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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 subsentence with one of the sametruthvalue 

Rules of Thumb for → 
‘If’ midsentence corresponds to ← ‘Only if’ corresponds to → 

What is an arityindex? 
Each predicate letter has an arityindex 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 nonempty set be reflexive? 
A complete and nonempty 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 