Fitch notation, also known as Fitch diagrams (named after Frederic Fitch), is a notational system for constructing formal proofs used in sentential logics and predicate logics. Fitch-style proofs arrange the sequence of sentences that make up the proof into rows. A unique feature of Fitch notation is that the degree of indentation of each row conveys which assumptions are active for that step.

Example

Each row in a Fitch-style proof is either: * an assumption or subproof assumption. * a sentence justified by the citation of (1) a

rule of inference In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of in ...

and (2) the prior line or lines of the proof that license that rule. Introducing a new assumption increases the level of indentation, and begins a new vertical "scope" bar that continues to indent subsequent lines until the assumption is discharged. This mechanism immediately conveys which assumptions are active for any given line in the proof, without the assumptions needing to be rewritten on every line (as with sequent-style proofs). The following example displays the main features of Fitch notation:

0 , __                         ssumption, want P iff not not P1 ,    , __ P                   ssumption, want not not P2 ,    ,    , __ not P           ssumption, for reductio3 ,    ,    ,    contradiction   ontradiction introduction: 1, 24 ,    ,    not not P           egation introduction: 2  , 
5 ,    , __ not not P           ssumption, want P6 ,    ,    P                   egation elimination: 5  , 
7 ,    P iff not not P         iconditional introduction: 1 - 4, 5 - 6

0. The null assumption, ''i.e.'', we are proving a tautology
1. Our first subproof: we assume the l.h.s. to show the r.h.s. follows
2. A subsubproof: we are free to assume what we want. Here we aim for a reductio ad absurdum
3. We now have a contradiction
4. We are allowed to prefix the statement that "caused" the contradiction with a not
5. Our second subproof: we assume the r.h.s. to show the l.h.s. follows
6. We invoke the rule that allows us to remove an even number of nots from a statement prefix
7. From 1 to 4 we have shown if P then not not P, from 5 to 6 we have shown P if not not P; hence we are allowed to introduce the biconditional

References

* Frederic Brenton Fitch, ''Symbolic Logic: An introduction'', Ronald Press Co., 1952. * Jon Barwise and John Etchemendy, Language, Proof and Logicbr>''1st edition as PDF''
Seven Bridges Press and CSLI, 1999.

External links

Fitch's Paradox of Knowability

{{Webarchive, url=https://web.archive.org/web/20061002151420/http://logik.phl.univie.ac.at/%7Echris/gateway/formular-uk-fitch.html , date=2006-10-02
A Web implementation of Fitch proof system (propositional and first-order) at proofmod.mindconnect.cc

The Jape general-purpose proof assistant
(see Jape)
Resources for typesetting proofs in Fitch notation with LaTeX
(see LaTeX)
FitchJS: An open source web app to construct proofs in Fitch notation (and export to LaTeX)

Natural deduction proof editor and checker in Fitch notation
Philosophical logic Logical calculi

Example

See also

References

External links