Validity is defined in a couple of ways. I like to define it like this: An argument is valid iff the superconjunction^{1} of all the premises and the negation of the conclusion is impossible. It sounds a bit unusual at first but it is worded that way to prevent confusion about modalities. A more common definition is: An argument is valid iff it is impossible for all the premises to be true and the conclusion false. I contend that mine is clearer when understood. Consider this argument:

n | Proposition | Explanation |

1 | P | Premise |

2 | P→Q | Premise |

3 | Q | From 1, 2, MP |

Anyone trained in logic will immediately recognize that this argument is valid (and the form valid too) for it has the form of MP and all arguments of that form are valid. To see this using my definition of valid above we can simply make the superconjunction of the argument and put it into a truth table:

P | Q | ¬Q | P→Q | P∧(P→Q)∧¬Q |

T | T | F | T | F |

T | F | T | F | F |

F | T | F | T | F |

F | F | T | T | F |

Note that I skipped a conjunction step.

The superconjunction comes out as impossible.

And here comes the tricky part. Suppose that the conclusion of an argument happens to be a necessary truth. Consider this argument:

n | Proposition | Explanation |

1 | P | Premise |

2 | ¬(Q∧¬Q) | From 1 |

Is that argument valid? It is according to my definition and the usual definition of valid. A truth table will show that:

P | Q | (Q∧¬Q) | P∧(Q∧¬Q) |

T | T | F | F |

T | F | F | F |

F | T | F | F |

F | F | F | F |

Note that (Q∧¬Q) is equivalent to ¬¬(Q∧¬Q) which is the negation of the conclusion in the argument above. I skipped a double negation step.

Realize that whatever is conjoined with an impossibility such as (Q∧¬Q) will come out as false. So no matter the argument structure, an argument with a necessary truth in the conclusion is valid. This doesn’t seem to bad again, after all, we already know that anything implies a necessary truth and that an impossibility implies anything. Truth tables show that:

Necessary truth as consequent:

P | □Q | P→□Q |

T | T | T |

F | T | T |

Note that Q is true on all rows.

An impossibility as antecedent:

□¬P | Q | □¬P→Q |

F | T | T |

F | F | T |

Note that P is false on all rows.

But now consider that in the realm mathematics all propositions are either necessary truths or impossibilities. Thus, any mathematical argument that happens to have a necessary truth in its conclusion is valid, no matter the form of it. That seems like an odd conclusion.

1A conjunction has the form P∧Q. A superconjunction has the form P∧Q∧…∧T. I use it to avoid talking about sets.

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