---
title: "History of logic"
chunk: 10/13
source: "https://en.wikipedia.org/wiki/History_of_logic"
category: "reference"
tags: "science, encyclopedia"
date_saved: "2026-05-05T03:59:53.898187+00:00"
instance: "kb-cron"
---

In English, "for all x, if Ax then Bx". Thus only singular propositions are of subject-predicate form, and they are irreducibly singular, i.e. not reducible to a general proposition. Universal and particular propositions, by contrast, are not of simple subject-predicate form at all. If "all mammals" were the logical subject of the sentence "all mammals are land-dwellers", then to negate the whole sentence we would have to negate the predicate to give "all mammals are not land-dwellers". But this is not the case. This functional analysis of ordinary-language sentences later had a great impact on philosophy and linguistics.
This means that in Frege's calculus, Boole's "primary" propositions can be represented in a different way from "secondary" propositions. "All inhabitants are either men or women" is

  
    
      
        ∀
        
        x
        
          
            (
          
        
        I
        (
        x
        )
        →
        
          
            (
          
        
        M
        (
        x
        )
        ∨
        W
        (
        x
        )
        
          
            )
          
        
        
          
            )
          
        
      
    
    {\displaystyle \forall \;x{\Big (}I(x)\rightarrow {\big (}M(x)\lor W(x){\big )}{\Big )}}
  

whereas "All the inhabitants are men or all the inhabitants are women" is

  
    
      
        ∀
        
        x
        
          
            (
          
        
        I
        (
        x
        )
        →
        M
        (
        x
        )
        
          
            )
          
        
        ∨
        ∀
        
        x
        
          
            (
          
        
        I
        (
        x
        )
        →
        W
        (
        x
        )
        
          
            )
          
        
      
    
    {\displaystyle \forall \;x{\big (}I(x)\rightarrow M(x){\big )}\lor \forall \;x{\big (}I(x)\rightarrow W(x){\big )}}
  

As Frege remarked in a critique of Boole's calculus:

"The real difference is that I avoid [the Boolean] division into two parts ... and give a homogeneous presentation of the lot. In Boole the two parts run alongside one another, so that one is like the mirror image of the other, but for that very reason stands in no organic relation to it."
As well as providing a unified and comprehensive system of logic, Frege's calculus also resolved the ancient problem of multiple generality. The ambiguity of "every girl kissed a boy" is difficult to express in traditional logic, but Frege's logic resolves this through the different scope of the quantifiers. Thus

  
    
      
        ∀
        
        x
        
          
            (
          
        
        G
        (
        x
        )
        →
        ∃
        
        y
        
          
            (
          
        
        B
        (
        y
        )
        ∧
        K
        (
        x
        ,
        y
        )
        
          
            )
          
        
        
          
            )
          
        
      
    
    {\displaystyle \forall \;x{\Big (}G(x)\rightarrow \exists \;y{\big (}B(y)\land K(x,y){\big )}{\Big )}}
  

means that to every girl there corresponds some boy (any one will do) who the girl kissed. But

  
    
      
        ∃
        
        x
        
          
            (
          
        
        B
        (
        x
        )
        ∧
        ∀
        
        y
        
          
            (
          
        
        G
        (
        y
        )
        →
        K
        (
        y
        ,
        x
        )
        
          
            )
          
        
        
          
            )
          
        
      
    
    {\displaystyle \exists \;x{\Big (}B(x)\land \forall \;y{\big (}G(y)\rightarrow K(y,x){\big )}{\Big )}}
  

means that there is some particular boy whom every girl kissed. Without this device, the project of logicism would have been doubtful or impossible. Using it, Frege provided a definition of the ancestral relation, of the many-to-one relation, and of mathematical induction.

This period overlaps with the work of what is known as the "mathematical school", which included Dedekind, Pasch, Peano, Hilbert, Zermelo, Huntington, Veblen and Heyting. Their objective was the axiomatisation of branches of mathematics like geometry, arithmetic, analysis and set theory. Most notable was Hilbert's Program, which sought to ground all of mathematics to a finite set of axioms, proving its consistency by "finitistic" means and providing a procedure which would decide the truth or falsity of any mathematical statement. The standard axiomatization of the natural numbers is named the Peano axioms eponymously. Peano maintained a clear distinction between mathematical and logical symbols. While unaware of Frege's work, he independently recreated his logical apparatus based on the work of Boole and Schröder.
The logicist project received a near-fatal setback with the discovery of a paradox in 1901 by Bertrand Russell. This proved Frege's naive set theory led to a contradiction. Frege's theory contained the axiom that for any formal criterion, there is a set of all objects that meet the criterion. Russell showed that a set containing exactly the sets that are not members of themselves would contradict its own definition (if it is not a member of itself, it is a member of itself, and if it is a member of itself, it is not). This contradiction is now known as Russell's paradox. One important method of resolving this paradox was proposed by Ernst Zermelo. Zermelo set theory was the first axiomatic set theory. It was developed into the now-canonical Zermelo–Fraenkel set theory (ZF). Russell's paradox symbolically is as follows:

  
    
      
        
          Let 
        
        R
        =
        {
        x
        ∣
        x
        ∉
        x
        }
        
          , then 
        
        R
        ∈
        R
        
        ⟺
        
        R
        ∉
        R
      
    
    {\displaystyle {\text{Let }}R=\{x\mid x\not \in x\}{\text{, then }}R\in R\iff R\not \in R}
  

The monumental Principia Mathematica, a three-volume work on the foundations of mathematics, written by Russell and Alfred North Whitehead and published 1910–1913 also included an attempt to resolve the paradox, by means of an elaborate system of types: a set of elements is of a different type than is each of its elements (set is not the element; one element is not the set) and one cannot speak of the "set of all sets". The Principia was an attempt to derive all mathematical truths from a well-defined set of axioms and inference rules in symbolic logic.

=== Metamathematical period ===