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33 Cards in this Set

  • Front
  • Back

What is an ordered pair?

An ordered pair comprises two components in a given order

What is a binary relation?

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

Define Reflexive

Iff every element in the set is related toitself,

∀x(Sx → Pxx)

Define Symmetric

Iff when its true that Rab its true that Rba

∀x∀y((Sx ⋀ Sy) → (Pxy ⋀ Pyx))

Define Asymmetric

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

∀x ¬∃y(Sx ⋀ Sy ⋀ Pxy ⋀ Pyx)

Define Antisymmetric

Iff no 2 different elements have symmetry

Define Transitive

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

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

Define Equivalence Relationship

Iff R is reflexive, symmetric and transitive onS

Define a Function

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

What characteristics does the empty set have?

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

What is an argument?

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

What is a valid argument?

A valid argument must have no hidden assumptions

What is a logically valid argument?

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

Define logically consistent

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

Define logically true

A sentence is logically true iff it is true under anyinterpretation

Define a contradiction

A sentence is a contradiction iff it is false underall interpretations

Define logically equivalent

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

Bracketing Conventions

⋀ and ⋁ bind more strongly than → and ↔,

You can drop the outer brackets

You can drop the brackets on strings of ⋀ and ⋁

What is the object language and the metalanguage?

L1 is the object language, English is themetalanguage

⋀ Truth Table

⋁ Truth Table

→ Truth Table

↔ Truth Table

What are truth functional connectives?

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

Rules of Thumb for →

‘If’ mid-sentence corresponds to ←

‘Only if’ corresponds to →

What is an arity-index?

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

What is a counterexample?

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

Complete and Reflexive Example

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

Complete and Symmetric Example

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

Transitive, Asymmetric and Not Complete

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

Complete and Not Symmetric

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

Can a complete and non-empty set be reflexive?

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>

Propositionally Valid

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