Existential Introduction

Kareem Khalifa

Department of Philosophy

Middlebury College

SURPRISE!

• Translate the following:

• Anyone who is respected respects his/her respecter. Hence, Bob is respected, as he respects someone.

• Let – R be a 2-place predicate = “…respects…” – b is a name for “Bob.”

• You have until 11:20 to complete this.

Overview

• Why this matters

• Predicate logic is just propositional logic on (minor) steroids

• The First New Rule of Inference:– Existential Introduction

• Sample Exercises

Why this matters

• Recall: there are valid inferences that predicate logic promised to render valid.

• This will also make you more adept at evaluating and offering reasons involving statements involving “some.”

Predicate Logic: An Extension of Propositional Logic• Recall: proofs are like games.

– You have an initial position (premises), and using a set of pre-established rules (of inference), you move towards a goal (conclusion)

• Proofs are still like games in predicate logic!– There are just four more basic rules you can

use.

The First Rule: Existential Introduction (I)

• English Example:– Khalifa is a professor. So someone is a professor.

• Logical Example:1.Pk A

2. xPx 1 I

• General rule:– Given any formula with a name (i.e. a-t) in it, add an

existential quantifier, and replace each name with a variable.

– See Nolt, p. 225 for fancier definition

Three more examples

• Al loves himself. 1. Laa A

…So someone loves Al. (1st Example)2. xLxa 1 I

…So Al loves someone. (2nd Example)2’. xLax 1 I

…So someone loves him/herself. (3rd Example)

2’’. xLxx 1 I

A fourth, fancier example

Al loves himself. So someone loves someone. 1. Laa A2. yLay 1 I3. xyLxy 2 I• IMPORTANT: when you apply I twice, you

always work from the inside out (right to left).• Otherwise, you’re breaking the rule that students

are perpetually tempted to break (applying rules to parts of propositions, as opposed to applying rules to the whole proposition).

Do NOT do the following

1. Lab A

2. xLxx 1 I

• Think about the inference you’ve just made:– Al loves Beth. So someone loves

him/herself.– Clearly invalid! Al may still love Beth, but Al

and Beth may nevertheless both hate themselves.

A closely related but legitimate move

1. Fa A2. Gb A3. xFx 1 I4. xGx 2 I5. xFx & xGx3,4 &I6. This is OK, because although it is using the same

variable x for two unrelated terms, it is not saying that one thing is both F and G.

7. Ex. Al is friendly; Beth is grumpy. So someone is friendly and someone is grumpy.

8. If this confuses you, it’s usually fine to use distinct variables (e.g. y on line 4), but you will still need to be able to interpret statements like line 5 correctly.

Some things to keep in mind…

• So long as the main operator is ~, v, &, , or , your proof strategies from propositional logic still hold good.

• If your main operator is , then consider proving a singular statement to which you can subsequently apply I.

Sample Exercises, Nolt 8.1.1

Lab ├ xLxb

1. Lab A

2. xLxb 1 I

8.1.2

#2. Lab ├ xLax

1. Lab A

2. xLax 1 I

8.1.3

#3 Laa ├ xLxx

1. Laa A

2. xLxx 1 I

8.1.7

#7 ~xyRxy ├ ~Rab

1. ~xyRxy A

2. |Rab H for ~I

3. |yRay 2 I

4. |xyRxy 3 I

5. |xyRxy & ~xyRxy 1,4 &I

6. ~Rab 2-5 ~I

8.1.9

#9 ├ Fa → xFx

1. | Fa H for →I

2. | xFx 1 I

3. Fa → xFx 1-2 →I