Logic symbols
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.1
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.
There are many ways to express the concepts listed below in english. I do not pretend to list all the ways here only some examples. For a precise, detailed, non-ambiguous english language that is useful when translating symbols back into english or just writing with extreme clarity, see this page.
| Name/group | My preferred symbolization | Other symbolizations used | Example | English translation(s)2 |
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Proposition logic |
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| Logical/strict implication/conditionality3 | ⇒ | →, ->, > | P⇒Q | If P, then Q must be the case.
If P, then it must be the case that Q |
| Logical/strict equivalency | ⇔ | ↔, <->, <> | P⇔Q | P is logically equivalent with Q |
| Material implication/conditionality | → | ⇒, ->, >, ⊃ | P→Q | If P, then Q.4
P means that Q. |
| Material equivalency | ↔ | ⇔, <->, <>, ≡ | P↔Q | P is materially equivalent with 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. |
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| Inference | ⊢ | :., | {P, P→Q}⊢Q | From the set of (P, and P materially implies Q} one (can) infer Q. |
| Exclusive disjunction; Xor5 | ⊕ | ∨ | P⊕Q | Either P or Q. And not both. |
| Definition | ≡ | =df | P≡Q | P is defined as Q.
P means Q. |
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Predicate logic |
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| 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. | |
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Unique Existential quantification6 |
(∃!x) |
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(∃!x)(Fx) |
There exists exactly one x such that x is F. |
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Modal/alethic logic |
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| Possibility | ◊ | <> | ◊P | P is possible;
Possibly P; It is possible, that P. |
| Necessity | □ | [] | □P | P is necessary;
Necessarily P; It is necessary that P. |
| Consistency7 | ◦ | 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
Strict Conditional (Wikipedia)
Modal Logic (Stanford Encyclopedia of Philosophy)
R. Bradley and N. Swartz Possible Worlds, 1979, chapter 1.
Notes
1Character Map. An innate windows program which displays a lot of symbols you can use. To access it open “Run” (Hotkey is windows key + R) or find it in the start menu. Then type in “charmap”.
2These are not the only way the logical concept or relation can be expressed on normal language. For instance the word “but” also implies the relation of conjunction but it also means something more. The word “and” is preferred because it does not carry other connotations.
3It 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.
4Careful though not to simply assume that the english conditional means the same as the material conditional. This seems not to be the case.
5It is rarely used. Definable as (P∨Q)∧¬(P∧Q). See Wikipedia for a longer explanation.
6It is rarely used. See Wikipedia for an explanation.
7These four are rarely used. For an explanation see R. Bradley and N. Swartz Possible Worlds, 1979, chapter 1.
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