Logic

Front 1

What is an ordered pair?

Back 1

An ordered pair comprises two components in a given order

Front 2

What is a binary relation?

Back 2

A set is a binary relation iff it only contains only orderedpairs

Front 3

Define Reflexive

Back 3

Iff every element in the set is related toitself,

∀x(Sx → Pxx)

Front 4

Define Symmetric

Back 4

Iff when its true that Rab its true that Rba

∀x∀y((Sx ⋀ Sy) → (Pxy ⋀ Pyx))

Front 5

Define Asymmetric

Back 5

Iff for no elements d,e, if {d,e} then {e,d}

∀x ¬∃y(Sx ⋀ Sy ⋀ Pxy ⋀ Pyx)

Front 6

Define Antisymmetric

Back 6

Iff no 2 different elements have symmetry

Front 7

Define Transitive

Back 7

Iff a is related to b, is related to c, isrelated to a

∀x∀y∀z(Sx ⋀ Sy ⋀ Sz ⋀ Pxy ⋀ Pyz → Pxz)

Front 8

Define Equivalence Relationship

Back 8

Iff R is reflexive, symmetric and transitive onS

Front 9

Define a Function

Back 9

A function is a sentence that assumes a value of true or false

Front 10

What characteristics does the empty set have?

Back 10

The relation ∅isreflexive on the empty set but on no other set, it is symmetric but alsoantisymmetric and asymmetric, it is also transitive

Front 11

What is an argument?

Back 11

An argument consists of a set of declarative sentences(premises) and a declarative sentence (conclusion)

Front 12

What is a valid argument?

Back 12

A valid argument must have no hidden assumptions

Front 13

What is a logically valid argument?

Back 13

An argument is logically valid iff there is nointerpretation where the premises are all true and the conclusion is false

Front 14

Define logically consistent

Back 14

A set of sentences is logically consistent iff thereis at least one interpretation under which all sentences of the set are true

Front 15

Define logically true

Back 15

A sentence is logically true iff it is true under anyinterpretation

Front 16

Define a contradiction

Back 16

A sentence is a contradiction iff it is false underall interpretations

Front 17

Define logically equivalent

Back 17

Sentences are logically equivalent iff they are trueunder exactly the same interpretations

Front 18

Bracketing Conventions

Back 18

⋀ and ⋁ bind more strongly than → and ↔,

You can drop the outer brackets

You can drop the brackets on strings of ⋀ and ⋁

Front 19

What is the object language and the metalanguage?

Back 19

L1 is the object language, English is themetalanguage

Front 20

⋀ Truth Table

Back 20

Front 21

⋁ Truth Table

Back 21

Front 22

→ Truth Table

Back 22

Front 23

↔ Truth Table

Back 23

Front 24

What are truth functional connectives?

Back 24

Truth functional iff truth value of a compound sentencecannot be changed when replacing a direct sub-sentence with one of the sametruth-value

Front 25

Rules of Thumb for →

Back 25

‘If’ mid-sentence corresponds to ←

‘Only if’ corresponds to →

Front 26

What is an arity-index?

Back 26

Each predicate letter has an arity-index that corresponds tothe number of designators the predicate expression can take

Front 27

What is a counterexample?

Back 27

A L2 structure Ais a counterexample to an argument iff all premises of the argument are true inA and the conclusion isfalse in A

Create a complete dictionary where the Premises are true andthe Conclusion is false, often use numbers

Front 28

Complete and Reflexive Example

Back 28

S = {1}, R = {<1,1>}

Front 29

Complete and Symmetric Example

Back 29

S = {1}, R = {<1,1>}

Front 30

Transitive, Asymmetric and Not Complete

Back 30

S = {1,2,3}, R = {<1,2>, <2,2>, <1,3>}

Front 31

Complete and Not Symmetric

Back 31

S = {1,2}, R = {<1,1>, <1,2>, <2,2>}

Front 32

Can a complete and non-empty set be reflexive?

Back 32

A complete and non-empty set must be reflexive as if a e R then <a,a> e R so it cannot be the case that there is some a e R such that <a,a> is not e R,a>,a>

Front 33

Propositionally Valid

Back 33

An argument is propositionally valid iff it's L1 formalisation is valid