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I know that $A\Rightarrow B$ is equivalent to $\neg A \lor B$. Also that $\neg (A\Rightarrow B)$ is given by $A \land \neg B$. But can $\neg (A\Rightarrow B)$ be written in terms of $A$, $B$, $\Rightarrow$, and $\neg$ alone (i.e. without the use of $\lor$ and $\land$)?

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  • $\begingroup$ An implication is an "or"-statement. The negation of an "or"-statement is an "and"-statement. So I would say no, not apart from what you already have, namely $\lnot(A\Rightarrow B)$. $\endgroup$ – Arthur Nov 13 '15 at 14:28
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    $\begingroup$ Wait a minute, haven't you just done so? $\neg(A\Rightarrow B)$ is obviously written in terms of only $\neg, \Rightarrow, A, B$. $\endgroup$ – BrianO Nov 13 '15 at 17:03
  • $\begingroup$ @BrianO I think he also wants to get rid of the parentheses. $\endgroup$ – Akiva Weinberger Nov 13 '15 at 19:14
  • $\begingroup$ Or, perhaps, to write it in terms of $A$, $B$, $\lnot A$, $\lnot B$, and $\Rightarrow$. $\endgroup$ – Akiva Weinberger Nov 13 '15 at 19:20
  • $\begingroup$ @AkivaWeinberger Yes, must be. I realized that after seeing Rob Arthan's answer. $\endgroup$ – BrianO Nov 13 '15 at 19:25
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I think you are asking for the principal connective to be an implication rather than a negation. If that's right, a suitable equivalent to $\lnot (A \Rightarrow B)$ is $(A \Rightarrow B) \Rightarrow \lnot(A \Rightarrow A)$.

Note that it is impossible to write $\lnot(A \Rightarrow B)$ in terms of implication alone. If $\phi(A, B)$ is equivalent to $\lnot(A \Rightarrow B)$ , then $\phi(A \Rightarrow A, A)$ is equivalent to $\lnot A$, so as $\lnot$ and $\Rightarrow$ are functionally complete, if $\phi$ is constructed with $\Rightarrow$ alone, then $\Rightarrow$ on its own would be functionally complete, but by Post's characterisation of functionally complete sets of connectives, it is not (see link above).

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I use Polish notation.

The formation rules go:

  1. All lower case letters of the Latin alphabet qualify as well-formed formulas.
  2. 0 consists of a well-formed formula standing for the constant false proposition.
  3. If $\alpha$ and $\beta$ qualify as well-formed formulas, then so does C$\alpha$$\beta$. C stands for implication, or equivalently conditional.

Now let us adopt the definition that the negation of a proposition Np, comes as abbreviation for Cp0, since C00 = 1 and N0 = 1, and C10 = 0 and N1 = 0.

Thus, NCab abbreviates CCab0.

More formally, we can take as a definition:

  1. C$\delta$Np$\delta$Cp0 where $\delta$ consists of a functioral variable of one argument (in other words $\delta$ comes as a variable which ranges over the four unary-truth functors of two-valued logic).

Substituting p with Cab and dropping $\delta$ or equivalently substituting $\delta$' with the apostrophe symbol ' where the apostrophe symbol ' consists of the argument of $\delta$ we have

  1. CNCabCCab0

as a thesis obtained from the definition. Thus, given NCab by definition we obtain CCab0. From 1. and the rule of uniform substitution for $\delta$ and the rule of uniform substitution for propositional variables we can also obtain NCab from CCab0.

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  • $\begingroup$ Using "false" strikes me as not in the spirit of the question . . . $\endgroup$ – Noah Schweber Nov 14 '15 at 6:20
  • $\begingroup$ @NoahSchweber You might speak correctly, but I can't tell at this point in time. $\endgroup$ – Doug Spoonwood Nov 16 '15 at 17:49

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