---
title: "Logic"
chunk: 10/11
source: "https://en.wikipedia.org/wiki/Logic"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:56:39.118518+00:00"
instance: "kb-cron"
---

The term "mathematical logic" is sometimes used as a synonym of "formal logic". But in a more restricted sense, it refers to the study of logic within mathematics. Major subareas include model theory, proof theory, set theory, and computability theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic. However, it can also include attempts to use logic to analyze mathematical reasoning or to establish logic-based foundations of mathematics. The latter was a major concern in early 20th-century mathematical logic, which pursued the program of logicism pioneered by philosopher-logicians such as Gottlob Frege, Alfred North Whitehead, and Bertrand Russell. Mathematical theories were supposed to be logical tautologies, and their program was to show this by means of a reduction of mathematics to logic. Many attempts to realize this program failed, from the crippling of Frege's project in his Grundgesetze by Russell's paradox, to the defeat of Hilbert's program by Gödel's incompleteness theorems.
Set theory originated in the study of the infinite by Georg Cantor, and it has been the source of many of the most challenging and important issues in mathematical logic. They include Cantor's theorem, the status of the Axiom of Choice, the question of the independence of the continuum hypothesis, and the modern debate on large cardinal axioms.
Computability theory is the branch of mathematical logic that studies effective procedures to solve calculation problems. One of its main goals is to understand whether it is possible to solve a given problem using an algorithm. For instance, given a certain claim about the positive integers, it examines whether an algorithm can be found to determine if this claim is true. Computability theory uses various theoretical tools and models, such as Turing machines, to explore this type of issue.

=== Computational logic ===

Computational logic is the branch of logic and computer science that studies how to implement mathematical reasoning and logical formalisms using computers. This includes, for example, automatic theorem provers, which employ rules of inference to construct a proof step by step from a set of premises to the intended conclusion without human intervention. Logic programming languages are designed specifically to express facts using logical formulas and to draw inferences from these facts. For example, Prolog is a logic programming language based on predicate logic. Computer scientists also apply concepts from logic to problems in computing. The works of Claude Shannon were influential in this regard. He showed how Boolean logic can be used to understand and implement computer circuits. This can be achieved using electronic logic gates, i.e. electronic circuits with one or more inputs and usually one output. The truth values of propositions are represented by voltage levels. In this way, logic functions can be simulated by applying the corresponding voltages to the inputs of the circuit and determining the value of the function by measuring the voltage of the output.

=== Formal semantics of natural language ===

Formal semantics is a subfield of logic, linguistics, and the philosophy of language. The discipline of semantics studies the meaning of language. Formal semantics uses formal tools from the fields of symbolic logic and mathematics to give precise theories of the meaning of natural language expressions. It understands meaning usually in relation to truth conditions, i.e. it examines in which situations a sentence would be true or false. One of its central methodological assumptions is the principle of compositionality. It states that the meaning of a complex expression is determined by the meanings of its parts and how they are combined. For example, the meaning of the verb phrase "walk and sing" depends on the meanings of the individual expressions "walk" and "sing". Many theories in formal semantics rely on model theory. This means that they employ set theory to construct a model and then interpret the meanings of expression in relation to the elements in this model. For example, the term "walk" may be interpreted as the set of all individuals in the model that share the property of walking. Early influential theorists in this field were Richard Montague and Barbara Partee, who focused their analysis on the English language.

=== Epistemology of logic ===
The epistemology of logic studies how one knows that an argument is valid or that a proposition is logically true. This includes questions like how to justify that modus ponens is a valid rule of inference or that contradictions are false. The traditionally dominant view is that this form of logical understanding belongs to knowledge a priori. In this regard, it is often argued that the mind has a special faculty to examine relations between pure ideas and that this faculty is also responsible for apprehending logical truths. A similar approach understands the rules of logic in terms of linguistic conventions. On this view, the laws of logic are trivial since they are true by definition: they just express the meanings of the logical vocabulary.
Some theorists, like Hilary Putnam and Penelope Maddy, object to the view that logic is knowable a priori. They hold instead that logical truths depend on the empirical world. This is usually combined with the claim that the laws of logic express universal regularities found in the structural features of the world. According to this view, they may be explored by studying general patterns of the fundamental sciences. For example, it has been argued that certain insights of quantum mechanics refute the principle of distributivity in classical logic, which states that the formula 
  
    
      
        A
        ∧
        (
        B
        ∨
        C
        )
      
    
    {\displaystyle A\land (B\lor C)}
  
 is equivalent to 
  
    
      
        (
        A
        ∧
        B
        )
        ∨
        (
        A
        ∧
        C
        )
      
    
    {\displaystyle (A\land B)\lor (A\land C)}
  
. This claim can be used as an empirical argument for the thesis that quantum logic is the correct logical system and should replace classical logic.

== History ==