Logic

Question 1

Set

Answer 1

A collection of objects. Usually outlined by curly brackets {like this}. Order does not matter

Question 2

Elements

Answer 2

Objects in a set

Question 3

Identical sets

Answer 3

Have the same elements

Question 4

The empty set

Answer 4

Contains no objects

Question 5

Ordered pairs

Answer 5

Have a designated first component and second component

Question 6

Binary relation

Answer 6

Contains only ordered pairs. Includes the empty set

Question 7

A binary relation R is reflexive on a set S iff

Answer 7

iff for all elements d of S the pair is an element of R

Question 8

A binary relation R is symmetric on a set S iff

Answer 8

iff for all elements d,e of S: if ∈ S then ∈ R

Question 9

A binary relation R is asymmetric on a set S iff

Answer 9

iff for no elements d,e of S: if ∈ S then ∈ R

Question 10

A binary relation R is antisymmetric on a set S iff

Answer 10

iff for no two different elements d,e of S: if ∈ R then ∈ R (there is only one symmetric pair)

Question 11

A binary relation R is transitive on a set S iff

Answer 11

iff for all elements d,e,f of S: if ∈ R and ∈ R, then ∈ R

Question 12

A binary relation R is a function iff

Answer 12

R is reflexive, transitive and symmetric on S

Question 13

A binary relation R is an equivalence relation iff

Answer 13

for all elements d,e, f : if ∈ R and ∈ R then e=f (turns all things into one thing)

Question 14

Domain of a function R

Answer 14

The set {d: there is an e such that ∈ R}

Question 15

Range of a function R

Answer 15

The set {e: there is an d such that ∈ R}

Question 16

Declarative sentences

Answer 16

Sentences that are true or false

Question 17

Logical validity

Answer 17

An argument is logically avalid iff there is no interpretation under which the premisses are all true and the conclusion is false. The truth of the premisses guarantees the truth of the conclusion. Logically valid arguments are deductively valid

Question 18

Logical consistency

Answer 18

A set of sentences is logically consistent if and only if there is at least one interpretation under which all sentences of the set are true. There is a structure that assigns either all true or all false values that make the sentence true overall.

Question 19

Validity defined in terms of consistency

Answer 19

An argument is valid if and only if the set obtained by adding the negation of the conclusion to the premises is inconsistent

Question 20

Logical truth

Answer 20

A sentence is logically true if and only if it is true under any interpretation

Question 21

Logical contradiction

Answer 21

A sentenge is a contradiction iff it is false under all interpretations

Question 22

Logical equivilence

Answer 22

Sentences are logically equivalent iff they are true under exactly the same interpretations

Question 23

Term

Answer 23

Definition

Question 24

L1 Sentence

Answer 24

All sentence letters are sentences of L1. Using connectors with sentence letters makes L1 sentences. Nothing else is a sentence of L1

Question 25

L1 Structure

Answer 25

An L1 structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1

Question 26

Logically true L1 sentence

Answer 26

Is true iff the sentence is true in all L1 structures

Question 27

L1 contradiction

Answer 27

Is the case if the sentence is not true in any L1 structure

Question 28

Logical equivalence

Answer 28

If both sentences true in exactly the same L1 structures

Question 29

Logically valid L1 sentence

Answer 29

If there is no L1 structure in which all the sentences in the premis are true and the concludingg sentence is false.

Question 30

L1 counterexample

Answer 30

Shows that the premises can be true and the conclusion false in some L1 structure

Question 31

Semantic consistency

Answer 31

Set of sentences that make up the premises are semantically consistent if there is an L1 structure that makes each sentence true. Hence the premises prove a conclusion in L1 only if the set containing all sentences in the premis and the negation of the conclusion is semantically inconsistent

Question 32

Truth Functionality

Answer 32

A connective is truth-functional if and only if the truth-value of the compound sentence cannot be changed by replacing a direct sub sentence with another sentence having the same truth value. Truth functional if depends only on the truth values of the direct sub sentences

Question 33

Types of sentences that cannot be translated by the arrow →

Answer 33

Subjunctives or counterfactuals that describe what would have happened under certain cirumstances

Question 34

Scope of a connective in L1

Answer 34

The scope of an occurance of a connective in a sentence of L2 is the occurance of the smallest subsentence of

Question 35

Arity index

Answer 35

The upper index of a predicate letter that tells you how many designators it takes. May have no upper index and hence be arity 0

Question 36

Constants in L2

Answer 36

a,b,c,a1,a2,b1,c1….

Question 37

Variables in L2

Answer 37

x,y,z,x1,y1,z1…

Question 38

Sentence of L2

Answer 38

A formula of L2 is a sentence of L2 if and only if no variable occurs freely in the formula

Question 39

The interpretation that is assigned to a symbol by a structure is called

Answer 39

Semantic valueExtension of the symbol

Question 40

L2 Structure assigns

Answer 40

Specifies a domain of discourse. Assigns elements of the domain of discourse to the constants as their semantic values. Assign sentence letters a T or F

Question 41

L2 structure definition

Answer 41

An L2 structure is an ordered pair where D is some non-empty set and I is a function from the set of all constants, sentence letters and predicate letters such that the value of every constant is an element of D, the value of every sentence letter is a truth-value T or F, and the value of every n-ary predicate letter is an n-ary relation

Question 42

Variable assignment over an L2 structure

Answer 42

Assigns an element of the domain DA of A to each variable. (a function from the set of all variables into D)

Question 43

Variable assignment satisfies ∃x∅ iff

Answer 43

thre is a variable assignment B satisfying ∅ that differs from a at most in the entry for x. ∅ may have free occurences of other variables than y; for this reason B mut agree with a on all variables with the possible exception of x

Question 44

Syntactic consistency

Answer 44

A set of L2 sentences ⌈ is syntactically consistent if and only if there is a sentence such that ⌈⊬ϕ. (It must be not the case that any sentence whatsoever can be proved from the premises)

Question 45

Scope of a quantifier or connective in L2

Answer 45

The scope of an occurrence of a quantifier or a connective in a sentence of L2 is the occurrence of the smallest L2 formula that contains that occurrence of the quantifier or connecctive and is part of the sentence.

Question 46

Elements with the same extension

Answer 46

Elements denoting the same object

Question 47

Is English an extensional language?

Answer 47

No. Mary may believe that she is in Paris but she does not believe that she is in the site of the 1998 FIFA World Cup final.

Question 48

Critera for the formalization of an English predicate

Answer 48

Only if an English predicate expresses a relation, can it be adequately formalized as a predicate letter.

Question 49

The validity of English arguments in predicate logic

Answer 49

An argument in English is valid in predicate logic if and only if its formalization in the language L2 of predicate logic is valid.

Question 50

When is a L2 sentence deemed logically true?

Answer 50

If it is provable via natural deduction

Question 51

When is an L2 sentence deemed valid?

Answer 51

If it is provable from certain premises (think of the critera for a valid argument)

Question 52

The type of identity that L= formalizes differently

Answer 52

Numerical identity (different to qualitative identity)