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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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Previously
Langer explains that a proposition can only be known via another proposition. (183). Implication is a relation that only holds among propositions. (183). Propositions are regarded as postulates. (185). A postulate needs to belong to the system, in the language of that system. A postulate should imply further propositions of that system. A postulate should not contradict any other accepted postulate, or any other proposition implied by another postulate. (185).
In other words, symbolic logic requires non-contradiction within its system in a universe of discourse.
Requirements
Coherence: Every proposition in the system must cohere to the established conceptual structure. (185). It must be in coherence with the rest.
Contributiveness : A postulate should contribute and have implication. (185-186).
Consistency: Most important states Langer (186). Two contradictory propositions (or premises) cannot contradict each other in a system. (186). The inconsistent is logically impossible. It is a fatal condition. (186). It is not logic at all. (186).
Independence: Postulates should be independent from each other. (186). If a proposition is deductible from a postulate already provided, then it is a theorem, a necessary fact, not another assumption. (186). Something provable in a theorem would be error to include as a postulate. (186). I would reason that within philosophy there would be plenty of debate on what is a proposition/premise within systems and what would be a theorem. Langer explains that when a theorem needs elucidation, any proposition implied by another proposition as granted and proved within a system is a theorem. (186-187).
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The Truth of the Proposition
Philosopher Langer writes that in the book, so far, nothing had been mentioned in regard to the truth of a proposition. (188). An implied proposition is true if all the premises are true. (188). The implied proposition could also be defined as the conclusion. If the premises are false, she opines that the proposition may or may not be true. (188).
There can be false premises and a true conclusion for a valid argument, but there cannot be true premise (s) and false conclusion with validity.
Validity is a set of premises supporting a conclusion. Technically in logic the premises do not have to be true, simply valid. Elements (1997: 33).
Therefore a valid deductive argument can have
False premises and a true conclusion (FT)
False premises and a false conclusion (FF)
True premises and a true conclusion (TT)
However
True premises and a false conclusion (TF) is invalid.
Valid arguments with all true premises are called sound arguments. These include a true conclusion.
Langer explains
Brutus killed Caesar ⊃ Caesar is dead. (188). (⊃ is means the same as).
Since the implied premise is true the proposition is also true (consequent). (188).
If
Brutus killed Caesar ˜ ⊃ Caesar is dead (my equation using not the same), this would not change the implication that Caesar was dead. (188). Brutus did not kill Caesar; Caesar died in another way.
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.
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