Some Basic Implication Laws and Top-Down Derivations

Our previous (3) (which characterizes implication from a single premise in terms of logical truth) can be now stated as:

(6) A|= B ⇐⇒ |= A → B

And this can be generalized to the following important law:

(7) Γ, A|= B ⇐⇒ Γ|= A → B

(6) is a particular case of (7), obtained when Γ is empty. Here is the proof of (7):

To prove the left-to-right direction, assume that Γ, A|= B and show: Γ |=

A → B, i.e., that it is impossible (by virtue of the logical elements) that all members of Γ are true and A→ B is not. Assume a case where all members of Γ get T. If A gets F, then A → B gets T (by the truth-table of → ).

And if A gets T, then all members of Γ, A get T; since we assumed that Γ, A|= B, B gets T. Again, by the truth-table of →, A → B gets T.

To prove the right-to-left direction, assume that Γ|= A → B, and show: Γ, A |=

B, i.e., that it is impossible (by virtue of the logical elements) that all mem-bers of Γ, A get T and B get F. So assume that all memmem-bers of Γ, A get T.

Then (i) all members of Γ get T and (ii) A gets T. Having assumed that Γ |= A → B, it follows that A → B gets T. Since also A gets T, it follows, by the truth-table of →, that B gets T.

Note that the argument relies on the logical elements of the sentences in Γ, A, B and on the truth-table of →, which is itself a logical element.

(7) provides a very useful way of establishing implications in which the conclusion is a con-ditional. We can refer to it by the (rather unwieldy) “conclusion-conditional law”. We also mark it by the following self-explanatory notation

(|=, →) .

Here is an illustration how (|=, →) can work. Suppose that we want to show that:

|= (A → (B → A))

Using (|=, →) (where Γ is empty and B is substituted by B → A) this reduces to showing that:

A|= B → A

Again, using (|=, →) (where Γ consists of A, A is substituted by B, and B by A), this reduces to:

A, B|= A

But this last implication is obvious, because the conclusion occurs as one of the premises.

Thus, we have established the logical truth of our original sentence.

The argument just given is an example of a top-down proof, or top-down derivation: We start with the claim to be proved and, working our way “backward”, we keep reducing it to other sufficient claims (i.e., claims that imply it), until we reduce it to obviously true claims. We can then turn the argument into a bottom-up proof: the familiar kind that starts with obviously true claims and moves forward, in a sequence of steps, until the desired claim is established.

The bottom-up proof is obtained by inverting the top-down derivation. In our last example the resulting bottom-up proof is:

A, B |= A obvious,

A |= B → A by (|=, →),

|= A → (B → A) by (|=, →).

The implications occurring in top-down derivations are referred to as goals. The derivation starts with the initial goal (the implication we want to prove) and proceeds stepwise by reducing goals to other goals, until all the required goals are self-evident. The top-down method figures prominently in the sequel. Besides (|=, →), we shall avail ourselves of other laws. It goes without saying that all the laws are general schemes holding for all possible values of the sentential variables. Here are some.

(8) If Γ|= A and every sentence that occurs in Γ occurs in Γ⁰, then Γ⁰ |= A.

(8) says that the addition of premises can only increase the class of implied consequences. This property is the monotonicity of logical implication, or of the logical consequence relation.³ Given our definition of implication, (8) is trivial (if it is impossible that all the sentences in Γ get T and A gets F, then, a fortiori, it is impossible that all sentences in Γ⁰ get T and A gets F).

Note: Monotonicity obtains for many types of implication, not necessarily logical, and it seems quite obvious: By adding more premises one can only get more consequences, not less. Yet we often employ reasonings that are not monotone. Conclusions established on the basis of some information may be withdrawn when some additional information is obtained.

The well-known example is: Being told that Twitty is a bird, one will conclude that Twitty can fly; but one will withdraw this conclusion if told, in addition, that Twitty is a penguin.

Inferences of that nature have been, in the last twenty years, the subject of considerable research by logicians, computer scientists and philosophers working in the area of belief change and artificial intelligence. Various formal systems have been proposed. They come under the title of non-monotonic logic.

Not as obvious as (8), but still quite easy, is the following law of consequence addition:

(9) If Γ|= A, then, for every sentence B:

Γ|= B ⇐⇒ Γ, A|= B

(9) means that any consequence of the given premises can be added as an additional premise, without changing class of consequences. Here is the proof.

The left-to-right direction of the ‘⇐⇒’ follows from monotonicity. For the right-to-left direction, assume that Γ, A |= B. If all sentences in Γ get T, then, by the initial assumption (that Γ |= A), A must get T; hence, all sentences in Γ, A get T; therefore B gets T. Thus, it is impossible that all sentences in Γ get T and B gets F.

The following generalization of (9) allows us to add as premises many consequences of the original list.

(9^∗) Assume that every sentence in ∆ is a consequence of Γ, then Γ and Γ, ∆ have the same consequences; that is, for every sentence C:

Γ|= C ⇐⇒ Γ, ∆|= C

3In mathematics, ‘monotone’ (or ‘positively monotone’) is used to describe relations in which an increase in one quantity does not cause a decrease in another related one. For example, 2·x is a monotone function of x, for it doesn’t become smaller as x becomes larger. But x²− 2x is not monotone, for, as x becomes larger it sometimes increases and sometimes decreases (e.g., it increases if x is increased from 1 to 2, but decreases if x is increased from 0 to 1).

(9^∗), can be proved by the same reasoning that proves (9). It can be also deduced by re-peated applications of (9). (Assume that ∆ = B1, . . . , Bm; then every Bi is a consequence of Γ. By (9), Γ and Γ, B1 have the same consequences. Since B2 is a consequence Γ, it is, by monotonicity, a consequence of Γ, B1; again, by (9), Γ, B1 and Γ, B1, B2 have the same consequences. Therefore Γ and Γ, B1, B2 have the same consequences, etc.)

(9^∗) implies the following generalization of the transitivity law of one-premise implications:

If (i) for every B in ∆, Γ|= B, and (ii) ∆|= C, then Γ|= C .

The argument is easy: By monotonicity, every consequence of ∆ is a consequence of Γ, ∆;

and, by (9^∗), Γ, ∆ has the same consequences as Γ.

(10) If every sentence of ∆ is a consequence of Γ and every sentence of Γ is a consequence of ∆, then Γ and ∆ have the same consequences.

(10) follows trivially from generalized transitivity: every consequence of ∆ is a consequence of Γ and, vice versa, every consequence of Γ is a consequence of ∆.

Equivalent Premise Lists: Call two premise lists, Γ and ∆, logically equivalent, or equivalent for short, if they have the same logical consequences.

If two premise lists are equivalent then every sentence of one list is a consequence of the other (because it is a consequence of the list in which it occurs). (10) says that the reverse direction holds as well.

By the truth-table of →, we get immediately:

(11) A, A→B |= B

We have also: B |= A→B. These two imply the following very useful law:

(12) Γ, A, A→B |= C ⇐⇒ Γ, A, B |= C

(12) is obtained, via (10), by observing that every sentence in one of the two premise lists is a consequence of the other. (Every sentence in Γ, A, B is a consequence of Γ, A, A→B, because A, A→ B |= B. And every sentence in Γ, A, A → B is a consequence of Γ, A, B, because B |= A→B.)

We shall call (12) disjoining. It allows us to disjoin a premise that is a conditional into its parts, provided that the antecedent is among the premises.

Note: Disjoining is related to what is known as modus ponens, or the rule of detachment, by which one can formally infer B from A and A→ B. Disjoining is different. It is the semantic law which justifies the use of modus ponens. The name ‘disjoining’ is not a current term.

(12) can be generalized to:

(12^∗) If A⁰ |= A, then

Γ, A⁰, A→ B |= C ⇐⇒ Γ, A⁰, B |= C

To show (12^∗), assume that A⁰ |= A. Then the addition of A to any list containing A⁰, yields an equivalent list. Hence, Γ, A⁰, A → B is equivalent to Γ, A⁰, A, A → B, which, by (12), is equivalent to Γ, A⁰, A, B. And this last list is equivalent to Γ, A⁰, B, since it is obtained from it by adding A.

Here is an example of a top-down derivation that uses some of the listed laws. We want to show that

|= [A → (B → C)] → [B → (A → C)]

Starting with this as our initial goal, we keep reducing each goal to another sufficient goal and we write the goals on separate, numbered lines. Indicated in the margin is the law (or laws) by which the preceding implication is reduced to the current one. The sign ‘√

’ marks obvious implications that need no further reductions.

1. |= [A → (B → C)] → [B → (A → C)] initial goal,

2. A→ (B → C) |= B → (A → C)] by (|=, →),

3. A→ (B → C), B |= A → C by (|=, →),

4. A→ (B → C), B, A |= C by (|=, →),

5. B → C, B, A |= C by disjoining,

6. C, B, A |= C by disjoining. √

Note that the reduction from 4. to 5. uses an instance of disjoining, whereby A→ (B → C), A is replaced by B→ C, A.

For the sake of brevity, we can write the three steps from 1. to 2., from 2. to 3., and from 3.

to 4. as a single step:

|= [A → (B → C)] → [B → (A → C)] initial goal, A → (B → C), B, A |= C by three applications of (|=, →).

In a similar way, we can write the steps from 4. to 5. and from 5. to 6. as a single step in which disjoining is applied twice.

The bottom-up proof of the initial goal is obtained by reversing the list: Start from 6. and end at 1., justifying each step by the indicated rule; 5. is obtained from 6. by disjoining, 4.

from 5.–by disjoining, 3. from 4. by (|=, →), etc.

From now on we will omit ‘initial goal’ in the margin of the first line.

Here is another example, where substitution-of-equivalents is used as well. The equivalences we use are:

B∨ C ≡ ¬B → C and A→ B ≡ ¬B → ¬A

The first equivalence is used in getting 2. from 1., the second–in getting 4. from 3.; in the first case substitution is applied to the conclusion, in the second case–to one of the premises.

1. A→ B, ¬A → C |= B ∨ C

2. A→ B, ¬A → C |= ¬B → C substitution of equivalents,

3. ¬B, A → B, ¬A → C |= C by (|=, →),

4. ¬B, ¬B → ¬A, ¬A → C |= C substitution of equivalents,

5. ¬B, ¬A, ¬A → C |= C by disjoining,

6. ¬B, ¬A, C |= C by disjoining. √

Homework 4.3 Using the laws introduced so far, prove, via top-down derivations, the following five implications. The goal should be reduced in the end to an obvious implication in which the conclusion is one of the premises. You can use substitution-of-equivalents based on simple equivalences of the kind given in the last example.

In the derivations of 4. and 5. you can use laws (12^∗) and (10), as well as the implications B |= A ∨ B and A, B |= A ∧ B.

1. |= [A → (B → C)] → [(A → B) → (A → C)]

2. ¬A → B, B → C |= ¬C → A 3. A→ (B ∨ C), ¬B |= A → C 4. (A∨ B) → (B → C) |= B → C 5. A∧B → C, B |= A → C