Exposition of condition 1 |

Arrow (1983: p. 15) | The social welfare function is defined for every admissible pair of individual orderings, R_{1}, R_{2}. |

Arrow (1970: p. 39) | In all possible choices, there must be a set S composed of three choices, which satisfies the conditions below: for any individual ordering T_{1},・・・, T_{n} of objects in S, there is an admissible individual ordering set R_{1},・・・, R_{n} for all objects, such that for every individual i, and all x, y in S, we have xRiy if and only if xTiy. |

Exposition of condition 2 |

Arrow (1983: p. 15) | If an alternative social state x arises or does not fall in the ordering of each individual without any other change in those orderings and if x was preferred to another alternative y before the change in individual orderings, then x is still preferred to y. |

Arrow (1970: p. 41) | Let R_{1},・・・, R_{n} and・・・, be two sets of individual orderings, R and Rʹ be the corresponding social ordering relationship, P and be the corresponding social preferences relationship. Suppose for individual i the two ordering relationships have the following connection: (1) for any two options x' and y' different from option x, x'R'iy' if and only if xRiy'; (2) for any y', if xRiy', then xR'iy'; For any y', if xPiy', then xP'iy'. Then, if xPy, then xP'y. |

Exposition of condition 3 |

Arrow (1983: p. 16) | (Independence of irrelevant alternatives). Let R_{1}, R_{2}, and be two sets of individual orderings. If for both individuals i and for all x and y in a given set of alternatives S, xR_{i}y if and only if, then the social choice made from S is the same whether the individual orderings are R_{1}, R_{2}, or (the author notes: this condition is known as the “independent candidate independence” conditions) |

Arrow (1970: p. 42) | Let R_{1}, R_{2}, and be two sets of individual orderings, C(S) and C'(S) be the corresponding social choice functions. If for some given environment S, for all individual i and all elements x and y, if the sufficient and necessary condition for xRiy is xR'iy, then the same is true for C(S) and C'(S) (i.e. independence or uncorrelatedness). |

Exposition of condition 4 |

Arrow (1983: p. 17) | The social welfare function is not to be imposed. |

Arrow (1970: p. 45) | Definition 5: if for two different objects x and y, and for any individual ranking set R_{1},・・・, R_{n}, we have xRy, then we call the social welfare function to be imposed. Condition 4: social welfare function is not imposed. |

Exposition of condition 5 |

Arrow (1983: p. 19) | (Non-dictatorship). The social welfare function is not to be dictatorial (non-dictatorial). |

Arrow (1970: p. 46) | Definition 6: if there is some individual i, such that for any x and y, and for any ordering R_{1}, R_{n} of other individuals except i, xPiy implies xPy, then the social welfare function is called dictatorial. Condition 5: Social welfare function is not arbitrary (i.e. social welfare function is authoritarian). |