﻿ 1.2 Conditional and biconditional

### 1.2 Conditional and biconditional

#### Conditional

Conditional is the logical connective that corresponds to “if …, then…”.1 Here is an example:

If there is life on Mars, then there is water on Mars.

The connective is symbolized by “→”2. If p symbolizes the sentence “There is life on Mars” and q the sentence “There is water on Mars”, the above sentence will be symbolized by “pq”. Besides to the logical connective itself (→), “conditional” is also called the sentence formed by it (pq).

The sentence that expresses the condition (the one after “if”) is called antecedent, and the other sentence (which expresses what is dependent on the condition) is called consequent. In the example, the antecedent is “There is life on Mars” and the consequent is “There is water on Mars”. In “pq”, the antecedent is “p” and the consequent is “q”.

In the symbolic language of logic, the antecedent is always before the consequent with the arrow of the conditional between them, as in “pq”. In English (and other natural languages) the order is not fixed – the antecedent may be in second place. “If there is life on Mars, then there is water on Mars” has exactly the same meaning as “There is water on Mars if there is life on Mars”. In the first sentence, the antecedent precedes the consequent, and in the second it is the other way around. Both are represented with “pq”. The relative position to the word “if” is what determines which sentence is the antecedent and which the consequent in an if-sentence, not the order of the sentences. The antecedent is the sentence after “if” no matter whether it comes first or second.

The truth table of the conditional is as follows:

 α β α → β T T T T F F F T T F F T

Let us see why and in what sense this table agrees with “if α, then β”. Take again “If there is life on Mars, then there is water on Mars” as an example. If it turns out that there is life on Mars but no water (2nd row of the table), obviously the sentence will be false. If it turns out that there is both life and water on Mars (1st raw), this agrees with the sentence, so it is true. As for the last two rows, whoever utters the sentence commits themselves to there being water on Mars only if there is life on Mars. The utterer is committed to nothing if there is no life on Mars. Therefore, in the cases corresponding to the last two rows of the table, when the condition of there being life on Mars is not met, whether or not there is water on Mars cannot falsify the sentence – it is true in these cases.

As the truth table shows, a conditional is false just in one case – when the antecedent (α) is true, and the consequent (β) is false. Therefore, “If α, then β” can be rephrased as “It is not true that α and not-β”, symbolically ¬(α∧¬β). The last sentence is equivalent to α→β because it excludes precisely the cases in which α→β is false and is true in all other cases, which are the cases in which α→β is true.

The last two rows of the truth table show that if the antecedent is false, the conditional is true whatever the truth value of the consequent. The table also shows (1st and 3rd rows) that if the consequent is true, the conditional is true whatever the truth value of the antecedent. For this reason, for example the sentence “If the Earth is a cube, then the moon is in my back pocket” must be true because the Earth is not a cube (the truth value of the second sentence is irrelevant). Similarly, “If 2 + 2 = 4, then Paris is the capital of France” must be true since the consequent “Paris is a capital of France” is true (the truth value of the first sentence is irrelevant). But usually, we do not use such sentences because we are able to affirm them only when we know that one of the constituent sentences is true (or false), and then it is more appropriate to affirm the constituent sentence itself (or its negation) because by connecting it with another sentence through “if …, then …”, we convey less information then we have. For example, if I have decided to go to the cinema no matter the weather and say to someone “If the weather is bad, I will go to the cinema”, I am not lying (the conditional is true because its consequent is true) but I am conveying less information, as I know that I will go to the cinema also if the weather is good. Therefore, as a rule, sentences of the form “If α, then β” are used only when the truth values of α and β are not known. In such cases, we commit ourselves to the truth of the conditional because we think there exists a certain relation between the states of affairs expressed by α and β – a causal relation, or a relation of logical inference, or something else. However, the reason(s) for asserting a sentence is something different from what the sentence expresses – “if α, then β” (i.e., α→β) expresses neither causal relation, nor logical inference, nor other relation between α and β except that it is not the case that α is true and β is false.

“Only if” and “if” mean different things. “Only if” also expresses a conditional but the places of the antecedent and the consequent are switched. Take as an example the sentence “Only if there is life on Mars, there is water on Mars”, which is the same as the example we used above except that “if” is replaced with “only if”. Symbolizing (as above) “There is life on Mars” with “p” and “There is water on Mars” with “q”, we get “Only if p, q”, which is the same as “q only if p”. What the last sentence affirms is that if p is not the case, q is also not case. In other words, the sentence excludes the possibility for q to be true when p is not true, i.e. it has the meaning of “¬(q∧¬p)” (“It is not the case that q is true and p is not true”). However, in the paragraph before the previous one, we saw that the last sentence is equivalent to “qp”. So, while “If p, then q” is symbolized by “pq”, “Only if p, q” is symbolized by “qp”. Adding “only” to “if” has the effect of the antecedent and the consequent switching their roles – the antecedent becoming consequent, and vice versa. As with “if”, the relative position to “only if” determines which is the antecedent and which the consequent, not the order of the sentences: “q only if p” means the same as “Only if p, q” and is symbolized by “qp”. So, we may use the following rule: the sentence after “if” is the antecedent of the conditional (the other sentence is the consequent); the sentence after “only if” is the consequent of the conditional (the other sentence is the antecedent).

 Form of the sentence in English Symbolization If p then q. p → q q if p. p → q Only if p q. q → p q only if p. q → p

“If” and “only if” by no means exhaust the ways to form a conditional. “In case”, “provided”, “on condition” etc. may also be used – “If / Provided / In case Bob apologizes to Alice, she will forgive him”.

On the other hand, not all if-sentences have the form α→β. An if-sentence may be a universal categorical sentence3, also called formal implication. Such is for example the sentence “If Bob likes something, Alice doesn’t like it”. Its logical form cannot be adequately represented in the language of propositional logic and therefore it treats it as atomic (simple). The sentence can be paraphrased as “Whatever we take, if Bob likes it, Alice doesn’t like it” and boils down to an unspecified number of conditionals obtained from the scheme “If Bob likes …, Alice doesn’t like …” by putting in place of the dots an arbitrary term – “If Bob likes swimming, Alice doesn’t like swimming”, “If Bob likes Anna, Alice doesn’t like Anna”, etc. The sentence is true if all these conditionals are true and is false if one or more of them are false.

Another type of if-sentences that do not have the form α→β are the counterfactual conditionals. Such is, for example, the sentence “If Japan had not attacked Pearl Harbor, the United States would not have entered the war”. Grammatically, this sentence differs from the examples used so far by its mood, which is subjunctive rather than indicative. In counterfactual conditionals, the sentence after “if” is false, which is why if they were of the form α→β, they would be true regardless of what they say. (We saw that a conditional is trivially true if its antecedent is false.) Obviously, the above sentence is not true just because Japan did attack Pearl Harbor; it is perhaps false. Like universal categorical sentences (previous paragraph), the logical form of counterfactual conditionals cannot be adequately represented in the language of propositional logic, in which they are treated as atomic sentences (symbolized with “p”, “q”, …). We will see how these sentences may be analyzed in the section on modal logic.

As mentioned above, an if-sentence and in particular a sentence of the form α→β does not express logical inference although it may happen that β follows logically from α. If β is a logical consequence of α, the conditional “If α, then β” will surely be true since it will not be possible for β to be false if α is true. However, “If α, then β” may also be true without being any connection between α and β. For example, the sentence “If 2+2=4, then Paris is the capital of France” is true because both sentences in it are true but of course neither is a logical consequence of the other.

Еxamples of conditionals:

 p → q p q If Bob was here, then the glove is his. Bob was here. It is Bob’s glove. You will do badly if you don’t listen to my advice. You will not listen to my advice. You will do badly. Alice will continue to be with Bob only if he stops drinking. Alice will continue to be with Bob. Bob will stop drinking. There will be a global legal order only if individual states give up their sovereignty. There will be a global legal order. Individual states will give up their sovereignty.

Biconditional is the logical connective corresponding to the phrase “if and only if”. We will also call so the sentences formed by this logical connective. An example is

Alice will forgive Bob if and only if he apologizes to her.

Obviously “α if and only if β” contains two sentences – “α if β” and “α only if β”. The first is the same as “If β, then α”, and for the second we have seen that (as it corresponds to α→β) it can be rephrased as “If α, then β”. So, it turns out that “α if and only if β” has the meaning of “If α, then β, and if β, then α”, which is symbolized by (α→β)∧(β→α). We could use this formula to represent biconditional sentences, but for the sake of brevity we will introduce the symbol “↔” for the connective of biconditional. So, a sentence of the form “α if and only if β” will be symbolized by α↔β4.

We just saw that the sentence “α if and only if β” contains the conditionals “If α, then β” and “If β, then α”. The first excludes the case in which α is true and β false (allowing all other cases) and the second excludes the case in which α is false and β true. So, together the two conditionals allow only the cases in which α and β both are true, or both are false. Therefore, a sentence of the form α↔β is true if and only if α and β have the same truth value. Thus, we get the following truth table for the biconditional:

 α β α ↔ β T T T T F F F T F F F T

A biconditional sentence is true when its constituent sentences have the same truth values (the first and the last row) and is false when they have different truth values (the other two rows).

As we saw, “α if and only if β” amounts to the assertion of the conditionals “If α, then β” and “If β, then α”. The second one may be rephrased as “β only if α” (we saw that by “only if” we form a conditional whose consequent is the sentence after this phrase). Тhe last sentence asserts that if α is not the case, then β is not the case, i.e. it is equivalent to “If not-α, then not-β”. So, replacing the “If β, then α” part of “α if and only if β” with “If not-α, then not-β”, we obtain another way to express a biconditional sentence: with “If α then β, and if not-α then not-β”, symbolically (α→β)∧(¬α→¬β). So, for example, the following two sentences have the same meaning:

Alice will forgive Bob if and only if he apologizes to her.

If Bob apologizes to Alice, she will forgive him, and if he does not apologize to her, she will not forgive him.

Strictly speaking, “only if” expresses conditional, not biconditional; biconditional is expressed by “if and only if”. However, in everyday usage the first “if” is often omitted and the remaining “only if” is meant with the sense of “if and only if”. Imagine, for example, that someone says, “Alice will forgive Bob only if he apologizes to her”. What the speaker is certainly saying is that if Bob does not apologize to Alice, she will not forgive him. However, it is quite possible that he or she also means that if Bob apologizes to Alice, she will forgive him, even though strictly speaking this is not included in what is being said. What the sentence literally says is that if Bob does not apologize, Alice will not forgive him, and it does not contain any commitment to what will happen if Bob does apologize (may be Alice will forgive him, may be not). So, if we want to be precise, we should use “if and only if” rather than “only if” when we want to express а biconditional. The need for precision in mathematics is the reason why “if and only if” is perhaps mostly used there.

Examples of biconditionals:

 p ↔ q p q Alice will go to the party if Bob goes too, but only then. Alice will go to the party. Bob will go to the party. The program will stop if and only if all tasks are completed. The program will stop. All tasks will be completed.

If a natural language sentence is complex enough so that it is not immediately evident how to symbolize it, it is advisable to do the symbolization step by step moving from the whole of the sentence to its parts, i.e. in the opposite direction of its formation. We will illustrate this systematic approach with the following sentence:

If the crisis continues and taxes are not increased, there will either be a budget deficit or neither wages will be increased nor there will be Christmas bonuses.

First, we ask what the form of the sentence is as a whole, i.e. is it an if-sentence (a conditional), an or-sentence (a disjunction), an and-sentence (a conjunction), etc? The sentence begins with “if”, which indicates that it contains a conditional, but the latter may or may not be the main logical connective. The antecedent of the conditional lies between “if” and the comma, and the consequent starts after the comma. Тhe main logical connective will be conditional (i.e. the sentence will be an if-sentence) if the consequent goes to the end of the sentence, and in this case it is intuitively clear that it is so. The consequent is the sentence “There will be either a budget deficit or neither wages will be increased nor there will be Christmas bonuses”. Since the whole sentence has the form of a conditional, as a first step of the symbolization, we write the conditional sign and on its left and right side we write the antecedent and the consequent of the conditional not yet symbolized, and enclosed in parentheses:

 (1) (The crisis will continue and taxes will not be increased) → (There will be either a budget deficit or neither wages will be increased nor there will be Christmas bonuses)

The antecedent and the consequent are enclosed in parentheses because they may be (and in this case in fact are) compound sentences, and after being symbolized compound sentences which are parts of other sentences are enclosed in parentheses (exceptions are sentences having the form of negation). If some of these not yet symbolized sentences turns out to be non-compound or a negation, we will remove the parentheses.

As a next step, we direct our attention to the left of the two not yet symbolized sentences (“The crisis will continue and taxes will not be increased”) considering its symbolization as a separate task. Again, we ask what the form of the sentence is as a whole. Obviously, it is a conjunction between the atomic (non-compound) sentence “The crisis will continue” and the negation of the atomic sentence “Taxes will be increased”. Therefore, symbolizing the first atomic sentence with “p” and the second with “q”, we replace in (1) “The crisis will continue and taxes will not be increased” with “p∧¬q”:

 (2) (p∧¬q) → (There will be either a budget deficit or neither wages will be increased nor there will be Christmas bonuses)

Continuing in the same way, we ask what the form of the not yet symbolized sentence “There will be either a budget deficit or neither wages will be increased nor there will be Christmas bonuses” is as a whole. It is an or-sentence – a disjunction between the sentences “There will be a budget deficit” and “Neither wages will be increased nor there will be Christmas bonuses”. The first of them is atomic. We symbolize it with “r” and replace the “or”-sentence with the disjunction between r and “Neither wages will be increased nor there will be Christmas bonuses”, the latter enclosed in parentheses:

 (3) (p∧¬q) → (r ∨ (Neither wages will be increased nor there will be Christmas bonuses))

The not yet symbolized sentence in (3) is a neither-nor-sentence. Earlier we saw that such sentences are conjunctions between negations. Here the negated sentences are atomic – “Wages will be increased” and “There will be Christmas bonuses”. Symbolizing the first with “s” and the second with “t”, we replace the neither-nor-sentence in (3) with “¬s∧¬t”:

 (4) (p∧¬q) → (r ∨ (¬s∧¬t))

For easier reading, we may replace some of the parentheses by brackets or braces. Summarizing, we have the following:

If the crisis continues and taxes are not increased, there will be either a budget deficit or neither wages will be increased nor there will be Christmas bonuses.

(p∧¬q) → [r ∨ (¬s ∧ ¬t)]

 p – The crisis will continue. q – Taxes will be increased. r – There will be a budget deficit. s – Wages will be increased. t – There will be Christmas bonuses.