Critical Thinking for Transformative Justice

M) and "is mortal" (here D): the
sentence is given by the judgement
A(M,D). In predicate logic, the
sentence involves the same two nonlogical
concepts, here analyzed as
and
, and the sentence is
given by ,
involving the logical connectives for
universal quantification and
implication.
• But equally, the modern view is
more powerful. Medieval logicians
recognized the problem of multiple
generality, where Aristotelian logic
is unable to satisfactorily render such
sentences as "Some guys have all the
luck", because both quantities "all"
and "some" may be relevant in an
inference, but the fixed scheme that
Aristotle used allows only one to
govern the inference. Just as linguists
recognize recursive structure in
natural languages, it appears that
logic needs recursive structure.
Deductive and inductive reasoning,
and abductive inference
Deductive reasoning concerns what follows
necessarily from given premises (if a, then
b). However, inductive reasoning—the
process of deriving a reliable generalization
from observations—has sometimes been
included in the study of logic. Similarly, it is
important to distinguish deductive validity
and inductive validity (called "cogency").
An inference is deductively valid if and only
if there is no possible situation in which all
the premises are true but the conclusion
false. An inductive argument can be neither
valid nor invalid; its premises give only
some degree of probability, but not
certainty, to its conclusion.
The notion of deductive validity can be
rigorously stated for systems of formal logic
in terms of the well-understood notions of
semantics. Inductive validity on the other
hand requires us to define a reliable
generalization of
some set of
observations. The
task of providing
this definition may
be approached in
various ways, some
less formal than
others; some of
these definitions
may use mathematical models of
probability. For the most part this discussion
of logic deals only with deductive logic.
Abduction is a form of logical inference that
goes from observation to a hypothesis that
accounts for the reliable data (observation)
and seeks to explain relevant evidence. The
American philosopher Charles Sanders
Peirce (1839–1914) first introduced the term
as "guessing". Peirce said that to abduce a
hypothetical explanation from an observed
surprising circumstance is to surmise that
may be true because then would be a
matter of course. Thus, to abduce from
involves determining that is sufficient (or
nearly sufficient), but not necessary, for .
Consistency, validity, soundness, and
completeness
Among the important properties that logical
systems can have:
• Consistency, which means that no
theorem of the system contradicts
another.
• Validity, which means that the
system's rules of proof never allow a
false inference from true premises. A
logical system has the property of
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