To give the meaning of a sentence is, at least, to say when it is true, and a sentence is true or false only in a model. Build a small model of individuals and extensions, evaluate a quantified sentence by evaluating its parts, and change one pair in one extension to turn the same sentence from true to false.
Core ideas
The mind half of this course evaluated a claim in a model, whether of worlds, of rows or of likelihoods, and found the philosophy in the choice of model. The language half asks the same of a sentence. This first lesson fixes what it is to evaluate one, and the four lessons after it only change which sentences are evaluated.
A finite model lists some individuals, says for each predicate which individuals (or which pairs) it holds of, and says which individual each name picks out. That list is everything the sentence is evaluated against, and nothing else is consulted.
An atomic sentence is true when its individuals are in the extension. The connectives work as in Arguments and Validity. A quantifier asks its part to be true of every individual, or of some. This is compositionality: the whole is computed from the parts and how they are put together.
Because the value is computed from the model, one sentence read in two models can be true in one and false in the other. Truth conditions say when it would be true; they do not say that it is.
The material
A definition, one model, one evaluation worked by hand, and the same sentence in a model with one pair removed.
The truth conditions of a sentence are the conditions under which it would be true. To know them is to know, for any situation described in enough detail, whether the sentence is true there. It is not to know whether the sentence is true.
A situation, for this lesson, is a finite model. It has a domain of individuals, an extension for each predicate, and a referent for each name. Here is a small one, with three individuals a, b and c. The predicate Planet holds of a and b. The two-place predicate Orbits holds of the pair a and b taken in that order, and of the pair b and c. The model is a table and not astronomy; it is only meant to be easy to hold in the head.
The rules for evaluating a sentence are short, and they are the whole of the method.
Take the sentence ∀x (Planet(x) → ∃y Orbits(x, y)), which says that every planet orbits something. Put a for x. Planet(a) is true, and ∃y Orbits(a, y) is true because putting b for y gives a pair in the extension, so the conditional is true. Put b for x: Planet(b) is true, c is the something it orbits, and the conditional is true. Put c for x: Planet(c) is false, so the conditional is true whatever c orbits. Every individual passes, and the lab prints True.
Now remove the pair b and c from the extension of Orbits, and change nothing else. The sentence is the same sentence, with the same words in the same order. At x = a it still passes. At x = b the planet b orbits nothing, ∃y Orbits(b, y) has no witness, the conditional is false, and the sentence is false. The lab prints False and names b as the place it fails. One pair in one extension decided the value.
That is the point about truth conditions. Someone who understands the sentence knows what it takes for it to be true, and that is the rule above applied to any model. They do not thereby know which model is the actual one. So “is this sentence true?” has no answer until a model has been named, and what we call a sentence's being true on its own is always a model left unspoken.
An existential sentence is the same idea with the quantifier turned over. ∃x (Planet(x) ∧ Orbits(x, x)) asks for a planet that orbits itself. No individual in the model is one, so there is no witness to report, and the value is False. A universal sentence is refuted by one individual and an existential one is confirmed by one; each is false when the other kind of individual is missing altogether.
Two quantifiers nest the same way. ∀x ∃y Loves(x, y) asks, for each individual, whether some individual is loved by it, and in a ring where a loves b, b loves c and c loves a every individual passes, so the lab prints True. The tile for the outermost quantifier counts the passes: 3 of 3 in the ring, 3 of 3 for the planets, and 2 of 3 once the pair b, c is gone, which is the failing individual seen from the other side. Whether the order of two quantifiers matters is its own question, and “Scope Ambiguity and Negation” puts the same ring to that use.
The limit is the usual one. The lab evaluates in the model you typed, and a model has the individuals and pairs you gave it. Whether the pairs describe the solar system, or anything, is not something it can check.
Interactive
A first-order sentence evaluated over a domain you can edit
Method
Five steps, the same for any sentence and any model.
List the individuals, the extension of each predicate, and the referent of each name. If the sentence uses a predicate the model does not list, the model is incomplete.
That is the last thing the sentence does, and it fixes what has to be checked first.
Put each individual for the variable in turn. A universal is refuted by one failure; an existential is confirmed by one success.
At each individual the part is a smaller sentence. Evaluate it the same way, down to atomic sentences, and read those off the extensions.
Name the witness or the failing individual, and the pair or membership in the extension that put it there.
The sentence is ∀x (Planet(x) → ∃y Orbits(x, y)), read at each individual in turn.
The sentence did not change, and neither did the domain. The value moved because the extension of Orbits did, which is what it is for a sentence to be true in a model and not otherwise.
Given a finite model and a quantified sentence, you should be able to evaluate it from the values of its parts, name the witness or the failing individual, and say which single change to an extension would reverse the value. An answer that gives only True or False has not shown the evaluation.
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