I have just started learning introductory discrete math and I am kind of confused. I can't see the difference between compound statements and statement forms. To me, they look the same according two these 2 definitions:

Compound statement:

A statement represented by a some combination of statement variables and connectives is called a compound statement.

Statement form:

Statement form or propositional form is an expression made up of statement variables such as p, q, r and logical connectives that becomes a statement when actual statements are substituted for the component statement variables.

Are they the same thing ? What am I missing here?

  • $\begingroup$ More or less the same... $\endgroup$ – Mauro ALLEGRANZA Mar 26 '17 at 12:28
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    $\begingroup$ More correctly : "Napoleon is a general and Fido is a dog" is a (compound) statement. The corresponding statement form is $p \land q$. $\endgroup$ – Mauro ALLEGRANZA Mar 26 '17 at 12:29
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    $\begingroup$ The compund statement is obtained from the statement form when the actual statements "Napoleon is a general" is substituted for $p$ and "Fido is a dog" for $q$. $\endgroup$ – Mauro ALLEGRANZA Mar 26 '17 at 12:31
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    $\begingroup$ We may say: a statement form is : either (i) a statement variable (atomic s.f.), or (ii) some combination of statement variables and connectives (compund s.f.). $\endgroup$ – Mauro ALLEGRANZA Mar 26 '17 at 12:32
  • $\begingroup$ That makes sense, thanks Mauro. $\endgroup$ – Zed Mar 26 '17 at 12:34

A compound statement: "It is raining, and if I don't find my umbrella, I will stay at home."

A statement form corresponding to the above: $r\land(\neg f \to s)$.

A statement that is not compound: "It is raining."

A statement form corresponding to the non-compound statement: $r$.

Don't sweat this too much. Once you get to leave the natural-language sentences behind and just focus on the mathematics, both of these concepts will mostly fade back into obscurity, and it will suddenly be okay to call $r\land (\neg f\to s)$ a statement anyway.


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