Vtyjkyuk

You are loading the positioning library, but not using it. Once one changes the syntax, it works out fine. In more detail, e.g. above of=Sig has to become in positioning syntax above=<distance> of Sig. I acknowledge a comment by Fato39 urging me to do that. This is the only change I did. There are by now some more sophisticated proposals here, among which IMHO Max' matrix approach may be the most convenient, closely followed by Zarko's chains, and of course no one has as nice colors as J. Leon V. Nox had the very first answer, and it is always easier to write something better after one sees what others did.

 documentclass[a4paper]amsart
 usepackagetikz
 usetikzlibraryshapes.geometric
 usetikzlibrarycalc
 usetikzlibrarypositioning
 % -----------------------------
 begindocument
 begintikzpicture[square/.style=regular polygon,regular polygon sides=4]
 % 1 alpha FF Sig
 node at (0.2,0.2) [square, draw, anchor=south west, minimum size=2cm, fill=blue!50] (Sig) verb!Sig!;
 node at (1,0.5) [draw=blue!50] tinyfrom sign; % text inside Sig
 node [square, draw, minimum size=2cm, align=left, fill=blue!50, above=0pt of Sig] (Cau) verb!Cau!;
 node [square, draw, minimum size=2cm, fill=blue!50, above=0pt of Cau] (Ef) verb!Ef!;
 node [square, draw, minimum size=2cm, fill=blue!50, above=0pt of Ef](Cor)verb!Cor!;
 % 1 alpha VF
 node [square, draw, minimum size=2cm, fill=green!50, right=0pt of Sig,](Cr)verb!Cr!;
 % 1 alpha VV
 node [square, draw, minimum size=2cm, fill=yellow!50, right=0pt of Cr](St)verb!St!;
 node [square, draw, minimum size=2cm, fill=yellow!50, above=0pt of St](Ax)verb!Ax!;
 % 1 alpha PV
 node [square, draw, minimum size=2cm, fill=purple!50, right=0pt of St](Pr)verb!Pr!;
 node [square, draw, minimum size=2cm, fill=orange!50, right=0pt of Pr](De)verb!De!;
 % Axes
 draw [dotted,thick] (-7,0) -- (7,0);
 draw [dotted,thick] (0,-7) -- (0,7);
 % The grid -- comment when ready
 draw[step=.5cm,gray,very thin] (-7,-7) grid (7,7);
 endtikzpicture
 enddocument

Further improvements:

1. Corrected the width by taking into account thr outer sep. (This is not really necessary.)


2. Used labels to place the small texts. Specifically, defined styles label A (above) and label B (below) for that.


3. Simplified the other styles to avoid unnecessary repetition.


4. Set the node distance to -pgflinewidth in order to have the same thickness of the border everywhere, and also to spare you from typing 0pt.

Here is the improved code.

 documentclass[a4paper]amsart
 usepackagetikz
 usetikzlibraryshapes.geometric
 usetikzlibrarycalc
 usetikzlibrarypositioning
 % -----------------------------
 begindocument
 begintikzpicture[square/.style=minimum
 size=2cm-2*pgfkeysvalueof/pgf/outer xsep,
 draw,regular polygon,regular polygon sides=4,
 label B/.style=label=[anchor=north west,font=tinysffamily,yshift=16pt,
 xshift=3pt,scale=0.75,align=left]below left:#1,
 label A/.style=label=[anchor=north,font=tinysffamily,yshift=-2pt,scale=0.75]above:#1,
 blue square/.style=square,fill=blue!50,label A=1 pre FF,
 yellow square/.style=square,fill=yellow!50,label A=1 pre W,
 orange square/.style=square,fill=orange!50,label A=1 pre PV,
 node distance=-pgflinewidth
 ]
 % 1 alpha FF Sig
 coordinate (O) at (0,0);
 node [above right= of O,blue square,label B=from sign] (Sig) verb!Sig!;
 node [blue square,above= of Sig,label B=from cause] (Cau) verb!Cau!;
 node [blue square,above= of Cau,label B=from effect] (Ef) verb!Ef!;
 node [blue square,above= of Ef,label B=from correlation](Cor)verb!Cor!;
 % 1 alpha VF
 node [square,fill=green!50,right= of Sig,label A=1 pre VF,label B=from
 criterion](Cr)verb!Cr!;
 % 1 alpha VV
 node [yellow square,right= of Cr,label B=from standard](St)verb!St!;
 node [yellow square,above= of St,label B=axiolic\ argument](Ax)verb!Ax!;
 % 1 alpha PV
 node [square,fill=purple!50,right= of St,label B=pragmatic\ argument,
 label A=pre PF](Pr)verb!Pr!;
 node [orange square,right= of Pr,label B=deontic\ argument](De)verb!De!;
 node [orange square,above= of De,label B=from evaluation](Ev)verb!Ev!;
 % Axes
 draw [dotted,thick] (-7,0) -- (7,0);
 draw [dotted,thick] (0,-7) -- (0,7);
 % The grid -- comment when ready
 draw[step=.5cm,gray,very thin] (-7,-7) grid (7,7);
 endtikzpicture
 enddocument

Altough you already have two very usable answers, I wanted to show an alternative that is (arguably) more readable and easier to adjust to future changes. Using Tikz matrix of nodes, you can assign styles to certain columns, which is very convenient in such a table. I added a sub=<text>, and super=<text> key that take care of the extra text in the cells.

The code that describes the contents of the table is a lot less cluttered by positioning settings, and only is

|[sub=from correlation]|Cor & & & & \
|[sub=from effect]| Ef & & & & \
|[sub=from cause]| Cau & & |[sub=axiologic argument]| Ax & & |[sub=from evaluation]| Ev \
|[sub=from sign]| Sig & |[sub=from criterion]| Cr & |[sub=from standard]| St & |[sub=pragmatic argument]| Pr & |[sub=deontic argument]| De \

Which results in:

Note that you could overwrite the additional text above the cells with the super= key, for every cell. So the default value is the one specified for the column number, but if for some reason you want a different value for a specific cell, you can simply overwrite it.

Complete MWE:

documentclass[tikz,margin=2mm]standalone

usetikzlibrarymatrix

begindocument
 begintikzpicture
 matrix[
 matrix of nodes,
 column sep=-pgflinewidth,
 row sep=-pgflinewidth,
 nodes=
 minimum size=2cm,
 draw,
 anchor=center,
 fill,
 text width=1.8cm,
 inner sep=1mm,
 font=ttfamilyLARGE,
 align=justify,
 ,
 sub/.style=execute at end node=bgroup\[-0.65baselineskip]tiny #1egroup,
 super/.style=execute at begin node=bgroupcolorblackfootnotesize #1 \[-0.15baselineskip]egroup,
 column 1/.style=fill=blue!50, super=1 Pre FF,
 column 2/.style=fill=green!50, super=1 Pre VF,
 column 3/.style=fill=yellow!50,super=1 Pre VV,
 column 4/.style=fill=purple!50,super=1 Pre PF,
 column 5/.style=fill=orange!50,super=1 Pre PV,
 ][sub=deontic argument];
 endtikzpicture
enddocument

Edit

I changed the styles slightly to incorporate the solution for the sub and sup texts by using labels. Credits to marmot and zarko (I'm not sure who was first). I have not finished all arguments, and it's quite late here so there could be some spelling errors in there. One more edit: I added a style for the delta quadrant. I trust you will be able to finish it:

documentclass[tikz,margin=2mm]standalone

usetikzlibrarymatrix
tikzset
 sup/.style=label=[minimum size=0,font=scriptsize,anchor=north west,shift=(-10mm-0.5pgflinewidth,-1.5mm)]#1,
 sub/.style=label=[minimum size=0,font=tiny,anchor=north west,shift=(-10mm-0.5pgflinewidth,-12mm)]#1,
 co1/.style=fill=blue!50,
 co2/.style=fill=green!50,
 co3/.style=fill=yellow!50,
 co4/.style=fill=purple!50,
 co5/.style=fill=orange!50,
 co6/.style=fill=red!50,
 my empty cell/.style=minimum size=2cm,fill=none,draw=none,
 table of arguments/.style=
 matrix of nodes,
 column sep=-pgflinewidth,
 row sep=-pgflinewidth,
 inner sep=1mm,
 nodes=
 minimum size=2cm,
 draw,
 anchor=center,
 fill,
 text width=1.8cm,
 font=ttfamilyLARGE,
 align=justify,
 ,
 ,
 alpha quadrant/.style=
 table of arguments,
 name=alpha,
 matrix anchor=south west,
 co1/.append style=sup=1 Pre FF,
 co2/.append style=sup=1 Pre VF,
 co3/.append style=sup=1 Pre VV,
 co4/.append style=sup=1 Pre PF,
 co5/.append style=sup=1 Pre PV,
 ,
 beta quadrant/.style=
 table of arguments,
 name=beta,
 matrix anchor=south east,
 co1/.append style=sup=1 Sub FF,
 co3/.append style=sup=1 Sub VV,
 co4/.append style=sup=1 Sub PF,
 co6/.append style=sup=1 Sub PP,
 ,
 gamma quadrant/.style=
 table of arguments,
 name=gamma,
 matrix anchor=north east,
 co2/.append style=sup=2 Sub VF,
 co3/.append style=sup=2 Sub VV,
 co4/.append style=sup=2 Sub PF,
 ,
 delta quadrant/.style=
 table of arguments,
 name=delta,
 matrix anchor=north west,
 co2/.append style=sup=2 pre VF,
 co3/.append style=sup=2 pre VV,
 co4/.append style=sup=2 pre PF,
 co5/.append style=sup=2 pre PV,
 co6/.append style=sup=2 pre PP,
 ,


begindocument
 begintikzpicture
 matrix[alpha quadrant][co3,sub=from standard];
 matrix[beta quadrant][co3,sub=from analogy];
 matrix[gamma quadrant][co4,sub=from consistency];
 matrix[delta quadrant][co2,sub=from authority];
 draw[dashed] (0,8) -- (0,-8)
 (-12,0) -- (12,0);
 endtikzpicture
enddocument

This is a little hacky solution, but it should work, if you create the table from left to right. I corrected the offset of the lines manually and created a lastcolor command that colors the nodeaboves correctly.

documentclass[a4paper]amsart
usepackagetikz
usetikzlibrarypositioning

newcommandcont[3]%
 parbox[t][1.5cm]1.5cm%
 tiny #2\[.4cm]%
 normalsize texttt#1\[.1cm]%
 fontsize35selectfont #3%


newcommandlastcolor
newcommandlastbasenode
newcommandlastnode
newcommandlastsup

newcommandbasenode[4]%
 renewcommandlastcolor#4
 renewcommandlastbasenode#1
 renewcommandlastnode#1
 renewcommandlastsup#2
 node (rect) at (0.2cm,0.2cm) [draw,
 thick,
 minimum width=2cm,
 minimum height=2cm,
 anchor=south west,
 fill=lastcolor,
 xshift=-.4pt,
 yshift=-.4pt] (#1) cont#1#2#3;


newcommandnodeabove[2]%
 node (rect) [draw,
 thick,
 minimum width=2cm,
 minimum height=2cm,
 anchor=south west,
 above=of lastnode.center,
 fill=lastcolor,
 yshift=-.4pt] (#1) cont#1lastsup#2;
 renewcommandlastnode#1


newcommandnoderight[4]%
 renewcommandlastcolor#4
 renewcommandlastsup#2
 node (rect) [draw,
 thick,
 minimum width=2cm,
 minimum height=2cm,
 anchor=south west,
 right=of lastbasenode.center,
 fill=lastcolor,
 xshift=-.4pt] (#1) cont#1lastsup#3;
 renewcommandlastbasenode#1
 renewcommandlastnode#1



% -----------------------------
begindocument
begintikzpicture
% 1 alpha FF Sig

basenode Sig1 pre FFfrom signblue!30
nodeaboveCau from cause
nodeaboveEf from effect
%
noderightCr1 pre VFfrom criteriongreen!30
%
noderightSt1 pre VVfrom standardyellow!50
nodeaboveEf axiologic\argument
%
noderightPr1 pre PFpragmatic\argumentpurple!20

draw [dotted,thick] (-7,0) -- (7,0);
draw [dotted,thick] (0,-7) -- (0,7);
% The grid -- comment when ready
draw[step=.5cm,gray,very thin] (-7,-7) grid (7,7);
endtikzpicture
enddocument

edit: Cleaner code.

edit (1):
ups, i didn't load last solution. now corrected ...

oh, one more answer: exploiting library chains and common definition of all square nodes:

documentclass[tikz, margin=3mm]standalone
usetikzlibrarychains,
 positioning

begindocument
 begintikzpicture[
node distance = 0pt,
 start chain = going above,
square/.style args = #1/#2/#3%
 rectangle, draw, semithick,
 fill=#1,
 minimum size=20mm, inner sep=2mm, outer sep=0mm,
 font=largesffamily, 
 label=[anchor=north]above:#2,
 label=[anchor=south,yshift=0.5ex]below:#3,
 on chain,
every label/.append style = %
 label distance=0pt, text depth=0.25ex, align=center,
 font=footnotesizesffamilylinespread0.84selectfont
 ]
% 1. column, from bottom to top
node (Sig) [square=blue!30/1 pre FF/from\ sign] Sig;
node (Cau) [square=blue!30/1 pre FF/from\ cause] Cau;
node (Ef) [square=blue!30/1 pre FF/from\ effect] Ef;
node (Cor) [square=blue!30/1 pre FF/from\ 
 correlation] Cor;
% 2. column, from bottom to top
node (Cr) [square=olive!30/1 pre VF/from\ criterion,
 right=of Sig] Cr;
% 3. column, from bottom to top
node (St) [square=yellow!30/1 pre W/from\ standard,
 right=of Cr] St;
node (Ax) [square=yellow!30/1 pre W/axiologic\ 
 argument] Ax;
% 4. column, from bottom to top
node (Pr) [square=purple!30/1 pre PF/pragmatic\ argument,
 right=of St] Pr;
% 5. column, from bottom to top
node (De) [square=orange!30/1 pre PV/deontic\ argument,
 right=of Pr] De;
node (Ev) [square=orange!30/1 pre PV/from\ 
 evaluation] Ev;
% Axes
draw [dotted,thick] ([xshift=-1mm] Cor.north west) |- ([yshift=-1mm] De.south east);
endtikzpicture
enddocument

edit (2):
similarly can be drawn "squares" in other quadrants. in this the above code should be devided into two parts:

• first is in scope where chains of node going above (start chain = going above). it is used in the first and second quadrant)


• first is in scope where chains of node going below (start chain = going below). it is used in the first and second quadrant)

a not complete example of this approach is:

 documentclass[tikz, margin=3mm]standalone
 usetikzlibrarychains,
 positioning
 % -----------------------------
 begindocument
 begintikzpicture[
 node distance = 0pt,
 square/.style args = #1/#2/#3%
 rectangle, draw, semithick,
 fill=#1,
 minimum size=20mm, inner sep=2mm, outer sep=0mm,
 font=largesffamily,
 label=[anchor=north]above:#2,
 label=[anchor=south,yshift=0.5ex]below:#3,
 on chain,
 every label/.append style = %
 label distance=0pt, text depth=0.25ex, align=center,
 font=footnotesizesffamilylinespread0.84selectfont
 ]
 beginscope[start chain = going above]
 % RIGHT (first quadrant)
 % 1. column, from bottom to top
 node (Sig) [square=blue!30/1 pre FF/from\ sign] Sig;
 node (Cau) [square=blue!30/1 pre FF/from\ cause] Cau;
 node (Ef) [square=blue!30/1 pre FF/from\ effect] Ef;
 node (Cor) [square=blue!30/1 pre FF/from\
 correlation] Cor;
 % 2. column, from bottom to top
 node (Cr) [square=olive!30/1 pre VF/from\ criterion,
 right=of Sig] Cr;
 % 3. column, from bottom to top
 node (St) [square=yellow!30/1 pre W/from\ standard,
 right=of Cr] St;
 node (Ax) [square=yellow!30/1 pre W/axiologic\
 argument] Ax;
 % 4. column, from bottom to top
 node (Pr) [square=purple!30/1 pre PF/pragmatic\ argument,
 right=of St] Pr;
 % 5. column, from bottom to top
 node (De) [square=orange!30/1 pre PV/deontic\ argument,
 right=of Pr] De;
 node (Ev) [square=orange!30/1 pre PV/from\
 evaluation] Ev;
 % LEFT (second quadrant)
 % 1. column, from bottom to top
 node (Sim) [square=blue!30/1 pre FF/from\ sign,
 left= 2mm of Sig] Sig;
 node (G) [square=blue!30/1 pre FF/from\ cause] G;
 node (Exa) [square=blue!30/1 pre FF/from\ effect] Cas;
 node (Cas) [square=blue!30/1 pre FF/from\
 correlation] Cor;

 endscope
 beginscope[start chain = going below]
 % 2. column, from top to bottom
 node (Au) [square=olive!30/2 pre VF/from\ authority,
 below=2mm of Cr] Cr;
 node (Po) [square=olive!30/2 pre VF/ad\
 xxxxx] Po;
 node (Comm)[square=olive!30/2 pre VF/from\
 environment] Comm;
 % 3. column, from bottom to top
 node (U) [square=yellow!30/3 pre W/from\ yyyyy,
 right=of Au] St;
 node (Be) [square=yellow!30/3 pre W/from\
 zzzzz] Be;
 endscope
 % Axes
 draw [dotted,thick]
 ([xshift=-1mm] Cor.north west) -- ([xshift=-1mm] Cor.west |- Comm.south);
 draw [dotted,thick]
 ([yshift=-1mm] Sim.south west) -- ([yshift=-1mm] De.south east);
 endtikzpicture
 enddocument

hopefully is now clear how to how to add missing squares. also you need to check the contents of squares.

An option using a drawing structure definition to reuse a drawing with better control of positions of the text nodes inside the box, and avoid to repeating writing all the commands for each box; then helped with the positioning library you could put the next box next to other by certain distance, in all cases 0, but you can complete all the drawing with less lines of code jus defining the distance below, above, right or left from another box and the combination like:

Arg[Box_Name][Box_desc][Box_type](below right=12pt and 0 of Another_box_name)(color)

RESULT:

MWE:

documentclass[tikz,border=15pt]standalone
usepackage[scaled]helvet
usetikzlibrary[positioning]
begindocument
 begintikzpicture[
 font=sffamily
 ]
 %Defining a drawing structure
 defArg[#1][#2][#3](#4)(#5)%1:Node name and label 2:description 3:type 4:position relative using positioning 5:color
 node[
 line width=1pt,
 draw,
 #4,
 fill=#5,
 rectangle,
 inner sep=0,
 outer sep=0,
 minimum size=2.5cm
 ](#1);
 draw[every node/.append style=anchor=west]
 (#1.center)++(-1,0) node LARGEbfseries #1
 (#1.center)++(-1,0.9) node scriptsizebfseries #3
 (#1.center)++(-1,-0.3) node[anchor= north west,align=left,font=scriptsizebfseriessffamily] #2;
 

 %Start drawing the thing...
 definecolormycyanHTMLA5C4E4
 definecolormygreenHTMLB2D096
 definecolormyyellowHTMLF9DA79
 definecolormypurpleHTMLB39BCD
 definecolormyorangeHTMLEBB58B

 Arg[Cor][from correlation][1 pre FF]()(mycyan)
 Arg[Ef][from effect][1 pre FF](below=0 of Cor)(mycyan)
 Arg[Cau][from cause][1 pre FF](below=0 of Ef)(mycyan)
 Arg[Sig][from sign][1 pre FF](below=0 of Cau)(mycyan)
 Arg[Cr][from criterion][1 pre VF](right=0 of Sig)(mygreen)
 Arg[St][from standard][1 pre W](right=0 of Cr)(myyellow)
 Arg[Ax][from axiologic \ argument][1 pre W](above=0 of St)(myyellow)
 Arg[Pr][from pragmatic \ argument][1 pre PF](right=0 of St)(mypurple)
 Arg[De][from deontic \ argument][1 pre PV](right=0 of Pr)(myorange)
 Arg[Ev][from evaluation \ argument][1 pre PV](above=0 of De)(myorange)
 %Fron the other quadrant-
 Arg[Em][from emotion][2 pre PP](below right=12pt and 0 of De)(myorange!70!red)

 draw[dash pattern=on 5.5pt off 3pt,ultra thick,black!60]
 (De.south east) ++(2.5cm,-6pt) coordinate (axisX)
 (Cor.north west) ++ (-6pt,0) coordinate (axisY)
 (axisX -| axisY) edge (axisX) edge (axisY); % Using edges to obtain corner with dash line on.

 endtikzpicture
enddocument