Deduction Rules and Strategy

Question 1

What is rule P?

Answer 1

Premise introduction.

Question 2

What is Rule D?

Answer 2

Discharge.
1) On a line (k), which is earlier than line (n), a premise R has been introduced,
2) On a line (m), which is also earlier than (n), and which has k among its premises #’s, there is a schema S.
What the rule allows us to write on line (n): a schema which is a conditional with antecedent R and consequent S.
Premise #’s: all the premise numbers of line (m) except for k.
Citation: k

Question 3

Rule TF

Answer 3

Truth-Functional Implication
Cond’ns on invoking this rule:
Schemata R1, R2, R3,…Ri (i≥1) occur on lines (m1), (m2), (m3),…(mi), all of which are earlier than line (n),
What the rule allows us to write on line (n):
any schema S which is truth-functionally implied by the conjunction of R1, R2, R3,…Ri.
Premise #’s: all the premise #’s of lines m1, m2, m3,…mi.
Citation: (m1),(m2),(m3),…(mi)

Question 4

Rule UI

Answer 4

Universal Instantiation
Cond’ns on invoking this rule: A schema of the form (∀u)R occurs on a line (m), which is earlier than line (n),
What the rule allows us to write on line (n): Any schema S which is an instance of the universal generalization on line (m).
Premise #’s: same as the premise #’s of line (m)
Citation: (m)

Question 5

Rule EG

Answer 5

Existential Generalization
Cond’ns on invoking this rule: A schema S occurs on a line (m), which is earlier than line (n),
What the rule allows us to write on line (n): Any schema of the form (∃u)R, which is such that S is an instance of it.
Premise #’s: same as the premise #’s of line (m)
Citation: (m)

Question 6

Rule CQ

Answer 6

Conversion of Quantifiers:
Cond’ns on invoking this rule:

(∀u)-R = -(∃u)R
(∃u)-R = -(∀u)R
-(∀u)R = (∃u)-R
-(∃u)R = (∀u)-R

Premise #’s: same as the premise #’s of line (m)
Citation: (m)

Question 7

Rule UG

Answer 7

Universal Generalization
What the rule allows us to write on line (n):
Any schema of the form (∀u)R, which is such that (i) S is a conservative instance of it (with instantial variable v), and (ii) the variable v does not occur free in any premise of line (m).
Premise #’s: same as the premise #’s of line (m)
Citation: (m)

Question 8

What is the first part of rule EI?

Answer 8

Existential Instantiation Introduction (EII)
Assume this rule is being invoked on line (n).
Cond’ns on invoking this rule:
A schema of the form (∃u)R occurs on a line (m), which is earlier than line (n).
What the rule allows us to write on line (n):
Any schema S which is a conservative instance of the existential generalization on line (m)
[We introduce two new definitions here:
v, which is the instantial variable here, is said to be the variable flagged on line (n);
S, which is the schema introduced by EII on line (n), is called an EI-premise.]
Premise #’s: same as the premise #’s of line (m), with the addition of n itself.
Citation:(m)v

Question 9

What is the second part of rule EI?

Answer 9

Existential Instantiation Elimination (EIE)
Assume this rule is being invoked on line (n).
Conditions on invoking this rule:
1) On an earlier line (j), a schema S was introduced by using rule EII, flagging variable v.
2) On a line (m), which is also earlier than (n), and whose premise #’s include j, there is a schema T.
This line obeys two restrictions:
a) v [the variable flagged on line (j)] must not occur free in T [the schema on line (m)].
b) and v also must not occur free in any premise of line (m), other than in line (j) itself.
What the rule allows us to write on line (n):
It allows us to write schema T again, i.e., the very same schema that already occurred on the earlier
line (m), but with the following change in premise #’s.
Premise #’s: all the

Question 10

What is the EIE shortcut?

Answer 10

Instead of rewriting schema T again on a later line (n), with one less premise, we can do the following on the earlier line (m):

i) In the premise #’s of line (m), draw a line through the premise # of the EI-premise being eliminated, that is, j.
ii) At the end of line (m), add a further citation [j], followed by “EIE”.

Question 11

What’s a good strategy to use when you are trying to deduce a conditional?

Answer 11

Use a conditional proof, or, use a conditional proof of its contrapositive.

Question 12

When you are starting with a disjunction, what’s a good strategy to use?

Answer 12

Argument by dilemma.

Question 13

When you are deducing a disjunction, what’s a good strategy to use?

Answer 13

Deduce the equivalent conditional and then transform it into its disjunctive form.

Question 14

What is the TF rule W?

Answer 14

W is weakening. It allows me to deduce…
R⊃S from S
R∨S from R
S∨R from R

Question 15

What is the TF rule DIL?

Answer 15

DIL allows me to deduce…

r from pVq as long as p->r and q->r.

Question 16

What is a flagged variable?

Answer 16

When we write down the EI premise, we flag (note to the side) the instantial variable of the original existential generalization. When doing an EIE, we check to see if the flagged variable is free in any of the premises listed for the schema under consideration except the line the EI premise occurs on.

Question 17

What is an EI premise?

Answer 17

When doing an existential instantiation we have two parts, EII and EIE. When doing the EII part, we introduce an EI premise. For the EI premise, we write down a conservative instance of an existential generalization and then flag (note to the side) the instantial variable of the original existential generalization on the citation line of the EI premise. EI premises have their own premise number.