Logical Implication

As noted in the introduction, logic was considered historically the science of correct reasoning, which uncovers and systematizes valid forms of inference. Generally, an inference starts with certain assumptions called premises and ends with a conclusion. It is not that required that the premises be true, but that they imply the conclusion; i.e., it should be impossible that the premises be true and the conclusion–false.

In general, implications are not grounded in pure logic. That Jack was at a certain hour in New York implies that he was not, shortly afterwards, in Toronto. This is not a logical implication.

It rests on the practical impossibility of covering the New York - Toronto distance in too short a time. If the time is sufficiently short, the impossibility may be traced to a physical

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law. And in the extreme case, it becomes the impossibility of being at the same time in two different places. But even this is not something that rests on pure logic.

We shall not address at this point what comes under “pure logic”. As in the cases of logical equivalence and logical truth (cf. chapter 2), we can say that a sentence logically implies another if it is impossible that the first be true and the second false, by virtue of the logical elements of the two sentences. In sentential logic the only logical elements are the connectives.

A logical implication that rests only on the meaning of the connectives is tautological. Here is the definition.

A tautologically implies B if there is no assignment of truth-values to the atomic sentences under which A gets T and B gets F.

As in the case of tautological equivalence, (cf. 2.2.0) there is no need to go to the level of the atomic sentences. That a sentence tautologically implies another can be seen by displaying their relevant sentential structure. The definition entails the following.

A tautologically implies B if and only if they can be written as sentential expressions (with each sentential variable standing for the same sentence in both) such that there is no assignment to the sentential variables under which the expression for A gets T, and the expression for B gets F.

This means that, in a truth-table containing columns for both, there is no row in which A’s value is T and B’s value is F.

If A is a contradiction, then there is no truth-value assignment (to the atomic sentences) under which it gets T. Hence, for all B, there is no assignment in which A gets T and B gets F. Consequently a contradiction implies tautologically all other sentences. By a similar argument, every sentence implies tautologically a tautology. We shall return to this later.

Note that tautological implication is a certain type of logical implication. In sentential logic the two are the same. But in more expressive systems, in particular in first-order logic, there are logical implications that are not tautological.

Terminology and Notation:

• ‘Logical’ in the context of sentential logic means tautological. In the present chapter, the terms are interchangeable. For the sake of brevity we often use ‘implication’ for logical implication.

• ‘|=’ denotes logical implication, that is:

A |= B

means that A logically implies B. If A |= B, we say that B is a logical consequence of A.

• ‘|=’ is a shorthand for ‘logically implies’. Like ‘≡’ it belongs to our English discourse, not to the formal system. To say that A|= B is to claim that A logically implies B.

Note: Terms such as ‘implication’ and ‘equivalence’ are used mostly with respect to sen-tences, or sentential expressions, of our formal system. But we use them also with respect to our own statements. E.g., we can say that A ≡ A⁰ implies that ¬A ≡ ¬A⁰, and we speak about the implication

A|= A⁰ =⇒ ¬A⁰ |= ¬A .

And here, ‘implication’, which is denoted by ‘=⇒’, refers to our own statements. Similarly, we may speak of the equivalence

A≡ B ⇐⇒ B ≡ A .

A similar ambiguity surrounds ‘consequence’. The intended meaning of these, and other two-level terms, should be clear from the context.

If two sentences are equivalent, then under all truth-value assignments (to the atomic com-ponents) they get the same truth-value. Hence they imply each other. Vice versa, if they imply each other, than there is no assignment under which one gets T and the other gets F; therefore they are equivalent. Hence, sentences are equivalent just when they imply each other:

(1) A≡ B ⇐⇒ A|= B and B |= A .

(1) shows how logical equivalence can be defined in terms of logical implication. On the other hand, using conjunction, we can express implication in terms of equivalence:

(2) A|= B ⇐⇒ A ≡ A∧B .

The argument for (2) is easy:

Assume that A|= B, then (i) if A gets T, so does B and so does A ∧ B; and (ii) if A gets F, then A∧ B gets F. Therefore A and A ∧ B always get the same value.

Vice versa, if A ≡ A ∧ B, it is impossible that A gets T and B gets F, for then A and A∧ B get different values.

One can, nonetheless, argue that implication is the more basic notion. It corresponds directly to inferring. Moreover, logical equivalence is reducible to it without employing connectives, but not vice versa.¹ As we shall presently see, the most basic notion is that of implication with one or more premises.

1The reason for treating, in this book, equivalence before implication is didactic. Its analogy with equality makes equivalence more accessible and enables one to use algebraic techniques.

Using conditional, we can express logical implication in terms of logical truth.

(3) A|= B ⇐⇒ A→ B is logically true.

Homework 4.1 Prove (3), via the same type of argument used in proving (2).

The following properties of implication are easily established.

Reflexivity: A|= A

Transitivity: If A|= B and B |= C, then A |= C.

Transitivity is of course intuitively implied by the very notion of implication. The detailed argument is trivial:²

Assuming that A |= B and B |= C, one has to show that there is no assignments under which A gets T and C gets F. So assume that A gets T. Then B must get T, because A |= B. But then, C must get T, because B |= C.

Logical implications are preserved when we substitute the sentences by logically equivalent ones:

(4) If A≡ A⁰ and B≡ B⁰, then A|= B iff A⁰ |= B⁰.

One can derive (4) immediately from the definitions, by noting that logical implication is defined in terms of possible truth-values and that logically equivalent sentences always have the same value.

((4) is also derivable from (1) via the transitivity of implication: If, A ≡ A⁰ then, by (1), A⁰ |= A. Similarly, if B ≡ B⁰, then B |= B⁰. If also A|= B, we get:

A⁰ |= A, A|= B, B |= B⁰

Applying transitivity twice we get A⁰ |= B⁰. In the same way we derive A |= B from A⁰ |= B⁰.) (4) implies that, in checking for logical implications, we are completely free to substitute sentences by logically equivalent ones. We can therefore use all the previous simplification techniques in order to reduce the problem to one that involves simpler sentences.

2It is trivial if we define implication by appealing to assignment to atomic sentences. If we want to bypass atomic sentences we have to show that the two implications A |= B and B |= C, can be founded on sentential expressions in which the three sentences are generated from the same stock of basic components (see also footnote 4 in chapter 2, page 32). This can be done by using unique readability.

Every case of logical equivalence is, by (1), a case of two logical implications, from left to right and from right to left. But generally implications are one-way. Here are a few easy examples in which the reverse implication does not hold in general.

(i) A∧ B |= A (ii) A|= A ∨ B (iii) B |= A → B (iv) ¬A |= A → B

(v) A∧ B |= A ↔ B (vi) ¬A ∧ ¬B |= A ↔ B

That the reverse implications do not hold in general can be seen by assigning the sentential variables truth-values, under which the left-hand side gets T and the right-hand side gets F. We can interpret them as standing for atomic sentences that can have these values. For example, in the case of (i), let A get T and let B get F.

Note that we can also force this assignment by interpreting the variables as standing for tautologies or contradictions. For example let A = C → C, B = C ∧ ¬C.

In particular cases, the right-to-left implication holds as well. For example:

If B is logically true, e.g. (B = C → C) then A |= A ∧ B.

If B is a logically false, then A → B |= ¬A .

Here, as an illustration, is the argument for the second statement.

Assume that B is logically false. If A → B gets T, then, since B gets F (being logically false), A must get F. Hence, ¬A gets T. There is, therefore, no assignment (to the atomic sentences) in which A → B gets T and ¬A doesn’t.

Homework 4.2 Find, for each of (i) - (vi) above, whether the reverse implication holds for all B, in each of the following cases:

(1) A is logically true. (2) A is logically false.

Altogether you have to check 12 cases. Prove every positive answer by an argument of the type given above. Prove every negative answer by a suitable counterexample.

Note: We can define a notion of logical implication that applies to sentential expressions.

This is completely analogous to the case of logical equivalence and logical truth (cf. 2.2). We shall discuss it in 4.3.1, in the more general context of implication from many premises.