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52 Cards in this Set

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This rule allows you to assert any conjunct of Pi of a conjunctive sentence P1 ^...^ Pi ^...^ Pn.
^ Elim (Conjunction elimination)
"Allows you to assert a conjunction P1 ^...^ Pn provided you have already established each of its constituent conjuncts P1 through Pn
^ Intro (Conjunction Introduction
"Allows you to conclude a sentence S from a disjunction P1 v...v Pn if you can prove S from each of P1 through Pn individually
v Elim (Disjunction elimination)
Proof by cases
v Elim (Disjunction elimination)
Two subproofs for each use
v Elim (Disjunction elimination)
What rule let's you assert P?
From ⌐⌐P to P
⌐ Elim (Negation elimination)
The rule of corresponds to the method of indirect proof or proof by contradiction. Like v Elim, it involves the use of a subproof, as will the formal analogs of all nontrivial methods of proof. The rule says that if you can prove a contradiction ┴ on the basis of an additional assumption P, then you are entitled to infer ⌐P from the original premises.
⌐ Intro (Negation introduction)
Indirect Proof
⌐ Intro (Negation introduction)
Proof by Contradiction
⌐ Intro (Negation introduction)
"very trivial, valid step"
⌐ Elim (Negation elimination)
simplest rule
^ Elim (Conjunction elimination)
What allows you to assert ⌐P?
|
| | P
| | .
| | .
| | ┴
| ⌐P
⌐ Intro (Negation introduction)
This rule of allows us to obtain the contradiction symbol if we have established an explicit contradiction in the form of some sentence P and its negation
┴ Intro (Contradiction Introduction)
What rule allows us to assert contradiction? What steps do we cite? What steps do we cite for negation introduction?

| A
| | ⌐A
| | ┴
| | ⌐⌐A ⌐ Intro
┴ Intro 1,2
⌐ Intro 2,3

"Ordinarily, you will only apply contradiction introduction in the context of a subproof, to show that the subproof's assumption leads to a contradiction. The only time you will be able to derive ┴ in your main proof is when the premises of your argument are themselves inconsistent. In fact, this is how we give a formal proof that a set of premises is inconsisten. A formal proof of inconsistency is a proof that derives ┴ at the main level of the proof."
-"As we remarked earlier, if in a proof, or more importantly in some subproof, you are able to establish a contradiction, then you are entitled to assert any FOL sentence P whatesoever. In our formal system, this is modeled by what rule?
┴ Elim (Contradiction elimination)
What rule allows you to assert P?

| ┴
| .
| .
| P
┴ Elim (Contradiction elimination)

"As we remarked earlier, if in a proof, or more importantly in some subproof, you are able to establish a contradiction, then you are entitled to assert any FOL sentence P whatesoever. In our formal system, this is modeled by the rule of ┴ Elimination."
What is modus ponens?
→ Elim (Conditional elimination)
If you have proven both P → Q then you can asert Q, citing what steps as justification?

| P → Q
| .
| .
| P
| .
| .
| Q
The two earlier steps, e.g, P → Q and P

| P → Q
| .
| .
| P
| .
| .
| Q
Requires us to construct a subproof with the assumption of P and try to prove Q
→ Intro (Conditional elimination)
If we succeed then we are allowed to discharge the assumption and conclude our desired conditional, citing the subproof as justification
→ Intro (Conditional elimination)
What rule of proof does this elicit?

|
| | P
| | .
| | .
| | Q
| P → Q
→ Intro (Conditional elimination)
What rule allows us to now assert Q?

| P ↔ Q (or Q ↔ P)
| .
| .
| P
| .
| .
| Q
↔ Elim (Bi-conditional Elimination)
This rule requires that you give two subproofs, one showing that Q follows from P, and one showing that P follows from Q.
↔ Intro (Bi-conditional Introduction)
This rule of proof does this elicit?

| | P
| | .
| | .
| | Q
|
| | Q
| | .
| | .
| | P
| P ↔ Q
↔ Intro (Bi-conditional Introduction)
What rule of proof does this elicit? Be prepared to cite the proper lines.

|
| | P
| | | ⌐P
| | | ┴ ┴ Intro 1,2
| | ⌐⌐P ⌐ Intro 2-3
|
| | ⌐⌐P
| | P ⌐ Elim 5
| P ↔ ⌐⌐P
↔ Intro 1-4,5-6
What lines do we cite for the first ┴ Intro?
|
| | P
| | | ⌐P
| | | ┴ ┴ Intro 1,2
| | ⌐⌐P ⌐ Intro
|
| | ⌐⌐P
| | P ⌐ Elim
| P ↔ ⌐⌐P ↔ Intro
┴ Intro 1,2
What lines do we cite for ↔ Intro?
|
| | P
| | | ⌐P
| | | ┴ ┴ Intro 1,2
| | ⌐⌐P ⌐ Intro
|
| | ⌐⌐P
| | P ⌐ Elim
| P ↔ ⌐⌐P ↔ Intro
↔ Intro 1-4,5-6
Define validity
An argument is valid if the conclusion must be true in any circumstance in which the premises are true
If S follows from the premises simply in virtue of the meaning of the truth-functional connectives, we are said to have what?
What is Tautological Consequence, Alex.
If two sentences are equal simpily in virtue of the meanings of the truth-functional connectives, they are what?
Tautological Equivalent
Two sentences are what if they have the same truth value in all possible circumstances?
Logically equivalent
We say that a sentence or claim is what if there is no logical reason it cannot be true?
Logically possible
A sentence that is a logical consequence of any set of premises
Logical truth
The Latin name for the rule that allows us to infer Q from P and P → Q
modus ponens
The first component clause of a conditional
Antecedent
The number of arguments a predicate takes
Arity
The most basic sentences in FOL, formed by a predicate followed by the right number of names. Correspond to the simplest sentence in English, e.g, "max saw claire"
Atomic sentences
FOL
First Order Logic
W proposition that is accepted as true about some domain and used to establish other truths about that domain
Axiom
The logical connectives conjunction, disjunction, and negation allow us to form complex claims from simpler claims and are known as what and named after whom?
Boolean connectives, after George Bool
A possible situation in which all the premises are true but the conclusion is false.
Counterexample
Finding this is sufficient to show that an argument is not logically valid.
Counterexample
Name two good methods of showing counterexample
1) truth-table
2) create a world
a.k.a indirect proof
Proof by contradiction
When are two sentences logically equivalent?
When they have the same truth values in all possible circumstances
How do we prove ⌐S by contradiction?
Assume S and prove a contradiction. IOW, we assume the negation of what we wish to prove and show that this assumption leads to a contradiction.
These show the way in which the truth value of a sentence is built up using truth-functional connectives depends on the truth values of the sentences components.
Truth Table
In a truth table, the far left column follows what repeating T/F order and the inner left column follows what repeating T/F order?
Far left: T on the top half, F on the bottom
Far right: T/F T/F
How do you calculate the number of rows in your truth table?
N^2
IOW, the number of atomic sentences squared.
wff
Well-formed formula
These are "grammmatical" expressions of FOL. They are defined inductively.
wff
Sentences of FOL are what?
wffs with no free variables