Logic Flashcards
Set
A collection of objects. Usually outlined by curly brackets {like this}. Order does not matter
Elements
Objects in a set
Identical sets
Have the same elements
The empty set
Contains no objects
Ordered pairs
Have a designated first component and second component
Binary relation
Contains only ordered pairs. Includes the empty set
A binary relation R is reflexive on a set S iff
iff for all elements d of S the pair is an element of R
A binary relation R is symmetric on a set S iff
iff for all elements d,e of S: if ∈ S then ∈ R
A binary relation R is asymmetric on a set S iff
iff for no elements d,e of S: if ∈ S then ∈ R
A binary relation R is antisymmetric on a set S iff
iff for no two different elements d,e of S: if ∈ R then ∈ R (there is only one symmetric pair)
A binary relation R is transitive on a set S iff
iff for all elements d,e,f of S: if ∈ R and ∈ R, then ∈ R
A binary relation R is a function iff
R is reflexive, transitive and symmetric on S
A binary relation R is an equivalence relation iff
for all elements d,e, f : if ∈ R and ∈ R then e=f (turns all things into one thing)
Domain of a function R
The set {d: there is an e such that ∈ R}
Range of a function R
The set {e: there is an d such that ∈ R}
Declarative sentences
Sentences that are true or false
Logical validity
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
Logical consistency
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.
Validity defined in terms of consistency
An argument is valid if and only if the set obtained by adding the negation of the conclusion to the premises is inconsistent
Logical truth
A sentence is logically true if and only if it is true under any interpretation
Logical contradiction
A sentenge is a contradiction iff it is false under all interpretations
Logical equivilence
Sentences are logically equivalent iff they are true under exactly the same interpretations
Term
Definition
L1 Sentence
All sentence letters are sentences of L1. Using connectors with sentence letters makes L1 sentences. Nothing else is a sentence of L1
L1 Structure
An L1 structure is an assignment of exactly one truth-value (T or F) to every sentence letter of L1
Logically true L1 sentence
Is true iff the sentence is true in all L1 structures
L1 contradiction
Is the case if the sentence is not true in any L1 structure
Logical equivalence
If both sentences true in exactly the same L1 structures
Logically valid L1 sentence
If there is no L1 structure in which all the sentences in the premis are true and the concludingg sentence is false.
L1 counterexample
Shows that the premises can be true and the conclusion false in some L1 structure
Semantic consistency
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
Truth Functionality
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
Types of sentences that cannot be translated by the arrow →
Subjunctives or counterfactuals that describe what would have happened under certain cirumstances
Scope of a connective in L1
The scope of an occurance of a connective in a sentence of L2 is the occurance of the smallest subsentence of
Arity index
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
Constants in L2
a,b,c,a1,a2,b1,c1….
Variables in L2
x,y,z,x1,y1,z1…
Sentence of L2
A formula of L2 is a sentence of L2 if and only if no variable occurs freely in the formula
The interpretation that is assigned to a symbol by a structure is called
Semantic valueExtension of the symbol
L2 Structure assigns
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
L2 structure definition
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
Variable assignment over an L2 structure
Assigns an element of the domain DA of A to each variable. (a function from the set of all variables into D)
Variable assignment satisfies ∃x∅ iff
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
Syntactic consistency
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)
Scope of a quantifier or connective in L2
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.
Elements with the same extension
Elements denoting the same object
Is English an extensional language?
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.
Critera for the formalization of an English predicate
Only if an English predicate expresses a relation, can it be adequately formalized as a predicate letter.
The validity of English arguments in predicate logic
An argument in English is valid in predicate logic if and only if its formalization in the language L2 of predicate logic is valid.
When is a L2 sentence deemed logically true?
If it is provable via natural deduction
When is an L2 sentence deemed valid?
If it is provable from certain premises (think of the critera for a valid argument)
The type of identity that L= formalizes differently
Numerical identity (different to qualitative identity)
