A woman says the following illogical sentences: "The eagle takes flight at three o’clock and flies here from General Electric. I get my check from the eagle; he goes from the mailbox to make a payment. He is a secret spy for my husband and makes a shotgun sound." Based on what she says, this woman is most likely experiencing and is displaying a symptom.

Schizophrenia is a mental disorder primarily characterized by hallucinations and delusions, as well as a disorganization in speech, abnormal motor behavior and negative symptoms.

Let's break down the first two:

Hallucinations: a perception without an external stimulus present. They come through the senses. An individual can hear voices (hearing hallucination) or see things (sight hallucination), for example.
Delusion: a false belief which the individual tends to accept as the truth and reality. It can be paranoid, fear of persecution, reference and grandiose, amongst others.

Mental disorders tend to have 2 types of symptoms:

Positive: an added trait, such as the hallucinations and delusions.
Negative: a removed trait, loss of it, such as apathy, no energy to do anything and anhedonia, amongst others.

In this case, the woman has the delusion that an eagle is a secret spy for her husband and has the hallucination that it makes a shotgun sound. Her belief that the eagle delivers her check is perceived as her reality.

The woman is displaying positive symptoms of schizophrenia, understood as the hallucinations and delusions.