Deduction Rules and Strategy Flashcards
What is rule P?
Premise introduction.
What is Rule D?
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
Rule TF
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)
Rule UI
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)
Rule EG
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)
Rule CQ
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)
Rule UG
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)
What is the first part of rule EI?
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
What is the second part of rule EI?
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
What is the EIE shortcut?
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”.
What’s a good strategy to use when you are trying to deduce a conditional?
Use a conditional proof, or, use a conditional proof of its contrapositive.
When you are starting with a disjunction, what’s a good strategy to use?
Argument by dilemma.
When you are deducing a disjunction, what’s a good strategy to use?
Deduce the equivalent conditional and then transform it into its disjunctive form.
What is the TF rule W?
W is weakening. It allows me to deduce…
R⊃S from S
R∨S from R
S∨R from R
What is the TF rule DIL?
DIL allows me to deduce…
r from pVq as long as p->r and q->r.
What is a flagged variable?
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.
What is an EI premise?
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.
