Formal Systems

Formal systems serve as an important foundation in the study of mathematics and logic. At its core, a formal system is a framework consisting of symbols and inference rules that are used to form statements within a specific language and derive new truths. In proof theory, which is a branch within the foundations of mathematics, formal systems are treated as syntactic models that focus on the structure of logical expressions without evaluating their content.

Components of formal systems

A formal system typically has a few essential components:

• Alphabet: A finite set of symbols.
• Language: A set of sentences or expressions constructed from an alphabet following specific grammatical rules.
• Axioms: A set of expressions accepted as a starting point or true without proof.
• Inference rules: logical rules that determine how new strings or expressions can be derived from existing strings or expressions.

Example: Let us explain with a simple hypothetical formal system.

An example alphabet might include the following symbols:

{ a, b, →, (, ) }

This language can be defined as a set of well-formed formulas. Examples of expressions in this language include:

(A → B), (B → (A → B))

The axioms for this system can be the following expressions:

(A → (B → A))

Example inference rule:

Modus ponens: "A → B" and from "A" you can infer "B".

Purpose and applications

The purpose of a formal system is to provide a rigorously defined means of representing and manipulating the syntax of mathematical and logical propositions. An important application of this is found in the mechanization of proofs, where the systematic application of rules allows the automatic derivation of conclusions from axioms and hypotheses.

Mathematics and logic

In mathematics and logic, formal systems allow the construction of proofs and provide a clear framework for understanding the validity of arguments. They are of fundamental importance in understanding how truths can be derived through pure logic.

Computer science

The influence of formal systems extends to fields such as computer science, particularly in the design of programming languages and proof verification software. They play an integral role in the development of algorithms and the design of compilers, where syntax and rule-based transformations are important.

Rules of inference

In formal systems, inference rules are important because they define how one can derive new truths. Let's take a deeper look at some common inference rules with examples.

Modus ponens

As mentioned earlier, modus ponens is a fundamental rule in many logical systems.

Rule:

"P → Q" and from "P", infer "Q".

Suppose,

P: It is raining.
Q: The ground is wet.

If “It is raining means the ground is wet” (P → Q) and “It is raining” (P), then you can conclude that “The ground is wet” (Q).

Example of logical conclusion

Consider a logical conclusion relating propositions A, B, and C:

1. A → B (If A, then B)
2. B → C (If B, then C)
3. A (A is true)

To get C use:

1. From steps 1 and 3, apply modus ponens to conclude that B is true.
2. From the assumptions of Step 2 and the conclusion of Step 1 (B is true), apply modus ponens to conclude that C is true.

Thus, starting with A and using logical rules, we have concluded C.

Axiomatic systems

The structure of axiomatic systems within a formal framework presents a form where the axioms are basic premises considered to be true, from which other truths are deduced.

• Example: Euclidean geometry

Euclidean geometry is a classical example of an axiomatic system. Here, the axioms are known as Euclidean postulates, such as "a straight line can be drawn between any two points."

Consistency, completeness and decidability

Understanding the properties of formal systems often revolves around consistency, completeness, and decidability.

Stability

A formal system is consistent if it does not yield contradictions. In other words, it should never prove both a statement and its negation.

Completeness

A formal system is complete if, for every statement in its language, either the statement or its negation is derivable within the system.

Decisiveness

A formal system is decidable if and only if there exists an algorithm that can determine whether a statement given in the system's language can be proven within that system.

Gödel's incompleteness theorems

Kurt Gödel's incompleteness theorems are important in the study of formal systems. They effectively assert that for any consistent, sufficiently expressive formal system, there are true statements that cannot be proven within the system.

The significance of Gödel's theorems is profound because they demonstrate the inherent limitations of formal systems, and challenge the notion of a completely formal description of all mathematical truths.

Visualization with logical connectives

An SVG representation showing common logical connectives can help to understand how expressions can be combined within a formal system:

In this diagram, the arrow represents logical implication.

Examples from predicate logic

Predicate logic extends formal systems by allowing expressions about objects. Consider the following examples involving predicates and quantifiers:

Let P(x) denote "x is a human" and Q(x) denote "x is mortal". Then, the expression:

∀x (p(x) → q(x))

This means "for all x, if x is human, then x is mortal," which expresses a universally quantified implication relevant in many logical inference scenarios.

Conclusion

Understanding formal systems in proof theory is fundamental to understanding how logical and mathematical frameworks are constructed and analyzed. Despite the idealization of complete and consistent systems, groundbreaking results such as Gödel's theorems show inherent complexities and limitations, making formal systems an ever-interesting area of research and study.

Formal systems remain an important concept in theoretical explorations and practical applications in mathematics, logic, computer science, and a variety of other fields.

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