In mathematical logic

Mathematical logic

Mathematical logic is a subfield of mathematics with close connections to foundations of mathematics, theoretical computer science and philosophical logic. The field includes both the mathematical study of logic and the applications of formal logic to other areas of mathematics...

, generalization (also universal generalization, GEN) is an inference rule

Rule of inference

In logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...

 of predicate calculus. It states that if has been derived, then can be derived.

Generalization with hypotheses

The full generalization rule allows for hypotheses to the left of the turnstile

Turnstile (symbol)

In mathematical logic and computer science the symbol \vdash has taken the name turnstile because of its resemblance to a typical turnstile if viewed from above. It is also referred to as tee and is often read as "yields", "proves", "satisfies" or "entails"...

, but with restrictions. Assume Γ is a set of formulas, φ a formula, and has been derived. The generalization rule states that can be derived if y is not mentioned in Γ and x does not occur in φ.

These restrictions are necessary for soundness. Without the first restriction, one could conclude from the hypothesis . Without the second restriction, one could make the following deduction:

1. (Hypothesis)
2. (Existential instantiation)
3. (Existential instantiation)
4. (Faulty universal generalization)

This purports to show that which is an unsound deduction.

Example of a proof

Prove: .

Proof:

Number	Formula	Justification
1		Hypothesis
2		Hypothesis
3		Axiom PRED-1
4		From (1) and (3) by Modus Ponens
5		Axiom PRED-1
6		From (2) and (5) by Modus Ponens
7		From (6) and (4) by Modus Ponens
8		From (7) by Generalization
9		Summary of (1) through (8)
10		From (9) by Deduction Theorem
11		From (10) by Deduction Theorem

In this proof, the deduction theorem was applicable in steps 10 and 11 because the formulas being moved have no free variables.

See also

• Categorical imperative
  Categorical imperative
  The Categorical Imperative is the central philosophical concept in the moral philosophy of Immanuel Kant, as well as modern deontological ethics...
  
   redirected from "Generalization in Ethics"
• Derivative (generalizations)
  Derivative (generalizations)
  The derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, and geometry.- Derivatives in analysis :...
  
  
• First-order logic
  First-order logic
  First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
  
  
• Hasty generalization
  Hasty generalization
  Hasty generalization is a logical fallacy of faulty generalization by reaching an inductive generalization based on insufficient evidence essentially making a hasty conclusion without considering all of the variables...
  
  
• Generalization error
  Generalization error
  The generalization error of a machine learning model is a function that measures how far the student machine is from the teacher machine in average over the entire set of possible data that can be generated by the teacher after each iteration of the learning process...
  
  
• Generalizations of Fibonacci numbers
  Generalizations of Fibonacci numbers
  In mathematics, the Fibonacci numbers form a sequence defined recursively by:The Fibonacci sequence has been studied extensively and generalized in many ways, for example, by starting with other numbers than 0 and 1, by adding more than two numbers to generate the next number, or by adding objects...
  
  
• Generalizations of Pauli matrices
  Generalizations of Pauli matrices
  In mathematics and physics, in particular quantum information, the term generalized Pauli matrices refers to families of matrices which generalize the properties of the Pauli matrices. In this article we give a few classes of such matrices....
  
  
• Universal instantiation
  Universal instantiation
  In logic universal instantiation is an inference from a truth about each member of a class of individuals to the truth about a particular individual of that class. It is generally given as a quantification rule for the universal quantifier but it can also be encoded in an axiom...