Unit 3 Definitions Flashcards
Define Logically Equivalent
When two statement forms have truth-values that agree in all instance. AKA, their truth tables agree in every row.
What is a Subformula?
A grammatically correct sentence that is a part of another grammatically correct sentence.
Identify the Subformula in this equation: (AvB) -> C
(AvB).
What is a Proof?
A proof is a sequence of statements, each of which is either a premise or a statement that is obtained from one or more earlier statements by applying one of the rules of inference.
What are the 10 Basic Inference Rules? (abbreviations)
~I, &E, vI, <->E, ->I, ~E, &I, vE, <->I, and ->E
What are the 3 Derived Rules? (abbreviations)
Modus Tollens, Hypothetical Syllogism, and Disjunctive Syllogism.
What are the 5 Equivalence Rules? (abbreviations)
DeMorgans, Double Negation, Transportaion, Material Implication, and Exportation.
What is the form for [~I]?
|p
|…
|q&~q
———
.: ~P
What are the two forms for [&E]?
.: p
.: q
What are the two forms for [vI]?
p
.: p v q
q
.: p v q
What are the two forms for [<->E]?
p <-> q
.: p -> q
p <-> q
.: q -> p
What is the form for [->I]?
|p
|…
|q
———
.: p -> q
What is the form for [~E]?
~~p
.: p
True or False: this is a valid form of the inference rule [~E]…
p
——
.: ~~p
FALSE. Inference rules do not go both ways. The only correct form for [~E] is:
~~p
——-
.: p
What is the form for [&I]?
p
q
——-
.: p&q
What is the form for [vE]?
p v q
p -> r
q -> r
———
r
What is the form for [<->I]?
p -> q
q -> p
———
.: p <-> q
What is the form for [->E]?
p -> q
p
——–
q
What is the form for Modus Tollens?
p -> q
~q
——–
~p
What is the form for Hypothetical Syllogism?
p -> q
q -> r
———
p -> r
What are the two forms for Disjunctive Syllogism?
p v q
~p
———
q
p v q
~q
———
p
What are the two forms for DeMorgans?
~(p&q) = ~p v ~q
~(pvq) = ~p & ~q
What is the form for [DN]?
p = ~~p
True or False: This is a valid form for Double Negation…
~~p
=
p
Yes. It is an equivalence rule so it can be used both ways. The order doesn’t matter because they are still able to be substituted for each other.
What is the form for Transportation?
p -> q = ~q -> ~p
What is the form for Material Implication?
p -> q = ~p v q
What is the form for [EXP]?
(p&q) -> r = p -> (q -> r)
Of the following, identify which represents an Instance:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k
1 is an instance. Sentence letters (representing statements) replaced the form’s variables.
Of the following, identify which represents a Substitution:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k
3 is a substitution. Sentence D takes the place of p and sentence K takes the place of q in the form structure.
Of the following, identify which represents a Form:
1) D -> K, D, .: K
2) p -> q, p, .: q
3) p : D, q : k
2 is a form. These are variables and don’t represent anything. They are meant to display a form’s structure so that they can later be replaced with sentence letters in proofs/truth tables/etc.
Are these sentence letters or variables:
A, B, C, D
Sentence letters, they represent statements!
Are these sentence letters or variables:
p, q, r
Variables, they stand as placeholders in form structures for sentence letters!
Why can’t Logicians just continue to use truth tables?
- truth tables don’t represent actual process (which are usually in steps)
- they don’t work for every deductively valid argument
- they can be too long
Is (~(A <-> E) v C) -> ~(D&E) an instance of the form p -> q?
Yes, substitue!
p: (~(A <-> E) v C)
q: ~(D&E)
True or False: All proofs are valid because each step must be verified by a rule.
True.
True or False: you can hypothesis whatever, whenever in proofs.
False. You can can (or rather should) only hypothesize if you are trying to prove [~I] or [->I].
Fill in the blank: Every _______ begins a new vertical line.
Hypothesis.
Can a line in a hypothesis’ scope be used/cited after it is has been discharged?
No. Any citing or usage must be done within the hypothesis’ scope, otherwise we would be wrongfully combining the real and hypothetical world.
True or False: Hypotheses can overlap.
False. A hypothesis made in the scope of another hypothesis needs to be discharged before the first one is.
What is Modus Tollens?
MT. From any statement p -> q and ~q, we may infer ~p.
p -> q
~q
——–
~p
What is Hypothetical Syllogism?
HS. From any statement p -> q and q -> r, we can infer p -> r.
p -> q
q -> r
———
p -> r
What is Disjunctive Syllogism?
DS. From any statement p v q and ~q, we may infer p.
p v q
~p
———
q
p v q
~q
———
p
Fill in the blank: The given statement for a theorem should placed in the ______ line of the proof.
Last, always. It is the conclusion!
Fill in the blank: A statement form is a theorem iff it is a _____.
Tautology.
True or False: All theorems tautologous.
True. Tautology <-> Theorem.
True or False: If you can prove p1…p2 .: C, then the related formula (p1…p2) -> C is a theorem.
True, it is also a tautology. And also vice versa.
What are the 3 features of Proofs of Theorems?
- always uses either [~I] or [->I]
- the first step is always a hypothesis
- reasoning always goes from bottom up
True or False: Basic and Derived rules can be applied to subformulas.
False. Basic and Derived rules can only be applied to whole lines; Equivalence rules can be applied to both subformulas and whole lines.
What is the difference between [DN] and [~E]?
~E only works on way: if you have ~~p then you can assume p. DN works both ways, if you have p you can assume ~~p and vice versa.
Fill in the blank: If you need to switch a wedge to an arrow, use _____.
MI.
True or False: Two statement forms are Logically Equivalent if they can be substituted for one another, even if the truth value changes.
False. They are logically equivalent if they can be substituted for each other WITHOUT the truth values changing.
True or False: Two statement forms, p and q, are Logically Equivalent because p <-> q is a tautology.
True.
Fill in the Blank: To show that p is a contradiction, you need to prove that ~p is a ______.
Tautology.
True or False: You can use proofs to prove that a statement form, like p, is a contingency.
False.
True or False: You can prove that an argument is invalid using a proof.
False, proofs only prove validity because it needs the approval/guidelines of the rules, never invalidity.
What is the proof strategy for a premise with a ~ as the major operator?
Try driving the ~ inwards using DeMorgans, MI, etc.
What is the proof strategy for a premise with a <-> as the major operator?
Try <->E.
What is the proof strategy for a premise with a -> as the major operator?
Look for ->E or Modus Tollens.
What is the proof strategy for a premise with a v as the major operator?
Look for Disjunctive Syllogism or vE.
What is the proof strategy for a premise with a & as the major operator?
Try &E.
What is the proof strategy for a conclusion with a <-> as the major operator?
Try <->I.
What is the proof strategy for a conclusion with a -> as the major operator?
Try ->I.
What is the proof strategy for a conclusion with a v as the major operator?
Try ->I, Material Implication, or Double Negation to get it.
What is the proof strategy for a conclusion with a & as the major operator?
Try &I to get.
What is the proof strategy for a conclusion with a ~ as the major operator?
1) Try ~I
2) if it is a complex statement (has subformulas) try driving ~ inwards using DeMorgans, MI, etc.
True or False: Every argument form whose conclusion is a tautology is valid.
True.
True or False: “A unless B” is equivalent to: “Either B or not A”.
False.
True or False: It is possible for a compound sentence to contain only one component sentence.
True.
What is Double Negation?
[DN]. Like [~E] except it goes both ways.
p :: ~~p
What is DeMorgans?
[DM].
~(p&q) :: ~p v ~q
~(pvq) :: ~p & ~q
What is Transportation?
[TRANS]. A rewriting of Modus Tollens.
p -> q :: ~q -> ~p
**keep in mind order and negations
What is Material Implication?
[MI]. Meant to switch arrows to wedges and vice versa.
p -> q :: ~p v q
What is Exportation?
[EXP].
(p&q) -> r :: p -> (q -> r)
What is this form called and what kind of rule is it:
p -> q :: ~q -> ~p
Transportation, equivalence rule.
What form is this and what kind of rule is it:
p -> q :: ~p v q
Material Implication, equivalence rule.
