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2.16: Universal Statements and Existential Commitment

  • Page ID
    205033
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    Consider the following inference:

    1. All S are P
    2. Therefore, some S are P

    Is this inference valid or invalid? As it turns out, this is an issue on which there has been much philosophical debate. On the one hand, it seems that many times when we make a universal statement, such as “all dogs are mammals,” we imply that there are dogs—i.e., that dogs exist. Thus, if we assert that all dogs are mammals, that implies that some dogs are mammals (just as if I say that everyone at the party was drunk, this implies that at least someone at the party was drunk). In general, it may seem that “all” implies “some” (since some is encompassed by all). This reasoning would support the idea that the above inference is valid: universal statements imply certain particular statements. Thus, statements of the form “all S are P” would imply that statements of the form “some S are P.” This is what is called “existential commitment.”

    In contrast to the reasoning just laid out, modern logicians reject existential commitment; they do not take statements of the form “all S are P” to imply that there exists anything in the “S” category. Why would they think this? One way of understanding why universal statements are interpreted in this way in modern logic is by considering laws such as the following:

    All trespassers will be fined.

    All bodies that are not acted on by any force are at rest.

    All passenger cars that can travel 770 mph are supersonic.

    The “S” terms in the above categorical statements are “trespassers,” “bodies that are not acted on by any force,” and “passenger cars that can travel 770 mph.” Now ask yourself: do these statements commit us to the existence of either trespassers or bodies not acted on by any force? No, they don’t. Just because we assert the rule that all trespassers will be fined, we do not necessarily commit ourselves to the claim that there are trespassers. Rather, what we are saying is anything that is a trespasser will be fined. But this can be true, even if there are no trespassers! Likewise, when Isaac Newton asserted that all bodies that are not acted on by any force remain at rest, he was not committing himself to the existence of “bodies not acted on by any force.” Rather, he was saying that anything that is a body not acted on by any force will remain in motion. But this can be true, even if there are no bodies not acted on by any force! (And there aren’t any such bodies, since even things that are stationary like your house or your car parked in the driveway are still acted on by forces such as gravity and friction.) Finally, in asserting that all passenger cars that can travel 770 mph are supersonic, we are not committing ourselves to the existence of any such car. Rather, we are only saying that were there any such car, it would be supersonic (i.e., it would travel faster than the speed of sound).

    For various reasons (that we will not discuss here), modern logic treats a universal categorical statement as a kind of conditional statement. Thus, a statement like,

    All passenger cars that can travel 770 mph are supersonic

    is interpreted as follows:

    For any x, if x is a passenger car that can travel 770 mph then x is supersonic.

    But since conditional statements do not assert either the antecedent or the consequent, the universal statement is not asserting the existence of passenger cars that can travel 770 mph. Rather, it is just saying that if there were passenger cars that could travel that fast, then those things would be supersonic.

    We will follow modern logic in denying existential commitment. That is, we will not interpret universal affirmative statements of the form “All S are P” as implying particular affirmative statements of the form “some S are P.” Likewise, we will not interpret universal negative statements of the form “no S are P” as implying particular negative statements of the form “some S are not P.” Thus, when constructing Venn diagrams, you can always rely on the fact that if there is no particular represented in the premise Venn (i.e., there is no asterisk), then if the conclusion Venn represents a particular (i.e., there is an asterisk), the argument will be invalid. This is so since no universal statement logically implies the existence of any particular. Conversely, if the premise Venn does represent a particular statement (i.e., it contains an asterisk), then if the conclusion doesn’t contain particular statement (i.e., doesn’t contain an asterisk), the argument will be invalid.

    Exercise

    Construct Venn diagrams to determine which of the following immediate categorical inferences are valid and which are invalid. Make sure you remember that we are not interpreting universal statements to imply existential commitment.

    1. All S are P; therefore, some S are P
    2. No S are P; therefore, some S are not P
    3. All S are P; therefore, some P are S
    4. No S are P; therefore, some P are not S

    This page titled 2.16: Universal Statements and Existential Commitment is shared under a CC BY 3.0 license and was authored, remixed, and/or curated by Matthew Van Cleave.