| (1) | A conjunction of two propositions A and B is true only if A is true and B is true. |
| (2) | □(A conjunction of two propositions A and B is true only if A is true and B is true). |
| (3) | □(A conjunction of two propositions A and B is true only if A is true and B is true) because conjunction just is that binary function of propositions the value of which is a true proposition iff both its arguments are true propositions. |
| (C1) | (3) provides an explanation of the necessity. |
| (C2) | The explanans in (3) is necessarily true. |
| (C3) | The explanation that (3) provides ‘neither appeals to nor otherwise presupposes its necessity’ in order to explain the necessity of the explanandum. |
| (3’) | □(A conjunction of two propositions A and B is true only if A is true and B is true) because |
| □(A conjunction is true iff both its conjuncts are true). |
| (E3) | By the nature of conjunction, a conjunction is true iff both its conjuncts are true.5 |
| (E3’) | □(A conjunction is true iff both its conjuncts are true). |
| (E3*) | The source of the necessity expressed in (2) is the nature of conjunction alone. |
| (Q1) | How does (E3) explain the necessity of (2)?6 |
| (Q2) | What is the connection between explaining the truth of a proposition by appeal to the nature of some entity and explaining its necessity? |
| (Q3) | How does (E3) explain the truth of (2)? |
| (NE3) | □(By the nature of conjunction, a conjunction is true iff both its conjuncts are true).8 |
| (E3) | By the nature of conjunction, a conjunction is true iff both its conjuncts are true. |
| (2) | □(A conjunction of two propositions A and B is true only if A is true and B is true). |
| (NC) | □(There is such a thing as the nature of conjunction). |
| (E4) | The nature of conjunction is a pure (second-level) property. |
| (E5) | By the nature of pure property, a pure property exists if there could be a suitable predicate. |
| (NE5) | □(By the nature of pure property, a pure property exists if there could be a suitable predicate). |
| (E3+) | Conjunction just is that binary function of propositions the value of which is a true proposition iff both its arguments are true propositions. |
| (N) | The nature of essence is such that, if something is true in virtue of an entity’s essence, then it is necessary. |
| (NN) | □(The nature of essence is such that, if something is true in virtue of an entity’s essence, then it is necessary.) |
| (F) | It is true in virtue of X’s nature that Φ(X) |