2.1.2. A truth function
We can begin to provide a single account of the laws outined in 2.1.1 by recalling our definition of entailment. In positive form, it says that a set Γ entails a sentence φ if and only if φ is T in every possible world in which each member of Γ is T. Restating our three laws in these terms, we have
φ is T in every possible world in which φ ∧ ψ is T
ψ is T in every possible world in which φ ∧ ψ is T
φ ∧ ψ is T in every possible world in which both φ and ψ are T
The last of these says that φ ∧ ψ is true in a possible world if both φ and ψ are true in that world. While the first two taken together tell us that φ ∧ ψ is true in a possible world only if both φ and ψ are true. In short, φ ∧ ψ is true in a possible world if and only if both φ and ψ are true.
This means that the truth value of the compound φ ∧ ψ is determined by the truth values of the components φ and ψ, a fact we can express in the truth table below.
| φ | ψ | φ | ∧ | ψ |
|---|---|---|---|---|
| T | T | T | ||
| T | F | F | ||
| F | T | F | ||
| F | F | F |
This table shows the contribution of conjunction to the truth conditions of compound sentences formed using it, for it tells us how to determine the truth value of a conjunction φ ∧ ψ in any possible world once we know the truth values of the conjuncts φ and ψ. And the table also shows what lies behind the general laws of entailment that led us to it: it is because conjunction makes this sort of contribution to the meaning of sentences that those laws hold. A particular way of associating an output truth value with input truth values is a truth function, so we can say that conjunction expresses a truth function, and we can say the laws of entailment stated above reflect the character of the truth function that conjunction expresses.
It is worth pausing a moment to look at the way in which the proposition that is expressed by a conjunction is related to the propositions that are expressed by its components. Since a conjunction is false whenever either component is false, it rules out any possibility ruled out by either component; and, since the possibilities ruled out are an indication of the information a sentence contains, we can say that a conjunction contains all the information contained in its components. This means that the effect of conjunction is to add up informational content. Now, more information means fewer possibilities left open and, looking at the table in these terms, we see that a conjunction leaves open only the possibilities left open by both components. This means that the range of possibilities it leaves open is the region in the full space of possibilities where the ranges of possibilities left open by the two components overlap.
For example, the sentence The number shown by the die is odd and less than 4 can be analyzed as the conjunction The number shown by the die is odd ∧ the number shown by the die is less than 4. The first component rules out possibilities where the die shows 2, 4, or 6 and the second rules out possibilities where it shows 4, 5, or 6. The conjunction rules out all these possibilities—that is, any possibility where the die shows 2, 4, 5, or 6. Looking at things in terms of the possibilities left open, the first component leaves open those where the die shows 1, 3, or 5 and the second leaves open those where it shows 1, 2, or 3. The conjunction leaves open a possibility when it is left open by both components; that is, it leaves open those where the die shows 1 or 3.
This is shown pictorially in Figure 2.1.2-1 below.
Here, each rectangle represents the space of all possible worlds. The die faces mark regions consisting of the possible worlds in which the die shows one or another number. In Figure 2.1.2-1A, the possibilities ruled out by the first component are at the bottom while those ruled out by the second component occupy the region at the right. The possibilities left open by the first component then form the region in the top half while those left open by the second are in the region at the left. Figure 2.1.2-1B shows the proposition expressed by the conjunction of these two sentences. The possibilities ruled out add up to form the shaded region; those left open are in the unhatched region at the top left where the ranges of possibilities left open by the original components overlap. These diagrams can be compared to the truth table for conjunction. The sort of worlds covered by first row of the table, worlds where both components are true, appear at the top left of the 2.1.2-1A; the other rows of the table correspond to the remaining three regions of the this diagram—those at the top right, the bottom left, and the bottom right, respectively.
Although the most fundamental approach to the deductive properties of the logical form will come through laws governing its role as the conclusion of an entailment or as one among its possibly many premises, specific characteristics of a logical form can often be highlighted most clear by its significance for relations between pairs of sentences, especially the positive relations of implication and equivalence. The following principles are some of the more important examples of this in the case of conjunction:
Commutativity. The order of conjuncts in a conjunction does not affect the content. That is, φ ∧ ψ ≃ ψ ∧ φ.
Associativity. When a conjunction is a conjunct of a larger conjunction, the way components are grouped does not affect the content. That is, φ ∧ (ψ ∧ χ) ≃ (φ ∧ ψ) ∧ χ.
Idempotence. Conjoining a sentence to itself does not change the content. That is, φ ∧ φ ≃ φ.
Covariance. A conjunction implies the result of replacing a component with anything that component implies. That is, if ψ ⊨ χ, then φ ∧ ψ ⊨ φ ∧ χ and ψ ∧ φ ⊨ χ ∧ φ.
The names of these principles are terms used for analogous principles in other contexts. For example, you may have encountered the first two as names of principles for addition and multiplication since order and grouping do not matter for these operations. Conjunction shares the third property with numerical operations that produce the maximum or minimum of a pair of numbers, and this is not surprising since, if we think of truth values as being ordered so that falsity comes below truth, the truth value of a conjunction is just the minimum of the truth values of its components.
The last property, covariance, says roughly that the content of a conjunction varies in the same direction as the content of its components. An analogous property holds for addition and the maximum and minimum operations (e.g., if y ≤ z then min(x, y) ≤ min(x, z)) but it doesn’t hold for multiplication when negative numbers are considered (e.g., −2 × 3 > −2 × 4 even though 3 ≤ 4). We cannot say that an increase or decrease in the content of one component will produce an actual increase or decrease, respectively, in the conjunction since information added or lost in a change to one component may be provided in any case by the other component. For example, although The sign had red letters on a blue background says more than does The sign had red letters, the conjunction The sign had red letters on a blue background, and the background was light blue is equivalent to The sign had red letters, and the background was light blue. (This is analogous to the fact that, min(2, 3) = min(2, 4) even though 3 < 4.) What can be said is that, if the content of one component of conjunction increases, the content of the conjunction must increase if it changes at all.
One consequence of covariance is the following principle:
Compositionality. Conjunctions are equivalent if their corresponding components are equivalent. That is, if φ ≃ φ′ and ψ ≃ ψ′, then φ ∧ ψ ≃ φ′ ∧ ψ′.
Although this follows from covariance (since equivalent components imply each other), it can hold when covariance does not. And compositionality is so fundamental that, if conjunction did not satisfy it, we might hesitate even to count it as a logical form. Since sentences are logically equivalent when they express the same proposition, this principle says that conjunctions cannot express different propositions unless there is some difference in the propositions expressed by their components. Understanding the meanings of sentences to be the propositions expressed, the principle of compositionality tell us that the meaning of a conjunction is composed out of the meanings of its components in the particular way we label conjunction.