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If, Only If and If and Only If

By the end: I can distinguish a sufficient condition from a necessary condition and a two-way rule.

Created by Denis M., British English teacher.

I’m studying:

Try the activities, then check your answers.

Playback speed

Choose a short recording.

Enough, but not necessarily required

In our fictional rules, Entry is granted if you have a blue token makes a blue token sufficient. The rule does not say that nobody else can enter.

If you have a blue token, entry is granted.

A blue token is enough under this rule.

The woman presents a single blue token at a small isolated entry gate; the man welcomes her through under a rule where this token is enough. The gate is open for her, but do not depict other routes as prohibited. Token and simple gate only. — Entry is granted if you have a blue token.The token is enough under this rule; other routes to entry remain possible.

Required, but not necessarily enough

Entry is granted only if you have a blue token makes the token necessary. It does not by itself promise entry to every token holder.

You can enter only if you have a blue token.

Entry implies possession of the token, not the reverse.

Test the rule with a refusal and an admission

A token holder is refusedEntry is granted only if you have a blue token.

This rule can still hold: it requires a token for entry but does not promise entry to every holder.

Someone without a token is admittedEntry is granted only if you have a blue token.

This violates the stipulated rule because the required condition is missing.

For the plain if rule, refusing a blue-token holder violates the guarantee. Admitting someone without a token can be allowed by another route. Keep these conclusions inside the fictional rules stated here.

Both directions

If and only if makes the condition both necessary and sufficient within the stated system. Test each direction separately before summarising.

The light is on if and only if the switch is up.

Switch up means light on; light on means switch up.

The same woman shows a blue token to the man at the entry gate, which remains closed while he considers a further condition. The token is required but is not shown guaranteeing admission. Use calm attentive expressions, not humiliation or threat. — Entry is granted only if you have a blue token.Entry implies a token, but a token alone is not promised to be enough.

Guided practice

Say or write your answer first. Compare form and meaning with the model.

1. Rule: Access if code. A valid code is present. What follows under the stipulated rule?

Compare a model and explanation

Access is granted.

The code is sufficient in this rule.

2. Rule: Access only if code. No code is present. What follows?

Compare a model and explanation

Access is not granted.

The necessary condition is missing.

3. Rule: Access only if code. A code is present. Is access settled?

Compare a model and explanation

No; the rule leaves that open.

A necessary condition alone need not be sufficient.

4. Rule: Access if code. No code is present. Is access ruled out?

Compare a model and explanation

No; the rule does not specify other possible routes.

Do not reverse an implication without permission.

5. Rewrite a two-way rule: The bell rings exactly when the lever is down.

Compare a model and explanation

The bell rings if and only if the lever is down.

Exactly when has the two-way meaning stipulated here.

6. Does Access only if code allow a code holder to be refused?

Compare a model and explanation

Yes.

Other necessary conditions could exist.

7. Which rule is violated if someone without a badge is admitted: Entry if badge or Entry only if badge?

Compare a model and explanation

Entry only if badge.

The plain if rule does not exclude other entry routes.

8. Explain why replacing if with if and only if can change a rule.

Compare a model and explanation

It adds the reverse restriction, making the condition necessary as well as sufficient.

That added restriction is substantive, not decorative.

A short conversation

Listen first, then open the transcript when useful.

Open the conversation transcript

A: Does this pass guarantee entry?

B: The rule only says you can enter only if you have a pass.

A: So having one is required, but it may not be enough?

B: Exactly. We need the rest of the rule before promising entry.

Listening checks

Listen to each extract or open its transcript. These three questions have a separate score.

1. What follows under the stated rule?

Read the listening transcript

In this fictional system, the lamp lights if a green button is pressed. The green button was pressed.

The lamp can never light another way.The lamp lights.Nothing follows.

2. What follows under this rule?

Read the listening transcript

A visitor may enter only if their name is listed. Omar’s name is not listed.

Omar may not enter.Omar is guaranteed entry.A listed person is always admitted.

3. What follows in this stipulated system?

Read the listening transcript

The marker is red if and only if the box is open. The marker is red.

The box must be closed.The marker alone can never tell us.The box is open.

Use it in your own words

Speaking 1

Invent separate sufficient and necessary conditions using clearly labelled fictional rules.

Reveal a possible response

In system A, a token guarantees entry. In system B, a token is required, but a booking may also be needed.

Success check: Do not merge the two systems’ rules.

Speaking 2

Explain a two-way rule and test both directions.

Reveal a possible response

The flag is up if and only if the gate is open. An open gate means the flag is up, and a raised flag means the gate is open.

Success check: State the stipulation before drawing conclusions.

Put it all together

A fictional game says A player may advance only if they have a star. One player has a star; another does not. Explain what is and is not settled, then propose a stronger rule that settles both cases.

Reveal a possible response

The player without a star cannot advance. The star holder’s position is not settled. Changing the rule to if and only if they have a star makes the star both necessary and sufficient.

Success check: Identify the added guarantee rather than presenting the rewrite as equivalent.

Take it with you

I can distinguish a sufficient condition from a necessary condition and a two-way rule.