Degrees of justification

A degree of justification (DOJ) quantifies how justified a truth-value assignment is in light of a debate. A DOJ always lies in the interval [0, 1] – and can be treated as a probability in the sense that it fulfils the Kolmogorov axioms ([Betz2012], Theorem 6).

See also

The concept was introduced to the theory of dialectical structures in [Betz2012].

taupy.analysis.doj.doj(pos, debate=None, conditional=None)

Returns the degree of justification for the position in pos relative to a debate, as defined in [Betz2012]. If debate is None, the debate stored in the Position object is used.

The conditional doj is returned if conditional is given another position of the same debate. When conditional is set, debate must be None.

Tip

taupy.doj() returns a fraction. If you want to use integers instead, call float(doj()).

Unconditional DOJs

Let $P$ be a (partial) position in the debate $\tau$ and $\mathrm{\Gamma }(\tau )$ the space of coherent and complete positions on $\tau$. Then, the degree of justification of $P$ in $\tau$ is defined as follows.

$$\text{doj}(P{)}_{\tau }:=\frac{\left|\{\gamma \in {\mathrm{\Gamma }}_{\tau }|P\subseteq \gamma \}\right|}{|{\mathrm{\Gamma }}_{\tau }|}$$

The DOJ of a position $P$ in a debate is equal to the proportion of positions in the debate’s SCCP that extend $P$, or have its truth-value assignments as a part. One can also understand this in terms of probability: if all complete and coherent positions in a debate $\tau$ were equally likely of being drawn, then how likely would the set of propositions $P$ be true according the drawn position?

from taupy import Argument, Debate, doj
from sympy.abc import a, b, c
# returns 3/7
doj({c: False}, debate=Debate(Argument(a&b, c)))

Note

The DOJ of an incoherent position always equals zero. The DOJ of a complete position equals $1/|{\mathrm{\Gamma }}_{\tau }|$, since there is exactly one item in the SCCP that extents that position – and this is the position itself. This means that DOJs are most informative for coherent partial positions – particularly for truth-value assignments of single sentences.

Conditional DOJs

We can not only ask the question of how well a position is justified given a debate simpliciter, but also how well it would be justified if some statements in the debate were taken for granted. Let $C$ be a set of propositions relative to which the justification of $P$ should be evaluated.

$$\text{doj}(P|C{)}_{\tau }:=\frac{|\{\gamma \in {\mathrm{\Gamma }}_{\tau }\cap C|P\subseteq \gamma \}|}{|\{\gamma \in {\mathrm{\Gamma }}_{\tau }\cap C\}|}$$

In taupy, a conditional DOJ is calculated with the conditional argument:

from taupy import Argument, Debate, Position, doj
from sympy.abc import a, b, c
pos1 = Position(Debate(Argument(a&b, c)), {a: True})
pos2 = Position(Debate(Argument(a&b, c)), {c: True})
# What is the degree of justification for pos1, conditional to pos2?
# Returns 1/2
doj(pos1, conditional=pos2)