Mathematica by Bertrand Russell and Alfred North Whitehead, and On Formally Undecidable Propositions of Principia Mathematica and Related Systems by Kurt Godel.
Topics in mathematical logic
The main areas of mathematical logic include model theory, proof theory and recursion theory (often now referred to as computability theory). Axiomatic set theory is sometimes considered too. There are many overlaps with computer science, since many early pioneers in computer science, such as Alan Turing, were mathematicians and logicians.
The Curry-Howard isomorphism between proofs and programs relates to proof theory; intuitionistic logic and linear logic are significant here. Calculi such as the lambda calculus and combinatory logic are nowadays studied mainly as idealized programming languages.
Some fundamental results
Some important results are:
- The set of valid first-order formulas is recursively enumerable. This follows from Gödel's completeness theorem (which establishes the equivalence of validity and provability), because the set of proofs for first-order logic formulas is recursively enumerable ("semi-decidable"). Therefore, there is a procedure that behaves as follows: Given a first-order formula as its input, the procedure eventually halts if the formula is valid or not valid, and runs forever otherwise. Some first-order theorem provers have this completeness property.
- The set of valid first-order formulas is not recursive, i.e., there is no algorithm for checking for universal validity. This follows from Gödel's incompleteness theorem.
- The set of all universally valid second-order formulas is not even recursively enumerable. This is also a consequence of Gödel's incompleteness theorem.
First-order languages and structures
Thus, in order to specify a language, it is often sufficient to specify only the collection of constant symbols, function symbols and relation symbols, since the first set of symbols is standard. The parentheses serve the only purpose of forming groups of symbols, and are not to be formally used when writing down functions and relations in formulas.
These symbols are just that, symbols. They don't stand for anything. They do not mean anything. However, that deviates further into semantics and linguistic issues not useful to the formalization of mathematical language, yet.
Yet, because it will indeed be necessary to get some meaning out of this formalization. The concept of model over a language provides with such a semantics.
Often, the word model is used for that of structure in this context. However, it is important to understand perhaps its motivation, as follows.
Terms, formulas and sentences
Hereafter, will denote a first-order language, will be an -structure with underlying universe set denoted by . Every formula will be understood to be an -formula.
In the case that is a sentence, that is, a formula with no free variables, the existence of a single v.a.f. for which immediately implies that .
Logical implication and truth
As a shortcut, when dealing with singletons, we often write instead of .
To say that a formula is valid really means that every -structure models .
The notion of substitutability of terms for variables corresponds to that of the preservation of truth after substitution is carried out in terms or formulas. Strictly speaking, substitution is always allowed, but substitutability will be imperative in order to yield a formula which meaning was not deformed by the substitution.
- A. S. Troelstra & H. Schwichtenberg (2000). Basic Proof Theory (Cambridge Tracts in Theoretical Computer Science) (2nd ed.). Cambridge University Press. ISBN 0521779111.
- George Boolos & Richard Jeffrey (1989). Computability and Logic (3rd ed.). Cambridge University Press. ISBN 0521007585.
- Elliott Mendelson (1997). Introduction to Mathematical Logic (4th ed.) Chapman & Hall.
- A. G. Hamilton (1988). Logic for Mathematicians Cambridge University Press.
- Model theory
- Computability logic
- Game semantics
- Provability logic
- Interpretability logic
- Sequent calculus
- Intuitionistic logic
- Predicate logic
- Theory of institutions
- Table of mathematical symbols
- Infinitary logicde:Mathematische Logik
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