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6.4: Proving the validity of immediate inferences

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    223863

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    Let’s take a second to look at the different immediate inferences using Venn diagrams. Why do they work? Why don’t the invalid inferences work?

    Square of Opposition

    Contradictories

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{1}\): A: All humans are mortal things
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{2}\): O: Some humans are not mortal things

    Taken together:

    z12.PNG

    There can’t be an X in a shaded area. We’ve “filled it with concrete” so there’s nothing there. What does this mean? These two propositions are claiming the opposite thing from one another. A is claiming that there are No things in region 1 (the left moon-shaped region), whereas O is claiming that there is at least one thing in region 1.

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{3}\): E: No humans are mortal things
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{4}\): I: Some humans are mortal things

    Taken together:

    z15.PNG

    Again, E is claiming that there is nothing in region 2 (the ellipse or football shape in the center) and I is claiming that there’s at least one thing in region 2.

    Contraries

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{5}\): A: All Humans are Mortal
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{6}\): E: No Humans are Mortal

    Taken together:

    z18.PNG

    Remember that these are only contrary on Aristotle’s assumption that the subject class has at least one member. Notice how in this diagram, the subject class (humans) must be empty. It is claiming that there are no humans. Since Aristotle assumes that the subject class is populated by at least one thing, it follows that A and E propositions cannot both be true (they are contraries).

    I won’t do subcontraries or subalternation because they’re more complicated and hopefully you get the idea by now.

    Other Immediate Inferences

    Conversion

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{7}\): E: No Dog is an Amphibian
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{8}\): E*: No Amphibian is a Dog

    See? They’re the same diagram! So the immediate inference is valid: we’re claiming the same thing in each proposition.

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{9}\): I: Some Dog is an Amphibian
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{10}\): I*: Some Amphibian is a Dog

    Again, they look the same since they’re both claiming that there’s at least one thing in region 2.

    But Conversion doesn’t work with A’s and O’s:

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{11}\): A: All Dogs are Amphibians
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{12}\): A*: All Amphibians are Dogs

    Notice how they are claiming almost opposite things. At any rate, A* is certainly not contained in the diagram for A. (That’s what validity looks like, the conclusion is already in the diagram for the premises).

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{13}\): O: Some Dogs are not Amphibians
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{14}\): O*: Some Amphibians are not Dogs

    Again, they are claiming basically the opposite thing from one another and as a result, O* is not already represented in the diagram for O.

    Obversion

    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{15}\): A: All Mooses (Meese?) are Animals
    fig-ch01_patchfile_01.jpg
    Figure \(\PageIndex{16}\): A*: No Mooses are Non-Animal

    There are no mooses in the non-animal group, which is the left moon-shaped region, or region 1. So we shade it in.

    All of the other Obversions end up with the same venn diagram too. Check it. Then give Contraposition a shot!


    This page titled 6.4: Proving the validity of immediate inferences is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Andrew Lavin via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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