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I have seen it said that MPC (Minimal Propositional Calculus) with LEM (Law of Excluded Middle; $A \lor \neg A$) is not equivalent to CPC (Classical Propositional Calculus), but I would like help finding a proof for this fact.

Notation

MPC means Minimal Propositional Calculus,

IPC means Intuitionistic Propositional Calculus,

CPC means Classical Propositional Calculus,

LEM means Law of Excluded Middle ($A \lor \neg A$),

DNE means Double Negation Elimination ($\neg \neg A \vdash A$),

EFQ means Ex Falso Quodlibet ($\bot \vdash A $).

Facts

I know that:

IPC = MPC + EFQ

CPC = MPC + DNE

CPC = IPC + LEM.

And so we see:

CPC = MPC + EFQ + LEM.

Goal

My goal is to show

CPC $\neq$ MPC + LEM.

Approach

This can be done by either showing that LEM does not imply EFQ:

$\neg A \lor A \not \vdash_{MPC} \bot \rightarrow B $,

equivalently,

$\not \models_{MPC} (\neg A \lor A) \rightarrow (\bot \rightarrow B)$

or that LEM does not imply DNE in MPC:

$\neg A \lor A \not \vdash_{MPC} \neg \neg B \rightarrow B $.

or equivalently

$\not \vdash_{MPC} (\neg A \lor A) \rightarrow (\neg \neg B \rightarrow B)$.

Difficulties

I am having trouble proving any of these. I am not sure how to prove that something is not a theorem; in CPC I would just use the contrapositive of completeness, but here that does seem to work; I would have thought to use some intermediate multi-valued Godel logic $G_n$ between CPC and IPC, but it seems like these statements are validities in $G_n$ for any $n$. Any advice for how to proceed would be appreciated.

Luke Reifenberg
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  • I haven't checked this in detail, but try $G_3$ with $\bot$ changed to be the other non-$\top$ element. – Rob Arthan Apr 06 '22 at 22:23
  • What do you mean by changing $\bot$ to be the other non- ⊤ element? $v(\bot)=0$ for all truth assignments $v$, by definition. So $\bot \rightarrow B$ will always evaluate to $1$ for any truth assignment. – Luke Reifenberg Apr 06 '22 at 23:20
  • In minimal logic, you don't have EFQ and there is no constraint on $\nu(\bot)$. – Rob Arthan Apr 06 '22 at 23:26
  • I am aware you don't have EFQ in MPC. In any Godel logic, you do have $v(\bot)=0$, since $\bot$ after all is just a simplification of $A \land \neg A$, which always evaluates to $0$ given the definition of $\neg$. – Luke Reifenberg Apr 06 '22 at 23:46
  • To be honest I am not sure what your suggestion is. Could you explain it in more detail please? – Luke Reifenberg Apr 06 '22 at 23:47
  • In a related question I was pointed to the following paper, which should also contain an answer to your question (cf. Proposition 3). The main intuition is that LEM fails to derive the explosion principle, whereas DNE can do that. – Léreau Apr 07 '22 at 08:21
  • I didn't thoroughly check the following. Seems to work, though. Consider the truth tables for CPC, but where $A\to B=0$ when $A,B=0$ and $\lnot A=1$ whatever the value of $A$. MPC+LEM is sound with respect to these tables. But they afford counterexamples to both EFQ and DNE. – Hermógenes Oliveira Apr 07 '22 at 12:04
  • @Léreau Proposition 3 does state what I want to show, but the linked paper doesn't seem to contain a proof, just a statement. Or am I mistaken? – Luke Reifenberg Apr 16 '22 at 01:50
  • @HermógenesOliveira I can see that editing the truth-tables for CPC gives counterexamples to EFQ and DNE, but could you show me how MPC+LEM is sound with respect to this semantics? Furthermore, why does showing that one rule #1 is sound with respect to an some semantics and rule #2 is not sound with respect to that semantics show that rule #2 is not derivable from rule #1? This may be a very basic question, sorry. – Luke Reifenberg Apr 16 '22 at 02:06
  • Soundness is shown by checking that the rules of MPC+LEM preserve the designated value under the semantics. Now, if $\not\Vdash EFQ$ in this semantics, then by soundness + modus tollens, $\not\vdash EFQ$. – Hermógenes Oliveira Apr 17 '22 at 11:44
  • @HermógenesOliveira It doesn't seem as if MPC+LEM preserve the designated value under the semantics. For example, $\rightarrow$ intro fails to preserve the designated truth value, in the case that $B$ is provable from $A$ (hence we can apply the intro rule) but $B$ and $A$ are both assigned false (in which case $A \rightarrow B$ is false. So I don't this semantics is sound with respect to MPC+LEM. – Luke Reifenberg Apr 17 '22 at 20:32
  • I'm sorry. You seem to be thinking about completeness. If neither $A$, nor $B$, nor $A\to B$ are being assigned "true", how is this supposed to show that the rule is failing to preserve truth? Another indication that there may be some confusion: is not that "the semantics is sound with respect to MPC+LEM", but the other way around. – Hermógenes Oliveira Apr 17 '22 at 23:56
  • No worries. I am just asking for the case of $\rightarrow$-Intro. In this situation, we do a subproof using A as an assumption and obtain B. From the semantic side, this means that if A is assigned true, then we assign B true. But it may well be the case that A gets assigned false, and B also gets assigned false. Nonetheless the hypothetical that if A true then B true might still hold. So we may have $A \rightarrow B$. But under your semantics $A \rightarrow B$ is false, and so $\rightarrow$-Intro does not preserve tautologies. Apologies for the phrasing error of soundness of x w.r.t y. – Luke Reifenberg Apr 18 '22 at 02:36
  • I think you may be thinking that I am talking about $\rightarrow$-Elim. I am talking about the introduction rule, $\rightarrow$-Intro, which makes no requirement on the truth-value of $A$ or $B$ besides the fact that $A$ being true gives $B$ as true. – Luke Reifenberg Apr 18 '22 at 02:39
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    As an explicit example, it seems that these MPC+LEM is not sound with respect to these semantics, as $\vdash A \rightarrow A$, but $ \not \models A \rightarrow A$ since under the truth assignment that sends A to false, $A \rightarrow A$ is false. – Luke Reifenberg Apr 18 '22 at 03:08
  • Yes. This last one is a proper counterexample to soundness of MPC+LEM with respect to the semantics that I proposed. – Hermógenes Oliveira Apr 18 '22 at 10:26

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