#analytic containment

[superclassical]

Take 2, Part 2

To show that the higher-degree containment logic described earlier has the desired relevant properties:

To show \(A\rightarrow B\vDash(C\rightarrow A)\rightarrow(C\rightarrow B)\):

Suppose \(\mathfrak{M}\vDash A\rightarrow B\) but \(\mathfrak{M}\nvDash(C\rightarrow A)\rightarrow(C\rightarrow B)\). Then there exists a \(\gamma\) such that \(X\in \bar{V}^{+}_{\gamma}(C\rightarrow A)\) but \(X\notin \bar{V}^{+}_{\gamma}(C\rightarrow B)\). As \(X\in \bar{V}^{+}_{0}(A\rightarrow B)\), any \(\pi\in\bar{V}^{+}_{\gamma}(A)\) is also a member of \(\bar{V}^{+}_{\gamma}(B)\), i.e., \(\bar{V}^{+}_{\gamma}(A)\subseteq\bar{V}^{+}_{\gamma}(B)\). The semantic constraint implies that  \(\bar{V}^{+}_{\gamma}(C\rightarrow A)\subseteq\bar{V}^{+}_{\gamma}(C\rightarrow B)\). [Note that trivially \(\bar{V}^{+}_{\gamma}\upharpoonright X=\bar{V}^{+}_{\gamma}\).] That \(X\in \bar{V}^{+}_{\gamma}(C\rightarrow A)\) thus entails that \(X\in \bar{V}^{+}_{\gamma}(C\rightarrow B)\). Contradiction.

To show \(A\leftrightarrow B\vDash(B\rightarrow C)\rightarrow(A\rightarrow C)\):

Suppose \(\mathfrak{M}\vDash A\leftrightarrow B\) but \(\mathfrak{M}\nvDash(B\rightarrow C)\rightarrow(A\rightarrow C)\). Then for some \(\gamma\),  \(X\in \bar{V}^{+}_{\gamma}(B\rightarrow C)\) but \(X\notin \bar{V}^{+}_{\gamma}(A\rightarrow C)\). Since \(X\in \bar{V}^{+}_{0}(A\leftrightarrow B)\), \(\bar{V}^{+}_{\gamma}(A)=\bar{V}^{+}_{\gamma}(B)\), whence by the constraint \(\bar{V}^{+}_{\gamma}(B\rightarrow C)=\bar{V}^{+}_{\gamma}(A\rightarrow C)\). This entails that \(X\in \bar{V}^{+}_{\gamma}(A\rightarrow C)\). Contradiction.

To show \(A\rightarrow B\nvDash(B\rightarrow C)\rightarrow(A\rightarrow C)\):

We show this by countermodel. Let \(\mathfrak{M}=(X,\Pi,V)\) where \(X=\lbrace a\rbrace\), \(\Pi=\lbrace X\rbrace\), and functions \(V_{0}\) and \(V_{1}\) are such that the former is empty and the latter maps \(B\) and \(B\rightarrow C\) to \(\lbrace X\rbrace\) and all other formulae to \(\varnothing\). Then \(X\in\bar{V}^{+}_{0}(A\rightarrow B)\) vacuously as there are no states at which \(A\) holds. Also, \(X\in\bar{V}_{1}^{+}(B\rightarrow C)\) and \(X\in\bar{V}_{1}^{+}(A\rightarrow C)\). That \(\bar{V}_{1}^{+}(A)\neq\bar{V}_{1}^{+}(B)\) permits this evaluation. Hence, \(X\notin\bar{V}_{0}^{+}((B\rightarrow C)\rightarrow(A\rightarrow C))\), providing the necessary countermodel.

There is still much work to do. But it's a reasonable--and natural--first pass. (Although, strictly speaking, it's the second pass.)

[superclassical]

Simplified Correia Models

Fabrice Correia provided the first (and only published) semantics for Angell's \(\mathsf{AC}\). Let \(\mathbf{At}\) be a set of atoms in some language. Then the semantics involve models \(\mathbf{v}\) composed of ordered pairs of sets of atoms \(<\Gamma,\Delta>\) such that

If \(<\Gamma,\Delta>\in\mathbf{v}\) and \(<\Gamma'\cup\Gamma'',\Delta'\cup\Delta''>\in\mathbf{v}\), then  

\(<\Gamma\cup\Gamma',\Delta\cup\Delta'>\in\mathbf{v}\)

The relation \(\vDash\) is then defined recursively:

\(\Gamma\vDash\Delta\) if \(<\Gamma,\Delta>\in\mathbf{v}\)

\(\Gamma\vDash\Delta,\neg A\) if \(\Gamma,A\vDash\Delta\)  

\(\Gamma,\neg A\vDash\Delta\) if \(\Gamma\vDash\Delta,A\)  

\(\Gamma\vDash\Delta,A\wedge B\) if \(\Gamma\vDash\Delta,A\)  and \(\Gamma\vDash\Delta,B\) 

\(\Gamma\vDash\Delta,A\vee B\) if \(\Gamma\vDash\Delta,A B\)  

\(\Gamma,A\wedge B\vDash\Delta\) if \(\Gamma ,A ,B\vDash\Delta\) 

\(\Gamma,A\vee B\vDash\Delta\) if \(\Gamma,A\vDash\Delta\) and \(\Gamma,B\vDash\Delta\)

Now, the semantics are a bit difficult to deploy.For one, the elements of the model are a bit unwieldy.

There is, however, an observation due to William Tait that ameliorates this situation somewhat. In proof theory, Tait noted that there wasn't a need to track both sides of a sequent in a Gentzen calculus. Rather, all information could be captured by shifting everything into the succedent position, adding a "\(\neg\)" to every formula in the antecedent.

Similarly, we can employ this in our case. Let \(\mathbf{Lit}\) be the set of literals---positive and negative atoms. Then consider sets \(\mathbf{v}^{\ast}\) of literals closed under the constraint that 

If \(\Gamma\in\mathbf{v}^{\ast}\) then \(\uparrow\Gamma\subseteq\mathbf{v}^{\ast}\)

where \(\uparrow\Gamma\) is the filter on \(\wp(\bigcup\mathbf{v})\) generated by \(\Gamma\), i.e., \(\mathbf{v}^{\ast}\) is closed under filters.

The semantics then becomes:

\(\vDash\Gamma\) if \(\Gamma\in\mathbf{v}^{\ast}\)

\(\vDash\Gamma,A\wedge B\) if  \(\vDash\Gamma,A\)   and \(\vDash\Gamma,B\) 

\(\vDash\Gamma,A\vee B\) if \(\vDash\Gamma,A,B\) 

\(\vDash\Gamma,\neg(A\wedge B)\) if \(\vDash\Gamma,\neg A,\neg B\) 

\(\vDash\Gamma,\neg(A\vee B)\) if \(\vDash\Gamma,\neg A\)  and \(\vDash\Gamma,\neg B\) 

And there you go; the price to pay for vastly simplifying the models and eliminating twenty percent of the recursive definition for turnstile is considering literals instead of atoms. Moreover, as intended, an apparent connection between this logic and Nelson's logic of constructible falsity emerges.

[superclassical]

To the 2nd Degree and Beyond

It was suggested to me that in order to punch past the first degree that the ternary relation semantics common in relevant logics might be useful. The relevant semantics involve a ternary relation \(R\) on points such that for a point \(w\), \(w\Vdash A\rightarrow B\) if and only if for all points \(x,y\) such that \(Rwxy\), if \(x\Vdash A\) then \(y\Vdash B\). A qualification is made such that for some distinguished point \(@\) (a "normal" world), \(R@xy\) holds if and only if \(x=y\). 

A stumbling block appears, however, if we naively try to employ such semantics. Note, for example, the principle known as suffixing, that \(A\rightarrow B\vdash(B\rightarrow C)\rightarrow (A\rightarrow C)\). Reading "\(\rightarrow\)" as containment, it is true that assuming that \(A\) contains \(B\) that should \(B\) likewise contain \(C\), \(A\) itself will contain \(C\); this is just to say that containment is transitive. But this does not entail that \(B\rightarrow C\) contains \(A\rightarrow C\); notably, the content in \(A\) not found in \(B\) will not be found in \(B\rightarrow C\).

But the ternary relation semantics confirm this immediately. Suppose \(@\Vdash A\rightarrow B\) but \(@\nVdash (B\rightarrow C)\rightarrow(A\rightarrow C)\). Then there is a point \(x\) with \(R@xx\) such that \(x\Vdash B\rightarrow C\) but \(x\nVdash A\rightarrow C\).

The latter point entails that there are \(u,v\) with \(Rxuv\) such that \(u\Vdash A\) and \(v\nVdash C\). Moreover, \(R@uu\), \(@\Vdash A\rightarrow B\), and \(u\Vdash A\) jointly entail that \(u\Vdash B\). Finally, that \(Rxuv\) and \(x\Vdash B\rightarrow C\) entails that \(v\Vdash C\). Contradiction.

The problem arrives because \(@\) "sees" the point \(u\). My proposed solution is adding a notion of rank to the points. Notably, for the normal point \(@\), consider the constraints that a) \(Rank(@)=0\) and b) \(R@xy\) if and only if both \(x=y\) and \(Rank(x)=0\). Finally, for any non-normal points \(w\), add the constraint that if \(Rwxy\) then \(Rank(x)=Rank(y)+1=Rank(w)+1\).

Looking at the above proof of suffixing, we can see that the proof is thereby interrupted. Inasmuch as \(x\) is non-normal, that \(Rxuv\) implies that \(Rank(u)=1\), precluding the necessary step that \(R@uu\).

Kinks still need to be worked out, but it is a promising start, I think.

[superclassical]

A First Degree Containment Logic

A feature of containment logics is the failure of disjunctive weakening, the principle of inference suggesting that from \(A\) one may infer \(A\vee B\) (i.e., \(A\vdash A\vee B\)). The motivation for rejecting this is that \(A\) may not "say" anything about \(B\), that is, it may not contain \(B\). In most treatments, the failure of this principle is given a syntactical explication; the succedent contains a parameter not contained in the antecedent.

But this is unenlightening. A fully semantic treatment can be provided for such a logic by asking what a disjunctive proposition really looks like. Suppose for a moment that disjunctions are in a sense really conjunctions--conjunctions of assertions of possibility, that to say "\(A\) or \(B\)" is to say "both \(A\) and \(B\) are possible.'' 

This can be modeled in the following way:

Let a model \(\mathbf{M}\) be a set of points \(W\) of order type \(\eta\)--effectively a copy of the rationals--along with two functions \(V^{+}\) and \(V^{-}\) mapping propositional parameters to subsets of \(W\). Recursively define a forcing relation between subsets of \(W\) and zero-degree formulae (formulae without instances of implication) by the following:

-\(X\Vdash A\) if \(X\subseteq V^{+}(A)\) for atoms \(A\)

-\(X\Vdash\neg A\) if \(X\subseteq V^{-}(A)\) for atoms \(A\)

-\(X\Vdash\neg\neg A\) if \(X\Vdash A\)

-\(X\Vdash A\wedge B\) if \(X\Vdash A\) and \(X\Vdash B\)

-\(X\Vdash \neg(A\vee B)\) if \(X\Vdash\neg A\) and \(X\Vdash\neg B\)

-\(X\Vdash A\vee B\) if there are nonempty sets \(X_{0}\) and \(X_{1}\) such that \(X_{0}\cup X_{1}=X\) and \(X_{0}\Vdash A\) and \(X_{1}\Vdash B\)

-\(X\Vdash \neg(A\wedge B)\) if there are nonempty sets \(X_{0}\) and \(X_{1}\) such that \(X_{0}\cup X_{1}=X\) and \(X_{0}\Vdash\neg A\) and \(X_{1}\Vdash\neg B\)

Then for formulae \(A,B\), say that \(A\vDash B\) if for any model \(\mathbf{M}\) that \(W\Vdash A\) implies that \(W\Vdash B\).

Despite the unorthodox treatment of disjunction, it can be checked that this modeling provides a number of reasonable features. De Morgan's laws hold, distributivity between conjunction and disjunction holds, associativity holds of each connective, and conjunctive simplification holds. Our "mock disjunction" really behaves as a disjunction.

Next up: moving past the first degree.

[superclassical]

Analyticity and Kant and Gödel I

I'm not a Kant scholar but I take it that there have been a few headaches over how Kant meant the term "analytic." It has been claimed that two notions of analyticity are afoot, which Kant mistakenly takes to be equivalent. The first we see in quotes from the first Critique such as:

In the analytic judgment we keep to the given concept, and seek to extract something from it. If it is to be affirmative, I ascribe to it only what is already thought in it.

On this account, a statement such as "all rectangles have four sides" is analytic because to confirm that the subject has the property one need only look to the concept of rectangularity itself. One will see that the property of four-sidedness is "contained" within it ("is already thought in it").

Kant, however, being motivated to counter some of Hume's theses, also employs the principle of contradiction- that no proposition and its negation may be simultaneously true- as a ready to hand technique to decide the truth of an analytic proposition:

[I]f the judgment is analytic... its truth can always be adequately known in accordance with the principle of contradiction.

Some of the debate, I think, centers around whether the two- analytic in virtue of "containment" and analytic in virtue of noncontradiction- are indeed one and the same. I think it's pretty clear that Kant intended them as equivalents; while the second quote indicates that the latter is necessary for the former, Kant explicitly suggests that it is "sufficient" as well. Kant's arithmetical examples betray a style of doing mathematics quite a bit different than the modern, axiomatic approach, so I don't think it is crazy to conflate the two. But I don't have a dog in the fight with respect to the historical question. (Though I will add that irrespective of their equivalence or inequivalence, I've always thought that Gödel's Theorems demonstrated quite nicely the existence of the synthetic a priori.)

I'd soon like to outline some strategies for the development of a formal approach to the notion of analyticity-as-containment and how this can be reconciled with a classical framework. For now, I'll leave with the below blatant appeal to authority. 

In Parry's 1933 dissertation (as relayed by Anderson and Belnap's Entailment), Gödel is quoted as saying:

"\(p\) implies \(q\) analytically" can be perhaps interpreted as "\(q\) is derivable from \(p\) and the logical axioms and does not include any other concepts than \(p\)" and, after this definition is more precise, it would be [wise?] to seek a completeness theorem [Vollständigkeitsbeweis] for the axioms of Parry, in the sense that all sentences valid for the above interpretation of \(\rightarrow\) are derivable. 

So, yes, Gödel is just as fallible as Kant. Nevertheless, the approach of retaining one logic (classical logic) to handle synthetic truths while employing a fragment thereof (in this case, Parry's \(\mathsf{PAI}\)) to track the analytic has a pretty distinguished provenance.