Franklin's Notes


Positive and conditional sentences

A positive sentence $\varphi$ is one which can be built up using only sentence symbols from $\mathscr{S}$, $\land$, and $\lor$. These sentences can be conceptualized as those which are never averse to making more statements true - that is, if $\varphi$ is true for some assignment of truth values of its sentence symbols, it can never become false when more of the sentence symbols become true. (It can also be thought of as a "monotone boolean function" on the sentence symbols.)

Proposition 1. For any models $A$ and $B$, $A\subset B$ if and only if all positive sentences which hold in $A$ also hold in $B$.

Proof. The "if" direction is easy - every sentence symbol is a positive sentence, so if all positive sentences holding in $A$ also hold in $B$, then $S\in B$ for all $S\in A$, and $A\subset B$.

The "only if" direction can be proven inductively. Suppose that $\varphi$ and $\psi$ are sentences that either hold in both $B$ and $A$, or only hold in $B$. Then $\varphi\land\psi$ and $\varphi\lor\psi$ also have the property that they either hold in both $B$ and $A$ or only in $B$. Hence, since sentence symbols $S$ have this property (so long as $A\subset B$), all positive sentences of $\mathscr{S}$, which are built up using $\land$ and $\lor$, must also have this property. $\blacksquare$

A set of sentences $\Sigma$ is called increasing iff $A\vDash\Sigma$ implies that $B\vDash\Sigma$ for all models $B$ such that $A\subset B$. In other words, making more sentence symbols true in a model can never cause a satisfied increasing theory to become unsatisfied.

Proposition 2. A consistent theory $\Gamma$ is increasing if and only if it has a set of positive axioms $\Delta$.

Proof. The "if" direction follows from the previous proposition. If $\Delta$ is a set of positive axioms for $\Gamma$, and $A,B$ are two models with $A\subset B$, then if $\Delta$ is satisfied by $A$, it must be satisfied by $B$, and since $\Delta\vdash\Gamma$ and therefore $\Delta\vDash\Gamma$, we have that $\Gamma$ must also be satisfied by $B$, making $\Gamma$ increasing.

Now, for the "only if" direction, we may let $\Delta$ be the set of all positive consequences of $\Gamma$. Let $\varphi\in\Gamma$. Then $\varphi$ is satisfied by $B$ if it is satisfied by $A$, for all models $A\subset B$, since $\Gamma$ is increasing. If some $\varphi$ is such that $(\neg\varphi)$ is valid, then $\Gamma$ is unsatisfiable, trivializing the claim, so let us ignore this case. Additionally, if all $\varphi\in\Gamma$ are valid, then the claim is also trivial, so we may assume not all sentences in $\Gamma$ are valid, and consider only those sentences $\varphi\in\Gamma$ which are not valid. Further, let ${S_1,...,S_n}$ be a subset of the sentence symbols including all of those which are used in $\varphi$, and consider the sentence where the disjunction ranges over all models $A'$ of ${S_1,...,S_n}$ satisfying $\varphi$ (of which there are finitely many). Because all supersets of models of $\varphi$ are also models of $\varphi$, we may conclude that $\varphi$ is equivalent to $\varphi'$. We may then use the set $\Delta={\varphi':\varphi\in\Gamma \text{ not valid}}$ as a set of positive axioms for $\Gamma$, since each statement in $\Delta$ is equivalent to a statement in $\Gamma$.

A conditional sentence $\varphi$ is one which takes the form $\varphi_1\land...\land\varphi_n$, where each $\varphi_i$ is in one of the following $3$ forms:
1. $S$ (a single sentence symbol)
2. $(\neg S_1)\lor...\lor(\neg S_p)$
3. $(\neg S_1)\lor...\lor(\neg S_p)\lor T$

and a set of sentences $\Sigma$ is said to be preserved under finite intersections if $A\vDash\Sigma$ and $B\vDash\Sigma$ together imply that $A\cap B\vDash\Sigma$. The syntactic property of conditionality and the semantic property of preservation under intersections are related in more or less the same ways as the properties of positivity and increasingness.

Proposition 3. A theory $\Gamma$ is preserved under finite intersections if and only if it is preserved under arbitrary intersections.

Proposition 4. A theory $\Gamma$ is preserved under intersections if and only if $\Gamma$ has a set of conditional axioms.

Proposition 5. A sentence $\varphi$ is preserved under intersections if and only if $\varphi$ is equivalent to a conditional sentence.

model-theory

sentential-logic

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