2.2 Entailment

The truth of certain propositions guarantees the truth of other propositions. In a world in which (13) is true, then (14) is also true.

(13) Fred is a bachelor.

(14) Fred is unmarried.

These are examples of one sentence entailing another (or technically, the proposition expressed by one sentence entailing the proposition expressed by another). Likewise, if (15) is true, then (16) must also be true.

(15) It's Thursday and it's cold out.

(16) It's Thursday.

On the progression from semantics to pragmatics, entailment is in the domain of semantics.

On one hand, entailment can be based on lexical relations. The lexical item 'bachelor' in (13) is associated with a lexically imposed fact that, by definition, all bachelors are unmarried. The combination of the presence of 'bachelor' and the lexical definition yields the entailed sentence in (14), as follows with (13) labeled as p and (14) labeled as q.

p: Fred is a bachelor
[all bachelors are unmarried]
q: Fred is unmarried

In a similar way, the combination of a statement about Kim being alive and the fact that nothing is both alive and dead yields the following entailment between p and q. Again it is based on a chain of reasoning stemming from the lexical item "alive".

p: Kim is alive
[nothing is both alive and dead]
q: Kim is not dead

• But the definition of 'alive' doesn't necessarily mention anything about being dead? [click to reveal]
  True, but the entailment arises based on a lexical relationship between two words--namely, the fact that "dead" is an antonym of "alive".

  • But that's not linguistic knowledge; that's real-world knowledge. [click]
    Yes, our lexical knowledge encodes a good deal of real-world knowledge, as well as purely linguistic constraints. Under some theories of grammar (e.g., HPSG), the lexicon does the bulk of the grammatical work by encoding all idiosyncratic syntactic/morphological/semantic information associated with a word into its lexical entry.

As a last example of lexically based entailment, p and q are linked via the (binary-gendered) assertion that all children are either male or female.

p: Sandy is a child
[every child is either a boy or a girl]
q: Sandy is either a boy or a girl

• Are all linguistics examples so heteronormative? [click]
  In response to a long tradition of the "John hit Mary" examples which abound in the literature (whenever an agent-patient verb is called for), there has been an occasional attempt to subvert this. My semantics professor, Polly Jacobson, opted to only use examples about her dogs ("Mitka licked Kolya") to avoid perpetuating the theme of violence and specifically violence against women.

Now consider how entailments can arise via sentential connectives (logical operators). For example, take a proposition p which is composed of two conjoined propositions. If p is true, then either of its conjuncts must also be true (see truth table in 2.1).

p: Fred is a bachelor and it's Thursday
['and' requires the truth of both conjuncts]
q: Fred is a bachelor

Entailment carries with it the interesting property that it does not survive negation. This means that the following pattern holds between p and q: If p is true, then q is also true, but if not p is true, then q is no longer true. So if you're in a world in which the proposition p that Fred is a bachelor is true, then the nature of entailments ensures that it must also be true in that world that q Fred is unmarried. The important addition is that if you're in a world in which not p holds, i.e., that it's not the case that Fred is a bachelor, then the proposition q that was entailed when p was true is no longer true.

if p is true, then q is also true:
p: Fred is a bachelor [True]
q: Fred is unmarried [True]

if not p is true, then q is no longer true:
¬p: Fred is not a bachelor [¬p is True]
q: Fred is unmarried [q is False]

• Does that make sense? [click]
  One way to think about these hypothetical truth values is to think of possible worlds. If you're in a world where p is not true (i.e., a world where Fred's bachelorhood is not the current state of affairs because, say, Fred has just gotten married), what else do you know definitively about that world? Well, one thing you know is that q is also not true; there's no way this guy Fred is unmarried in that world.

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To go on to section 2.3 "Beyond entailment", click here.