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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