Use LEFT and RIGHT arrow keys to navigate between flashcards;
Use UP and DOWN arrow keys to flip the card;
H to show hint;
A reads text to speech;
52 Cards in this Set
- Front
- Back
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
|