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Here are the logic symbols I use, roughly in order of frequency, along with a
brief description of what they do. The main logical paradigms used by philosophers are
sentential logic (that is the logic of sentences), predicate logic and modal logic
(which deals with necessity and possibility). Each is roughly a superset of the former.
 
∀ [predicate]
- the universal or 'for all' quantifier. This is used to specify that some
statement applies to all things which match.
- Example: (∀x)(Fx→Gx) means for all things x, all x's that are F are G.
e.g. for all things x, if x is frosty (F), x is cold (G).
 
∃ [predicate]
- the existential or 'for at least one' quantifier. This is used to
specify that some statement applies to at least one thing.
- Example: (∃x)(Fx) means for at least one thing x, x is F.
e.g. there is at least one dog that can bark.
 
¬ [sentential]
- negation. ¬P just means "not P".
in programming, this would be !expr.
 
∧ [sentential]
- logical and. Think about it the same way you would
([Expr foo] && [Expr bar]) in C/Java-style languages.
 
∨ [sentential]
- logical or. Equivalent to "||" in C/Java-style programming languages.
 
→ [sentential]
- conditional. P→Q means "if P, then Q"
 
▢ [modal]
- a square is the symbol for 'necessarily'.
Example: ▢P means it's necessarily so that P.
 
◇ [modal]
- a diamond is the symbol for 'possibly'.
Example: ◇Q→P means "if Q is possibly true, then P is true"
 
∴ [sentential]
- therefore. Often used in both sentential and predicate logic to specify
a conclusion in a multiple step logical formulation.
 
Many philosophers since Russell also use set theory to express their ideas:
- union
- intersection
- subset of
- superset of
 
Further reading
---------------
 
Basic logic textbooks: Samuel Guttenplan's "Languages of Logic", Wilfrid Hodge's
"Logic" or John Pollock's "Technical Methods in Philosophy".
 
Online resources:
http://philosophy.hku.hk/think/sl/
http://www.rbjones.com/rbjpub/logic/log019.htm
http://www.jgsee.kmutt.ac.th/exell/Logic/Logic31.htm
http://plato.stanford.edu/entries/set-theory/primer.html

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