資料來源 : Free On-Line Dictionary of Computing

natural deduction
     
        A set of rules expressing how valid {proof}s may be
        constructed in {predicate logic}.
     
        A horizontal line separates premises (above) from conclusions
        (below).  Vertical ellipsis (dots) stand for a series of
        applications of the rules.  "T" is the constant "true" and "F"
        is the constant "false" (sometimes written with a {LaTeX}
        {\perp}).
     
        "^" is the AND (conjunction) operator, "v" is the inclusive OR
        (disjunction) operator and "/" is NOT (negation or
        {complement}, normally written with a {LaTeX} {\neg}).
     
        P, Q, P1, P2, etc. stand for {proposition}s such as "Socrates
        was a man".  P[x] is a proposition possibly containing
        instances of the variable x, e.g. "x can fly".
     
        A proof (a sequence of applications of the rules) may be
        enclosed in a box.  A boxed proof produces conclusions that
        are only valid given the assumptions made inside the box,
        however, the proof demonstrates certain relationships which
        are valid outside the box.  For example, the box below
        labelled "Implication introduction" starts by assuming P,
        which need not be a true {propoistion} so long as it can be
        used to derive Q.
     
        Truth introduction:
     
         -
         T
     
        (Truth is free).
     
        Binary AND introduction:
     
         -----------
         | .  | .  |
         | .  | .  |
         | Q1 | Q2 |
         -----------
           Q1 ^ Q2
     
        (If we can derive both Q1 and Q2 then Q1^Q2 is true).
     
        N-ary AND introduction:
     
         ----------------
         | .  | .. | .  |
         | .  | .. | .  |
         | Q1 | .. | Qn |
         ----------------
          Q1^..^Qi^..^Qn
     
        Other n-ary rules follow the binary versions similarly.
     
        Quantified AND introduction:
     
         ---------
         | x  .  |
         |    .  |
         |  Q[x] |
         ---------
         For all x . Q[x]
     
        (If we can prove Q for arbitrary x then Q is true for all x).
     
        Falsity elimination:
     
         F
         -
         Q
     
        (Falsity opens the floodgates).
     
        OR elimination:
     
           P1 v P2
         -----------
         | P1 | P2 |
         | .  | .  |
         | .  | .  |
         | Q  | Q  |
         -----------
              Q
     
        (Given P1 v P2, if Q follows from both then Q is true).
     
        Exists elimination:
     
         Exists x . P[x]
         -----------
         | x  P[x] |
         |     .   |
         |     .   |
         |     Q   |
         -----------
               Q
     
        (If Q follows from P[x] for arbitrary x and such an x exists
        then Q is true).
     
        OR introduction 1:
     
            P1
         -------
         P1 v P2
     
        (If P1 is true then P1 OR anything is true).
     
        OR introduction 2:
     
            P2
         -------
         P1 v P2
     
        (If P2 is true then anything OR P2 is true).  Similar
        symmetries apply to ^ rules.
     
        Exists introduction:
     
             P[a]
         -------------
         Exists x.P[x]
     
        (If P is true for "a" then it is true for all x).
     
        AND elimination 1:
     
         P1 ^ P2
         -------
            P1
     
        (If P1 and P2 are true then P1 is true).
     
        For all elimination:
     
         For all x . P[x]
         ----------------
               P[a]
     
        (If P is true for all x then it is true for "a").
     
        For all implication introduction:
     
         -----------
         | x  P[x] |
         |     .   |
         |     .   |
         |    Q[x] |
         -----------
         For all x . P[x] -> Q[x]
     
        (If Q follows from P for arbitrary x then Q follows from P for
        all x).
     
        Implication introduction:
     
         -----
         | P |
         | . |
         | . |
         | Q |
         -----
         P -> Q
     
        (If Q follows from P then P implies Q).
     
        NOT introduction:
     
         -----
         | P |
         | . |
         | . |
         | F |
         -----
          / P
     
        (If falsity follows from P then P is false).
     
        NOT-NOT:
     
         //P
         ---
          P
     
        (If it is not the case that P is not true then P is true).
     
        For all implies exists:
     
         P[a]   For all x . P[x] -> Q[x]
         -------------------------------
        	      Q[a]
     
        (If P is true for given "a" and P implies Q for all x then Q
        is true for a).
     
        Implication elimination, modus ponens:
     
         P   P -> Q
         ----------
              Q
     
        (If P and P implies Q then Q).
     
        NOT elimination, contradiction:
     
         P   /P
         ------
           F
     
        (If P is true and P is not true then false is true).
     
        (1995-01-16)