Term
|
Definition
Operator;
material implication;
implies; if .. then;
A ⇒ B is true just in the case that either A is false or B is true, or both.
→ may mean the same as ⇒ (the symbol may also indicate the domain and codomain of a function; see table of mathematical symbols).
⊃ may mean the same as ⇒ (the symbol may also mean superset);
x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2); |
|
|
Term
|
Definition
Operator;
material equivalence;
if and only if; iff; means the same as;
A ⇔ B is true just in case either both A and B are false, or both A and B are true;
x + 5 = y +2 ⇔ x + 3 = y; |
|
|
Term
|
Definition
Operator;
negation;
not;
The statement ¬A is true if and only if A is false.
A slash placed through another operator is the same as "¬" placed in front;
¬(¬A) ⇔ A
x ≠ y ⇔ ¬(x = y); |
|
|
Term
|
Definition
Operator;
logical conjunction;
and;
The statement A ∧ B is true if A and B are both true; else it is false;
n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number; |
|
|
Term
|
Definition
Operator;
logical disjunction;
or;
The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false;
n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3
when n is a natural number; |
|
|
Term
|
Definition
Operator;
exclusive disjunction;
xor;
The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same;
(¬A) ⊕ A is always true, A ⊕ A is always false; |
|
|
Term
|
Definition
Truth-Values;
Tautology;
top, velum;
The statement ⊤ is unconditionally true;
A ⇒ ⊤ is always true; |
|
|
Term
|
Definition
Truth-Values;
Contradiction;
bottom, falsum;
The statement ⊥ is unconditionally false;
⊥ ⇒ A is always true; |
|
|
Term
|
Definition
Quantifier;
universal quantification;
for all; for any; for each
∀ x: P(x) or (x) P(x) means P(x) is true for all x;
∀ n ∈ N: n2 ≥ n; |
|
|
Term
|
Definition
Quantifier;
existential quantification;
there exists;
∃ x: P(x) means there is at least one x such that P(x) is true;
∃ n ∈ N: n is even; |
|
|
Term
|
Definition
Quantifier;
uniqueness quantification;
there exists exactly one;
∃! x: P(x) means there is exactly one x such that P(x) is true;
∃! n ∈ N: n + 5 = 2n; |
|
|
Term
|
Definition
Relation;
definition;
is defined as;
x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).
P :⇔ Q means P is defined to be logically equivalent to Q;
cosh x := (1/2)(exp x + exp (−x))
A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B); |
|
|
Term
|
Definition
precedence grouping;
Perform the operations inside the parentheses first;
(8/4)/2 = 2/2 = 1, but
8/(4/2) = 8/2 = 4; |
|
|
Term
|
Definition
Relation;
Turnstile;
provable;
x ⊢ y means y is provable from x (in some specified formal system);
A → B ⊢ ¬B → ¬A; |
|
|
Term
|
Definition
Relation;
double turnstile;
entails;
x ⊨ y means x semantically entails y;
A → B ⊨ ¬B → ¬A; |
|
|
Term
|
Definition
Operator;
alternative denial;
not both;
Means “not both”. Sometimes written as ↑; |
|
|
Term
|
Definition
Operator;
joint denial;
neither; nor;
Means “neither/nor”; |
|
|
Term
|
Definition
Relation;
therefore;
Used to signify the conclusion of an argument. Usually taken to mean implication, but often used to present arguments in which the premises do not deductively imply the conclusion; |
|
|
Term
|
Definition
Relation;
forces;
A relationship between possible worlds and sentences in modal logic; |
|
|
Term
|
Definition
Set Theory;
membership;
Denotes membership in a set. If a ∈ Γ, then a is a member (or an element) of set Γ; |
|
|
Term
|
Definition
Set Theory;
union;
Used to join sets. If S and T are sets of formula, S ∪ T is a set containing all members of both; |
|
|
Term
|
Definition
Set Theory;
intersection;
The overlap between sets. If S and T are sets of formula, S ∩ T is a set containing those elemenets that are members of both; |
|
|
Term
|
Definition
Set Theory;
subset;
A subset is a set containing some or all elements of another set; |
|
|
Term
|
Definition
Set Theory;
proper subset;
A proper subset contains some, but not all, elements of another set; |
|
|
Term
|
Definition
Set Theory;
set equality;
Two sets are equal if they contain exactly the same elements; |
|
|
Term
|
Definition
Set Theory;
absolute complement;
∁(S) is the set of all things that are not in the set S. Sometimes written as C(S), S or SC; |
|
|
Term
|
Definition
Set Theory;
relative complement;
T - S is the set of all elements in T that are not also in S. Sometimes written as T \ S; |
|
|
Term
|
Definition
Set Theory;
empty set;
The set containing no elements; |
|
|
Term
|
Definition
Modality;
necessarily;
Used only in modal logic systems. Sometimes expressed as [] where the symbol is unavailable; |
|
|
Term
|
Definition
Modality;
possibly;
Used only in modal logic systems. Sometimes expressed as <> where the symbol is unavailable; |
|
|
Term
|
Definition
Non-Logical;
propositions;
Uppercase Roman letters signify individual propositions. For example, P may symbolize the proposition “Pat is ridiculous”. P and Q are traditionally used in most examples; |
|
|
Term
|
Definition
Non-Logical;
formulae;
Lowercase Greek letters signify formulae, which may be themselves a proposition (P), a formula (P ∧ Q) or several connected formulae (φ ∧ ρ); |
|
|
Term
|
Definition
Non-Logical;
variables;
Lowercase Roman letters towards the end of the alphabet are used to signify variables. In logical systems, these are usually coupled with a quantifier, ∀ or ∃, in order to signify some or all of some unspecified subject or object. By convention, these begin with x, but any other letter may be used if needed, so long as they are defined as a variable by a quantifier; |
|
|
Term
|
Definition
Non-Logical;
constants;
Lowercase Roman letters, when not assigned by a quantifier, signifiy a constant, usually a proper noun. For instance, the letter “j” may be used to signify “Jerry”. Constants are given a meaning before they are used in logical expressions; |
|
|
Term
|
Definition
Non-Logical;
predicate symbols;
Uppercase Roman letters appear again to indicate predicate relationships between variables and/or constants, coupled with one or more variable places which may be filled by variables or constants. For instance, we may definite the relation “x is green” as Gx, and “x likes y” as Lxy. To differentiate them from propositions, they are often presented in italics, so while P may be a proposition, Px is a predicate relation for x. Predicate symbols are non-logical — they describe relations but have neither operational function nor truth value in themselves; |
|
|
Term
|
Definition
Non-Logical;
sets of formulae; possible worlds;
Uppercase Greek letters are used, by convention, to refer to sets of formulae. Γ is usually used to represent the first site, since it is the first that does not look like Roman letters. (For instance, the uppercase Alpha (Α) looks identical to the Roman letter “A”);
In modal logic, uppercase greek letters are also used to represent possible worlds. Alternatively, an uppercase W with a subscript numeral is sometimes used, representing worlds as W0, W1, and so on. |
|
|
Term
|
Definition
Non-Logical;
sets;
Curly brackets are generally used when detailing the contents of a set, such as a set of formulae, or a set of possible worlds in modal logic. For instance, Γ = { α, β, γ, δ }; |
|
|
