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I'm doing some homework in which I'm converting textual descriptions of logic statements to their respective symbolic representation.

If one reads It is false that if p then q.

I was wondering the typical representation of this.

i.e. Is it ~p -> q or ~(p -> q). Essentially what I'm asking is typically when converting a textual representation to a symbolic representation, do we assume that the negation applies to the entire expression, or simply the entity immediately following the negation.

Thanks.

Brugsen
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    The second one is right. – Cure May 07 '14 at 03:20
  • Negations only apply to well-formed formulas, or expressions which can get rendered into well-formed formulas (check the formation rules). So, a negation can only apply to an entire expression, NOT an entity which immediately follows it, since the entity immediately following a negation could be a word like "if" or "both", or what follows $\lnot$ could be a parenthesis symbol or something else also. The formation rules tell us that a negation only applies to an expression. – Doug Spoonwood May 07 '14 at 18:38

1 Answers1

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Read it as

It is false that: "if $p$, then $q$".

So the (entire) statement $(p\implies q)$ should be logically negated.

JohnD
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    Is that a commonly accepted convention? Or just a personal interpretation? Any references? – K1. May 07 '14 at 04:35
  • "If p, then q" means (p $\rightarrow$ q). So, we have the negation of (p $\rightarrow$ q), $\lnot$(p $\rightarrow$ q). The entity that follows the expression is "if", which only stops after the "then ...". The negation applies to the string of characters such that that string qualifies as a wff. For instance, if we use polish notation we could write $\lnot$$\rightarrow$pq. But, we build wffs up from the atoms, so $\rightarrow$pq is a wff, and thus the $\lnot$ must apply $\rightarrow$pq, NOT just $\rightarrow$. – Doug Spoonwood May 07 '14 at 18:35