Logical symbols

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Logical symbols

An overview

About

This page is meant as a resource where one can find the logical symbols which are used when one is writing philosophy (especially logic). These are either hard or impossible to find in Charmap.
A small program has been created by a friend of mine to help get the right symbols faster and easier. The program is compatible with whatever symbols you might want to use. You can find it here.

Name/group	My preferred symbolization	Other symbolizations used	Example	English translation(s)¹
Proposition logic
Logical/strict implication/conditionality²	⇒	→, ->, >,	P⇒Q	It must be the case that if P, then Q Necessarily, if P, then Q⁷
Logical/strict equivalency	⇔	↔, <->, <>	P⇔Q	P is logically equivalent with Q Necessarily, P if and only if Q Necessarily, P iff Q
Material implication/conditionality	→	⇒, ->, >, ⊃	P→Q	If P, then Q.³ P means that Q
Material equivalency	↔	⇔, <->, <>, ≡	P↔Q	P is materially equivalent with Q P if and only iff Q P iff Q
Negation	¬	~, !	¬P ~P	Not-P It is not the case that P.
Conjunction	∧	&, ͦ	P∧Q	P and Q.
Disjunction	∨		P∨Q	P or Q. Either P or Q.
Inference	⊢	:.,	{P, P→Q}⊢Q	From the set of (P, and P materially implies Q} one (can) infer Q.
Exclusive disjunction; Xor⁴	⊕	⊻	P⊕Q	Either P or Q and not both.
Definition	≡	=df	P≡Q	P is defined as Q. P means Q.
Predicate logic
Universal quantification	(∀x)	(x)	(∀x)(Fx)	For all x, x is F; For any x, x is F. All x.
Existential quantification	(∃x)		(∃x)(Fx)	There exists (at least one) x such that x is F.
Unique Existential quantification⁵	(∃!x)		(∃!x)(Fx)	There exists exactly one x such that x is F.
Modal/alethic logic
Possibility	◊	<>	◊P	P is possible; Possibly P; It is possible, that P.
Necessity	□	[]	□P	P is necessary; Necessarily P; It is necessary that P.
Consistency⁶	◦		P◦Q	P is consistent with Q.
Inconsistency	Φ		PΦQ	P is inconsistent with Q.
Contingency	∇		∇P	P is contingent.
Noncontingency	Δ		ΔP	P is non-contingent.

General references

Logical symbols (Wikipedia)

Modal Logic (Stanford Encyclopedia of Philosophy)

R. Bradley and N. Swartz Possible Worlds, 1979.

Notes

1These are not the only way the logical concept or relation can be expressed on normal language. For instance the word “but” is also used to express the concept of conjunction but it also means something more (like that the following is unexpected). The word “and” is preferred because it carries as few connotations as possible without inventing a new word for the concept. See Swartz and Bradley (1979) p. for a number of examples of other uses of "and", "but", "or" etc.

2It is common to use the slim arrows to mean logical implication and logical equivalency, however, since these are rarely used by me, and I prefer to use the slim arrows in general (they are more accessible), then I use the slim arrows for material implication and material equivalency. I use the fat arrows for logical implication and equivalency instead.

Another symbol is also commonly used for material implication and that is the horseshoe (⊃). There are three reasons why I don't use this for material implication. One, I dislike its visuals. Two, it is not easily accessible (like through charmap.exe on a windows computer). Three, the symbol is also used in set theory and I diswant to add more confusion.

3Careful though not to simply assume that the english conditional means the same as the material conditional. This seems not to be the case.

4It is rarely used. Definable as (P∨Q)∧¬(P∧Q). See Wikipedia for a longer explanation.

5It is rarely used. See Wikipedia for an explanation.

6These four symbols are rarely used. For an explanation see R. Bradley and N. Swartz Possible Worlds, 1979, chapter 1.

7 However a common translation is "If P, then it must be the case Q" which is dangerous as it leads on to believe that the form is P→□Q and thus to commit the modal fallacy.