QL vs. PL Semantics
In PL, the smallest unit of analysis was the complex proposition. We could determine the truth value of complex propositions by looking at the truth-value of their parts, noting how those parts were put together using our truth-functional operators. When it comes to simple propositions in PL, all we can do is say that such propositions are truth or false. We can't give truth conditions for simple sentences.
To sum up, in PL we showed that we could give truth conditions for a proposition like:
| Fred is tired or Mathilda is happy. |
noting that is a disjunction. The truth value of the whole disjunction is specified by the truth-table for disjunction. But PL cannot give us truth-conditions for:
| Someone is tired. |
since there are no operators in this sentence. In QL we sharpen our analytical focus a bit. That is, we note that there are "parts" of this sentence, though the parts are smaller than sentences. In particular, we note that "Someone is tired" is an existentially quantified sentence, which we can translate as follows:
| ∃xTx |
Now we can ask: Under what conditions is this proposition true? Since the proposition states that someone is tired, or there is an x, such that x is tired, the truth or falsity of this proposition depends on whether there is anything at all, and if there is something, whether it has the property of being tired. So the truth value depends on our universe or domain of discourse. Remember that we abbreviate individuals by the use of individual constants. So we can specify a domain of discourse by constructing a set with individuals referred to by individual constants.
Let's look at how we interpret our proposition in various domains of discourse:
| domain | proposition | truth conditions |
| {} | ∃xTx | false; the proposition says that something is Tx, but there is nothing to be Tx! |
| {a} | Ta | true if for the one member of the domain, it has the property Tx |
| {a, b} | Ta v Tb | true if for at least one member of the domain, it has the property Tx. |
We know, then, that this simple existentially quantified wff is false in an empty domain. We know that it can be true in a domain of one individual, (a) when Ta is true. And it can be true in a domain of two individuals (a, b) when Ta is true or Tb is true. This is a lot less tidy than the semantics for PL, but that's what happens when we use a more fine-grained logical analysis of propositions!
Question: What are the truth-conditions for $xTx in a domain of three individuals?
We just introduced the semantics for a simple existential sentence in QL. Now we look at a simple universal sentence.
| Everyone is enthusiastic. |
Using "Ex" for "x is enthusiastic" we translate this sentence as:
| (x)Ex |
| domain | proposition | truth conditions |
| {} | (x)Ex | true; the proposition says that everything is Ex, and everything is - or better put, it's not the case that something is not Ex. |
| {a} | Ea | true if for all members of the domain, it has the property Ex |
| {a, b} | Ea & Eb | true if for each member of the domain, it has the property Ex. |
Now that we've translated the universally quantified wff as a conjunction in this domain, we haven't given the truth conditions until we give the truth-conditions for the conjunction. But we know how do that! We simply do a truth-table for the conjunction. So the truth-conditions for (x)Ex are provided in the truth-table:
| Ea | Eb | (Ea & Eb) |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
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