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LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy)
The continuation of text review:
Key symbols
≡df = Equivalence by definition
: = Equal (s)
ε = Epsilon and means is
⊃ = Is the same as
⊨ is Entails
˜ = Not
∃ = There exists
∃! = There exists
∴ = Therefore
· = Therefore
< = Is included
v = a logical inclusive disjunction (disjunction is the relationship between two distinct alternatives). x = variable
· = Conjunction meaning And 0
= Null class
cls = Class
int = Interpretation
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Philosopher Langer continues that logic does not bond for any fact. (189). It stands for the conceptual possibility of a system. (189). In other words, a valid system is not necessarily presenting truth, but it is logical. The valid system with premises and conclusions is logically consistent. It represents not factual certainty, but instead logical certainty or validity. (189).
Langer demonstrates the following as logical:
Napoleon discovered America
Napoleon died before 1500 A.D. (189).
Conclusion
America was discovered before 1500 A.D. (189).
These two premises imply that America was discovered before 1500 and Langer opines that a third proposition that would be derived (a conclusion, my add) would also be logical and valid. (189).
Indeed the first two premises are historically false. (189). They are still logically consistent, while the consequent is true that America was discovered before 1500 A.D. (189).
Langer then states:
Columbus discovered America (Note this text was written in 1953 and revised in 1967 with older historical data and assumptions)
Columbus died after 1490
Therefore
America was discovered after 1490 (189).
The author explains that the two premises were true and the conclusion were true. (189-190). But because she used the term therefore, this is an untrue and unsound assertion. (190).
The term therefore makes the argument not sound (true) and the conclusion is logically, false and invalid.
I do not think this is adequately explained enough in the textbook. She explains that therefore expresses the only untrue assertion. (190). I deduce the term is presenting a certainty which is philosophically false. The term therefore, should have been excluded.
A system built on false premises may be as logically valid as one based on facts. (190). False propositions may imply true conclusions. (190).
The only case that is not logical is true premises that imply a false conclusion. (190). As I have documented in archives:
Therefore a valid deductive argument can have
False premises and a true conclusion (s) (FT)
False premises and a false conclusion (s) (FF)
True premises and a true conclusion (s) (TT)
However
True premises and a false conclusion (s) (TF) is invalid.
True premises and valid logical relations will lead to true conclusions, but the truth of premises cannot be established by logic, states philosopher, Langer. (190). This would be (TT). Validity is the only concern, as there could be (FT), (FF) premises and conclusions which were also logically valid.
CONWAY DAVID A. AND RONALD MUNSON (1997) The Elements of Reasoning, Wadsworth Publishing Company, New York.
LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York.
