Zeroth-Order

Zeroth-Order_Logic 2007-04-07T20:05:41Z Proteus fmt boiler   [[zh:零阶逻辑]] Other Languages : ([http://zh.wikipedia.org/wiki/%E9%9B%B6%E9%98%B6%E9%80%BB%E8%BE%91 中文 : 零阶逻辑]) '''''Zeroth order logic''''' is a term in popular use among practitioners for the subject matter otherwise known as [[boolean function]]s, [[monadic logic|monadic predicate logic]], [[propositional calculus]], or sentential calculus. One of the advantages of this terminology is that it institutes a higher level of abstraction in which the more inessential differences between these various subjects can be subsumed under the pertinent [[isomorphism]]s. By way of initial orientation, Table 1 lists equivalent expressions for the sixteen functions of concrete type ''X'' × ''Y'' → '''B''' and abstract type '''B''' × '''B''' → '''B''' in a number of different languages for zeroth order logic. {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%" |+ '''Table 1. Propositional Forms on Two Variables''' |- style="background:paleturquoise" ! style="width:15%" | L1 ! style="width:15%" | L2 ! style="width:15%" | L3 ! style="width:15%" | L4 ! style="width:15%" | L5 ! style="width:15%" | L6 |- style="background:paleturquoise" | | align="right" | x : | 1 1 0 0 | | | |- style="background:paleturquoise" | | align="right" | y : | 1 0 1 0 | | | |- | f0 || f0000 || 0 0 0 0 || ( ) || false || 0 |- | f1 || f0001 || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y |- | f2 || f0010 || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y |- | f3 || f0011 || 0 0 1 1 || (x) || not x || ¬x |- | f4 || f0100 || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y |- | f5 || f0101 || 0 1 0 1 || (y) || not y || ¬y |- | f6 || f0110 || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y |- | f7 || f0111 || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y |- | f8 || f1000 || 1 0 0 0 || x y || x and y || x ∧ y |- | f9 || f1001 || 1 0 0 1 || ((x, y)) || x equal to y || x = y |- | f10 || f1010 || 1 0 1 0 || y || y || y |- | f11 || f1011 || 1 0 1 1 || (x (y)) || not x without y || x → y |- | f12 || f1100 || 1 1 0 0 || x || x || x |- | f13 || f1101 || 1 1 0 1 || ((x) y) || not y without x || x ← y |- | f14 || f1110 || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y |- | f15 || f1111 || 1 1 1 1 || (( )) || true || 1 |} These six languages for the sixteen boolean functions are conveniently described in the following order: * Language '''L3''' describes each boolean function ''f'' : '''B'''2 → '''B''' by means of the sequence of four boolean values (''f''(1,1), ''f''(1,0), ''f''(0,1), ''f''(0,0)). Such a sequence, perhaps in another order, and perhaps with the logical values ''F'' and ''T'' instead of the boolean values 0 and 1, respectively, would normally be displayed as a column in a [[truth table]]. * Language '''L2''' lists the sixteen functions in the form '''fi''', where the index '''i''' is a [[bit string]] formed from the sequence of boolean values in '''L3'''. * Language '''L1''' notates the boolean functions '''fi''' with an index '''i''' that is the decimal equivalent of the binary numeral index in '''L2'''. * Language '''L4''' expresses the sixteen functions in terms of logical [[conjunction]], indicated by concatenating function names or proposition expressions in the manner of products, plus the family of ''[[minimal negation operator]]s'', the first few of which are given in the following variant notations: <math>\begin{matrix} (\ ) & = & 0 & = & \mbox{false} \\ (x) & = & \tilde{x} & = & x' \\ (x, y) & = & \tilde{x}y \lor x\tilde{y} & = & x'y \lor xy' \\ (x, y, z) & = & \tilde{x}yz \lor x\tilde{y}z \lor xy\tilde{z} & = & x'yz \lor xy'z \lor xyz' \end{matrix}</math> It may also be noted that <math>(x, y)\!</math> is the same function as <math>x + y\!</math> and <math>x \ne y</math>, and that the inclusive disjunctions indicated for <math>(x, y)\!</math> and for <math>(x, y, z)\!</math> may be replaced with exclusive disjunctions without affecting the meaning, because the terms disjoined are already disjoint. However, the function <math>(x, y, z)\!</math> is not the same thing as the function <math>x + y + z\!</math>. * Language '''L5''' lists ordinary language expressions for the sixteen functions. Many other paraphrases are possible, but these afford a sample of the simplest equivalents. * Language '''L6''' expresses the sixteen functions in one of several notations that are commonly used in formal logic. ==See also== ===Logical operators=== {| |+   | valign=top | * [[Exclusive disjunction]] * [[Logical conjunction]] * [[Logical disjunction]] * [[Logical equality]] | valign=top | * [[Logical implication]] * [[Logical NAND]] * [[Logical NNOR]] * [[Negation]] |} ===Related topics=== {| |+   | valign=top | * [[Ampheck]] * [[Boolean algebra]] * [[Boolean algebra topics]] * [[Boolean domain]] * [[Boolean function]] * [[Boolean logic]] | valign=top | * [[Boolean-valued function]] * [[Entitative graph]] * [[Existential graph]] * [[First order logic]] * [[Indicator function]] * [[Logical graph]] | valign=top | * [[Logical value]] * [[Minimal negation operator]] * [[Monadic predicate calculus]] * [[Operation (mathematics)|Operation]] * [[Propositional calculus]] * [[Truth table]] |}  ''Some content adapted from the [[Wikinfo]] article "[http://getwiki.net/url.php?wikinfo=Zeroth_order_logic Zeroth order logic]" under the [[GNU Free Documentation License]].''