Transcript Existential Introduction

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
2. xPx
A
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
A
2. yLay
1 I
3. xyLxy
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
2. xLxx
•
A
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.
2.
3.
4.
5.
•
•
•
Fa
A
Gb
A
xFx
1 I
xGx
2 I
xFx & xGx 3,4 &I
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.
Ex. Al is friendly; Beth is grumpy. So someone is
friendly and someone is grumpy.
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
2. xLxb
A
1 I
8.1.2
#2. Lab ├ xLax
1. Lab
2. xLax
A
1 I
8.1.3
#3 Laa ├ xLxx
1. Laa
2. xLxx
A
1 I
8.1.7
#7 ~xyRxy ├ ~Rab
1. ~xyRxy
2. |Rab
3. |yRay
4. |xyRxy
5. |xyRxy & ~xyRxy
6. ~Rab
A
H for ~I
2 I
3 I
1,4 &I
2-5 ~I
8.1.9
#9 ├ Fa → xFx
1. | Fa
2. | xFx
3. Fa → xFx
H for →I
1 I
1-2 →I