Symbol
Name
How to read with example
Example
∼
Negation
Not: “Not X”
∼X
∧
Conjunction
And: “X and Y”
X∧Y
∨
Disjunction
Or: “X or Y”
X∨Y
⊃
Implication
If… then: “If X then Y”
X⊃Y
Biconditional
If and only if: “X if and only if Y”
XY
A, B, C,…
Propositions
( ) [ ] { }
Brackets
Documentation of Proper Notation
Propositional logic
Calculation order
∼ Negation
∧, ∨ Conjunction & Disjunction
⊃ Implication
Biconditional
Symbol
Meaning:
⊭
Logically invalid
⊨
Logically valid
The form of reasoning is logically valid if not possible that the premises are true and at the same time, conclusion is false. Otherwise, the reasoning is logically invalid.
In other words: the form of reasoning is logically valid if (if and only if) premises and negation of the conclusion are contradictory (i.e., inconsistent).
Logical validity is in other words:
Premises
Conclusion
Validity
True
True
Valid
True
Untrue
Invalid
Untrue
True
Valid
Untrue
Untrue
Valid
Symbol
Meaning:
How to read:
Example
≡
Logical equivalence
The propositions 𝑋 and 𝑌 are logically
equivalent if and only if their truth values are identical. We then denote 𝑋≡𝑌.
𝑋≡𝑌
∴
“therefore”, or “from which follows”
⊢
Deduction sign, read from left to right.
⊬
Invalid deduction sign, read from left to right.
⊣⊢
In two-way rules, reasoning can
happen in either direction, sentences
has the same meaning.
First-order Logic
Symbol
Name
How to read
Example
∀
Universal quantifier
Case A applies to all elements.
(∀𝑥)𝐴(𝑥)
∃
Existence quantifier
Case A applies to at least for some elements. There is at least one such element which has property A.
(∃𝑥)𝐴(𝑥)
Scope of quantifiers
• The quantifier binds immediately to the right expression.
• The range of the quantifiers can be adjusted by brackets (analogous to negation):
(∀𝑥)[𝑁𝑥⊃∼𝑃(𝑥)]
No minister is president.
(∀𝑥)𝑁(𝑥)⊃∼𝑃(𝑥)
If no one is a minister, then someone / he is not president.
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