Fine on the Possibility of Vagueness

Abstract

In his paper ‘The possibility of vagueness’ (Fine in Synthese 194(10):3699–3725, 2017), Kit Fine proposes a new logic of vagueness, CL, that promises to provide both a solution to the sorites paradox and a way to avoid the impossibility result from Fine (Philos Perspect 22(1):111–136, 2008). The present paper presents a challenge to his new theory of vagueness. I argue that the possibility theorem stated in Fine (Synthese 194(10):3699–3725, 2017), as well as his solution to the sorites paradox, fail in certain reasonable extensions of the language of CL. More specifically, I show that if we extend the language with any negation operator that obeys reductio ad absurdum, we can prove a new impossibility result that makes the kind of indeterminacy that Fine takes to be a hallmark of vagueness impossible. I show that such negation operators can be conservatively added to CL and examine some of the philosophical consequences of this result. Moreover, I demonstrate that we can define a particular negation operator that behaves exactly like intuitionistic negation in a natural and unobjectionable propositionally quantified extension of CL. Since intuitionistic negation obeys reductio, the new impossibility result holds in this propositionally quantified extension of CL. In addition, the sorites paradox resurfaces for the new negation.

Appendices

Appendix 1

This appendix establishes the Impossibility Theorem from Sect. 31.2. Let $\mathscr {L}_{{\textbf {CL+}}}$ be the language of $\textbf{CL}$ supplemented with a unary negation operator $\sim$. A consequence relation, $\vdash$, for a language $\mathscr {L}$, is a relation between sets of $\mathscr {L}$-formulas and $\mathscr {L}$-formulas that is substitution invariant and closed under the usual structural rules of reflexivity, monotonicity and cut. When $\vdash$ is a consequence relation, we write $\varGamma \vdash \phi$ for $\langle \varGamma ,\phi \rangle \in \;\vdash$, and $\phi _{1},\ldots ,\phi _{n}\vdash \psi$ for $\lbrace \phi _{1},\ldots ,\phi _{n}\rbrace \vdash \psi$. Let $\vdash _{\textbf{CL}}:=\lbrace \langle \varGamma ,\phi \rangle :\varGamma \vDash _{\textbf{CL}}\phi , \varGamma \subseteq \mathscr {L}_{\textbf{CL}}, \phi \in \mathscr {L}_{\textbf{CL}}\rbrace$. We define $\vdash _{\mathbf {CL+}}$ to be the smallest consequence relation for $\mathscr {L}_{{\textbf {CL+}}}$ that contains $\vdash _{\textbf{CL}}$ and satisfies the following condition:

• (RAA) If $\varGamma ,\phi \vdash _{\mathbf {CL+}}\psi$ and $\varGamma ,\phi \vdash _{\mathbf {CL+}}{\sim }\psi$ for some $\psi$, then $\varGamma \vdash _{\mathbf {CL+}}{\sim }\phi$;

Thus, $\vdash _{\mathbf {CL+}}$ is closed under reductio ad absurdum for $\sim$. We define the logic $\mathbf {CL+}$ to be the consequence relation $\vdash _{\mathbf {CL+}}$. Note that the fact that $\mathbf {CL+}$ contains $\textbf{CL}$ and is substitution invariant guarantees that every $\mathscr {L}_{\mathbf {CL+}}$-substitution instance of a valid argument of $\textbf{CL}$ is in $\mathbf {CL+}$. This entails, for example, that $p,\lnot p\vdash _{\mathbf {CL+}}{\sim }p$, this being a $\mathscr {L}_{\mathbf {CL+}}$-substitution instance of ex falso quodlibet, which is a valid rule of $\textbf{CL}$.²² I will henceforth write $\vdash$ instead of $\vdash _{\mathbf {CL+}}$ whenever no confusion is likely to arise.

A set of formulas $\varGamma$ of $\mathscr {L}_{{\textbf {CL+}}}$ is inconsistent just in case $\varGamma \vdash \phi$ and $\varGamma \vdash {\sim }\phi$, for some formula $\phi$. $\varGamma$ is consistent just in case it is not inconsistent.

Lemma 2

Let $\varGamma$ be a consistent set of formulas. Then if $\varGamma \cup \lbrace \phi \rbrace$ is inconsistent, $\varGamma \cup \lbrace {\sim }\phi \rbrace$ is consistent.

Proof

Suppose that $\varGamma \cup \lbrace \phi \rbrace$ is inconsistent. Then $\varGamma ,\phi \vdash \psi \wedge {\sim }\psi$, for some $\psi$. By (RAA), $\varGamma \vdash {\sim }\phi$. Now suppose for reductio that $\varGamma \cup \lbrace {\sim }\phi \rbrace$ is inconsistent. Then $\varGamma ,{\sim }\phi \vdash \psi \wedge {\sim }\psi$, for some $\psi$. Hence, by (RAA), $\varGamma \vdash {\sim }{\sim }\phi$. Hence, $\varGamma$ is inconsistent, contrary to our hypothesis. Thus, $\varGamma \cup \lbrace {\sim }\phi \rbrace$ is consistent.

Lemma 3

For all formulas of $\mathscr {L}_{\mathbf {CL+}}$, $\lnot \phi \vdash {\sim }\phi$.

Proof

By reflexivity and monotonicity, $\phi ,\lnot \phi \vdash \phi$. By ex falso quodlibet for $\lnot$, $\phi ,\lnot \phi \vdash {\sim }\phi$. Now by (RAA), $\lnot \phi \vdash {\sim }\phi$.

Note that, as a consequence of Theorem 1, the converse entailment does not hold. The definitions of the notions of an extremal response and a sharp response are as in the main text, although they should now be understood with respect to the language $\mathscr {L}_{\mathbf {CL+}}$, and the notion of inconsistency appealed to in condition (ii) of the definition of a sharp response is now to be understood as $\mathbf {CL+}$-inconsistency.

We can now state and prove the Impossibility Theorem from the main text.

Theorem 2

(Impossibility) Let $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$, $n\ge 0$, be any formulas of $\mathscr {L}_{\mathbf {CL+}}$. Then there is no set of formulas $\varDelta _{0}$ which is consistent with the extremal response $\lbrace \phi _{0},\lnot \phi _{n+1}\rbrace$ and yet inconsistent with every sharp response to $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$.

Proof

The proof is similar to the proof of the Impossibility Theorem in Fine (2008). Let $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$ be any set of formulas and let $\varDelta _{0}$ be a set of formulas consistent with the extremal response $\lbrace \phi _{0},\lnot \phi _{n+1}\rbrace$. We show that $\varDelta _{0}$ is consistent with a sharp response to $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$ by extending $\varDelta _{0}$. Let $\varDelta _{1}=\varDelta _{0}\cup \lbrace \phi _{0},{\sim }\phi _{n+1}\rbrace$. For $k=1,2,\ldots ,n$, we define:

$$\varDelta _{k+1}={\left\{ \begin{array}{ll} \varDelta _{k}\cup \lbrace \phi _{k}\rbrace , &{}\text {if consistent} \\ \varDelta _{k}\cup \lbrace {\sim }\phi _{k}\rbrace , &{} \text {otherwise} \end{array}\right. }$$

Note that by Lemma 2, $\varDelta _{k}$ is consistent for $k=1,\ldots ,n+1$. The consistency of $\varDelta _{1}$ follows from Lemma 3. For suppose $\varDelta _{1}$ is inconsistent. Then (i) $\varDelta _{0},\phi _{0},{\sim }\phi _{n+1}\vdash \psi \wedge {\sim }\psi$, for some $\psi$. From this, by (RAA) and monotonicity, we obtain: (ii) $\varDelta _{0},\phi _{0},\lnot \phi _{n+1}\vdash {\sim \sim }\phi _{n+1}$. By Lemma 3 and monotonicity: (iii) $\varDelta _{0},\phi _{0},\lnot \phi _{n+1}\vdash {\sim }\phi _{n+1}$, in contradiction to the consistency of $\varDelta _{0}\cup \lbrace \phi _{0},\lnot \phi _{n+1}\rbrace$. Note further that for every $k\le n+1$, either $\phi _{k}\in \varDelta _{n+1}$ or ${\sim }\phi _{k}\in \varDelta _{n+1}$.

We can now define a collective response $\psi _{0}(p), \psi _{1}(p),\ldots ,\psi _{n+1}(p)$ and a corresponding collective response $\psi _{0}(\phi _{0}), \psi _{1}(\phi _{1}),\ldots ,\psi _{n+1}(\phi _{n+1})$ to $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$. Where $k=0,1,\ldots ,n+1$:

$$\psi _{k}(p)={\left\{ \begin{array}{ll} p, &{} \text{ if } \phi _{k}\in \varDelta _{n+1}\\ {\sim }p, &{} \text{ if } {\sim }\phi _{k}\in \varDelta _{n+1} \end{array}\right. }$$

We now get:

• (1) $\varDelta _{n+1}\vdash \psi _{k}(\phi _{k})$, for $k=0,1,\ldots ,n+1$.

The claim follows trivially from the reflexivity and monotonicity of $\vdash$. Since $\varDelta _{n+1}$ is consistent it follows from (1) that $\varDelta _{n+1}$ is consistent with the response $\psi _{0}(\phi _{0}), \psi _{1}(\phi _{1}),\ldots ,\psi _{n+1}(\phi _{n+1})$ to $\phi _{0},\phi _{1},\ldots ,\phi _{n+1}$. Hence, $\varDelta _{0}$ is also consistent with this response, since it is a subset of $\varDelta _{n+1}$.

It remains to show that the response $\psi _{0}(p), \psi _{1}(p),\ldots ,\psi _{n+1}(p)$ is sharp. The first condition for being a sharp response is satisfied, since $\psi _{0}(p)=p\ne {\sim }p=\psi _{n+1}(p)$. The second condition says that any two individual responses are either the same or inconsistent. But every response is either of the form p or of the form ${\sim }p$. So the second condition is also satisfied. This concludes the proof.

Appendix 2

This appendix establishes the Embedding Theorem from Sect. 31.3. The first step is to show that with the conditional $\rightarrow$ defined as $(\phi \rightarrow \psi )=^{df}\exists p(p\wedge ((\phi \wedge p)\supset \psi ))$, where p is the first variable not occurring free in either $\phi$ or $\psi$ and the negation $\lnot ^{i}$ defined as $\lnot ^{i}\phi =^{df}\phi \rightarrow \bot$, we can derive all the axioms and rules of $\textbf{IL}$ in $\mathbf {CL\pi }$. In what follows, we will often use the convention that $\wedge$ and $\vee$ are given precedence over $\supset$ and $\rightarrow$ so that, for example, $\phi \wedge \psi \supset \xi$ abbreviates $(\phi \wedge \psi )\supset \xi$ and $\phi \rightarrow \psi \vee \xi$ abbreviates $\phi \rightarrow (\psi \vee \xi )$. Outer brackets of conjunctions and disjunctions are thus often omitted.

We will use the axiomatization of $\textbf{CL}$ from Fine (2017, p. 3723f). Note that compatibilist negation $\lnot$ is a defined connective in this axiomatization: $\lnot \phi =^{df}\phi \supset \bot$.

In what follows, $\vdash$ denotes derivability in $\mathbf {CL\pi }$, which is obtained by adding the following axioms and rules for the propositional quantifier to $\textbf{CL}$:

We need to show that all of the following are provable in $\textbf{CL}\pi$:

We state without proof the following two elementary lemmas:

Lemma 4

$\vdash \phi \supset \phi$

Lemma 5

If $\vdash \phi$, then $\vdash \psi \supset \phi$

The use of associativity and commutativity of conjunction and disjunction will be tacit in the proofs to follow.²³ The proofs of these principles themselves are routine. The following central lemma will be useful:

Lemma 6

$\vdash (\phi \supset \psi )\supset (\phi \rightarrow \psi )$

Proof

• 1. $(\phi \supset \psi )\supset (\phi \supset \psi )$           [Lemma 4]

• 2. $\phi \wedge (\phi \supset \psi )\supset \psi$           [A1]

• 3. $(\phi \supset \psi )\supset (\phi \wedge (\phi \supset \psi )\supset \psi )$           [2., Lemma 5]

• 4. $((\phi \supset \psi )\supset (\phi \supset \psi ))\wedge ((\phi \supset \psi )\supset (\phi \wedge (\phi \supset \psi )\supset \psi ))$           [1., 3., R2]

• 5. $(\phi \supset \psi )\supset (\phi \supset \psi )\wedge (\phi \wedge (\phi \supset \psi )\supset \psi )$           [4., A8, R1]

• 6. $(\phi \supset \psi )\wedge (\phi \wedge (\phi \supset \psi )\supset \psi )\supset \exists p(p\wedge (\phi \wedge p\supset \psi ))$           [$\exists 1$]

• 7. $(\phi \supset \psi )\supset \exists p(p\wedge (\phi \wedge p\supset \psi ))$           [5., 6., R3]

• 8. $(\phi \supset \psi )\supset (\phi \rightarrow \psi )$           [7., Def.]

From this and the corresponding axioms of CL we immediately get IL4, IL5, IL6 and IL7. To prove MP$\rightarrow$, we first show:

Lemma 7

$\vdash \phi \wedge (\phi \rightarrow \psi )\supset \psi$

Proof

Let p be a variable not free in $\phi$ or $\psi$.

• 1. $(\phi \wedge p)\wedge (\phi \wedge p\supset \psi )\supset \psi$           [A1]

• 2. $\exists p((\phi \wedge p)\wedge (\phi \wedge p\supset \psi ))\supset \psi$           [1. $\exists 2$]

• 3. $\phi \wedge \exists p(p\wedge (\phi \wedge p\supset \psi ))\supset \exists p((\phi \wedge p)\wedge (\phi \wedge p\supset \psi ))$           [$\exists \wedge$]

• 4. $\phi \wedge (\phi \rightarrow \psi )\supset \psi$           [3., 2., R3, Def.]

From Lemma 7 we get MP$\rightarrow$ from R2 and R1:

Lemma 8

(MP$\rightarrow$) If $\vdash \phi$ and $\vdash \phi \rightarrow \psi$, then $\vdash \psi$.

Next, we prove IL1 and IL3. We first state without proof the following auxiliary lemma:

Lemma 9

If $\vdash \phi$ and $\vdash \psi \wedge \phi \supset \xi$, then $\vdash \psi \supset \xi$.

Lemma 10

(IL1) $\vdash \phi \rightarrow (\psi \rightarrow \phi )$

Proof

• 1. $\phi \wedge (\phi \wedge \psi \supset \phi )\supset (\psi \rightarrow \phi )$           [$\exists 1$, Def]

• 2. $\phi \supset (\psi \rightarrow \phi )$           [1., A6, Lemma 9]

• 3. $(\phi \supset (\psi \rightarrow \phi ))\supset (\phi \rightarrow (\psi \rightarrow \phi ))$           [Lemma 6]

• 4. $\phi \rightarrow (\psi \rightarrow \phi )$           [2., 3., R1]

Lemma 11

(IL3) $\vdash \phi \rightarrow (\psi \rightarrow \phi \wedge \psi )$

Proof

• 1. $\phi \wedge (\phi \wedge \psi \supset \phi \wedge \psi )\supset (\psi \rightarrow \phi \wedge \psi )$          [$\exists 1$, Def.]

• 2. $\phi \supset (\psi \rightarrow \phi \wedge \psi )$           [1., Lemma 9]

• 3. $\phi \rightarrow (\psi \rightarrow \phi \wedge \psi )$           [2., Lemma 6, R1]

For IL2 and IL8 we need the following

Lemma 12

(Exp.) $\vdash ((\phi \wedge \psi )\rightarrow \xi )\supset (\phi \rightarrow (\psi \rightarrow \xi ))$

Proof

Let p be a variable not free in $\phi$, $\psi$ or $\xi$.

• 1. $\exists p(p\wedge ((\phi \wedge \psi )\wedge p\supset \xi ))\wedge \phi \supset \exists p((p\wedge \phi )\wedge (p\wedge (\phi \wedge \psi )\supset \xi ))$           [$\exists \wedge$]

• 2. $((p\wedge \phi )\wedge ((p\wedge \phi )\wedge \psi \supset \xi ))\supset \exists q(q\wedge (q\wedge \psi \supset \xi ))$           [$\exists 1$]

• 3. $\exists p((p\wedge \phi )\wedge ((p\wedge \phi )\wedge \psi \supset \xi ))\supset \exists q(q\wedge (q\wedge \psi \supset \xi ))$           [2., $\exists 2$]

• 4. $(((\phi \wedge \psi )\rightarrow \xi )\wedge \phi )\supset (\psi \rightarrow \xi )$           [1., 3., R3]

• 5. $(((\phi \wedge \psi )\rightarrow \xi )\wedge ((((\phi \wedge \psi )\rightarrow \xi )\wedge \phi )\supset (\psi \rightarrow \xi )))\supset (\phi \rightarrow (\psi \rightarrow \xi ))$           [$\exists 1$, Def.]

• 6. $((\phi \wedge \psi )\rightarrow \xi )\supset (\phi \rightarrow (\psi \rightarrow \xi ))$           [4., 5., Lemma 9]

Lemma 13

(IL2) $\vdash (\phi \rightarrow \psi )\rightarrow ((\phi \rightarrow (\psi \rightarrow \xi ))\rightarrow (\phi \rightarrow \xi ))$

Proof

• 1. $\phi \wedge (\phi \rightarrow \psi )\supset \psi$           [Lemma 7]

• 2. $\phi \wedge (\phi \rightarrow (\psi \rightarrow \xi ))\supset (\psi \rightarrow \xi )$           [Lemma 7]

• 3. $\psi \wedge (\psi \rightarrow \xi )\supset \xi$           [Lemma 7]

• 4. $(\phi \wedge (\phi \rightarrow \psi ))\wedge (\phi \rightarrow (\psi \rightarrow \xi ))\supset \xi$           [A6-8, 1.-3., R3]

• 5. $(\phi \rightarrow \psi )\rightarrow ((\phi \rightarrow (\psi \rightarrow \xi ))\rightarrow (\phi \rightarrow \xi ))$           [4., Lemma 6, R1, Exp. (twice)]

For the proof of IL8 we require another auxiliary lemma.

Lemma 14

$\vdash (\xi \wedge \chi )\wedge (\phi \vee \psi )\supset (\phi \wedge \chi )\vee (\psi \wedge \xi )$

Proof

• 1. $\chi \wedge (\phi \vee \psi )\supset (\chi \wedge \phi )\vee (\chi \wedge \psi )$          [A12]

• 2. $\xi \wedge (\chi \wedge (\phi \vee \psi ) )\supset \xi \wedge ((\chi \wedge \phi )\vee (\chi \wedge \psi ))$           [1., A6-8, R3]

• 3. $\xi \wedge ((\chi \wedge \phi )\vee (\chi \wedge \psi ))\supset (\xi \wedge (\chi \wedge \phi ))\vee (\xi \wedge (\chi \wedge \psi ))$           [A12]

• 4. $\xi \wedge (\chi \wedge \phi )\supset (\phi \wedge \chi )\vee (\psi \wedge \xi )$           [A7, A9, R3]

• 5. $\xi \wedge (\chi \wedge \psi )\supset (\phi \wedge \chi )\vee (\psi \wedge \xi )$           [A7, A8, R3]

• 6. $(\xi \wedge (\chi \wedge \phi ))\vee (\xi \wedge (\chi \wedge \psi ))\supset (\phi \wedge \chi )\vee (\psi \wedge \xi )$           [4., 5., A11, R2, R1]

• 7. $\xi \wedge (\chi \wedge (\phi \vee \psi ) )\supset (\phi \wedge \chi )\vee (\psi \wedge \xi )$           [2., 3., 6., R3]

Lemma 15

(IL8) $\vdash (\phi \rightarrow \tau )\rightarrow ((\psi \rightarrow \tau )\rightarrow (\phi \vee \psi \rightarrow \tau ))$

Proof

• 1. $(\phi \wedge (\phi \rightarrow \tau )\supset \tau )\wedge (\psi \wedge (\psi \rightarrow \tau )\supset \tau )\supset$ $((\phi \wedge (\phi \rightarrow \tau ))\vee (\psi \wedge (\psi \rightarrow \tau ))\supset \tau )$           [A11]

• 2. $(\phi \wedge (\phi \rightarrow \tau ))\vee (\psi \wedge (\psi \rightarrow \tau ))\supset \tau$           [Lemma 7, 1., R1, R2]

• 3. $((\phi \rightarrow \tau )\wedge (\psi \rightarrow \tau ))\wedge (\phi \vee \psi )\supset$ $(\phi \wedge (\phi \rightarrow \tau ))\vee (\psi \wedge (\psi \rightarrow \tau ))$           [Lemma 14]

• 4. $((\phi \rightarrow \tau )\wedge (\psi \rightarrow \tau ))\wedge (\phi \vee \psi )\supset \tau$           [2., 3., R3]

• 5. $((\phi \rightarrow \tau )\wedge (\psi \rightarrow \tau ))\wedge (\phi \vee \psi )\rightarrow \tau$           [4., Lemma 6, R1]

• 6. $(\phi \rightarrow \tau )\rightarrow ((\psi \rightarrow \tau )\rightarrow (\phi \vee \psi \rightarrow \tau ))$           [5., Exp. (twice)]

It only remains to show the two negation axioms IL9 and IL10. But IL9 is just an instance of IL2, with $\bot$ for $\xi$. IL10 may be proved as follows:

Lemma 16

(IL10) $\vdash (\lnot ^{i}\phi )\rightarrow (\phi \rightarrow \psi )$

Proof

• 1. $(\phi \rightarrow \bot )\wedge \phi \supset \bot$           [Lemma 7]

• 2. $\bot \supset \psi$           [A13]

• 3. $(\phi \rightarrow \bot )\wedge \phi \supset \psi$           [1., 2., R3]

• 4. $(\phi \rightarrow \bot )\wedge \phi \rightarrow \psi$           [3., Lemma 6, R1]

• 5. $(\lnot ^{i}\phi )\rightarrow (\phi \rightarrow \psi )$           [4., Exp., R1, Def.]

To state the first part of our central result, we define a translation $'$ from $\mathscr {L}_{\textbf{IL}}$ into $\mathscr {L}_{\textbf{CL}\pi }$ by taking $(\phi \rightarrow ^{i}\psi )'=\exists p(p\wedge ((\phi '\wedge p)\supset \psi '))$ and $(\lnot ^{i}\phi )'=\exists p(p\wedge ((\phi '\wedge p)\supset \bot ))$, and translating conjunctions, disjunctions and atomic formulas homophonically.²⁴

Corollary 2

If $\vdash _{\textbf{IL}}\phi$, then $\vdash _{\mathbf {CL\pi }}\phi '$.

In fact, what we have proven is significantly stronger than Corollary 2. For we have shown that the intuitionistic axioms and rules hold unrestrictedly for all formulas of $\mathscr {L}_{\mathbf {CL\pi }}$ (with $\rightarrow$ and $\lnot ^{i}$ understood as defined connectives), and not only for translations of formulas of $\mathscr {L}_{\textbf{IL}}$.

Next we show that the converse of Corollary 2 holds as well, so that the translation of a formula of $\mathscr {L}_{\textbf{IL}}$ is a theorem of $\mathbf {CL\pi }$ only if its inverse image is an intuitionistic theorem. This is what justifies saying that the embedding of $\textbf{IL}$ into $\mathbf {CL\pi }$ is exact.

Let $\mathbf {IL\pi }$ be the propositionally quantified intuitionistic logic whose language is the same as $\mathscr {L}_{\textbf{IL}}$ with the addition of the propositional quantifier $\exists$, and which results from $\textbf{IL}$ by adding the following axioms and rules for the quantifiers:

We will make use of the following auxiliary translation from $\mathscr {L}_{\mathbf {CL\pi }}$ to $\mathscr {L}_{\mathbf {IL\pi }}$. Define h to be the function from $\mathscr {L}_{\mathbf {CL\pi }}$ to $\mathscr {L}_{\mathbf {IL\pi }}$ that replaces every $\supset$ by $\rightarrow ^{i}$, and leaving everything else the same (i.e. translating everything else homophonically). For example, the formula $p\supset q$ gets translated into $p\rightarrow ^{i}q$.

We prove the following three claims, for $\phi$ in $\mathscr {L}_{\textbf{IL}}$:

Lemma 17

If $\vdash _{\mathbf {CL\pi }}\phi$, then $\vdash _{\mathbf {IL\pi }}h(\phi )$

Proof

The result follows immediately from the observation that every theorem of CL is a theorem of IL [Theorem 2 in Fine (2017)], the fact that $\exists 1$ and $\exists 2$ are common to both logics (with $\rightarrow ^{i}$ in place of $\supset$) and the fact that the intuitionistic analog of $\exists \wedge$ is provable in $\textbf{IL}\pi$.

Lemma 18

For $\phi$ in $\mathscr {L}_{\textbf{IL}}$, if $\vdash _{\mathbf {IL\pi }}h(\phi ')$, then $\vdash _{\mathbf {IL\pi }}\phi$.

Proof

We observe that the following equivalences are easily provable in $\mathbf {IL\pi }$:

• (i) $\vdash _{\mathbf {IL\pi }}(\phi \rightarrow ^{i}\psi )\leftrightarrow ^{i}\exists p(p\wedge ((p\wedge \phi )\rightarrow ^{i}\psi ))$

• (ii) $\vdash _{\mathbf {IL\pi }}\lnot ^{i}\phi \leftrightarrow ^{i}\exists p(p\wedge ((p\wedge \phi )\rightarrow ^{i}\bot ))$

Now suppose that $\phi$ is in $\mathscr {L}_{\textbf{IL}}$ and $\vdash _{\mathbf {IL\pi }}h(\phi ')$. We obtain a proof of $\phi$ from a proof of $h(\phi ')$ by noting that $\phi$ is logically equivalent to its translation $h(\phi ')$ by virtue of (i) and (ii) and the fact that substitution of logical equivalents preserves theoremhood in $\mathbf {IL\pi }$.

Finally, we show that $\mathbf {IL\pi }$ is a conservative extension of $\textbf{IL}$.

Lemma 19

For $\phi \in \mathscr {L}_{\textbf{IL}}$, if $\vdash _{\mathbf {IL\pi }}\phi$, then $\vdash _{\textbf{IL}}\phi$.

Proof

We prove the contrapositive. Suppose $\nvdash _{\textbf{IL}}\phi$. Then by the completeness of the Kripke semantics for IL there is a Kripke model $\mathscr {M}=\langle W,\le ,V\rangle$ for IL in which $\phi$ fails to hold: $\mathscr {M}\nvDash \phi$. But $\mathscr {M}$ can also be regarded as an $\textbf{IL}\pi$-model in the familiar way; we only need to extend the truth definition by a clause for the existential quantifier [see, for instance, Kremer (1997)]. So we still have $\mathscr {M}\nvDash \phi$. Hence by the soundness of this semantics for $\mathbf {IL\pi }$, we obtain $\nvdash _{\mathbf {IL\pi }}\phi$.

Corollary 3

For $\phi \in \mathscr {L}_{\textbf{IL}}$, if $\vdash _{\mathbf {CL\pi }}\phi '$, then $\vdash _{\textbf{IL}}\phi$.

Corollaries 2 and 3 imply

Theorem 3

(Embedding) $\mathbf {CL\pi }$ exactly embeds $\textbf{IL}$ under translation.