The principles of logical proof

LANGER, SUSANNE K (1953)(1967) An Introduction to Symbolic Logic, Dover Publications, New York. (Philosophy).

The review continues... Me learning symbolic logic continues:

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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March 14, 2019

Langer explains that the propositions using tautology will use no exponents. (215). In other words, in multiplication, there will not be a smaller exponent number present, to the right of the base number. (215). This is in the context of multiplication.

Therefore, z x z = z, and z2, z3 and related, etcetera in not used. (My example, based on Langer (1953)(1967: 215). In a similar way with addition 2z cannot be arrived at with z + z = z. With addition, 23, 34 etcetera is not arrived at. (My example, based on Langer (1953)(1967: 215).

Summary

Cited

'A calculus is any system wherein we may calculate from some given properties of our elements to others not explicitly stated.' (235).

Calculus is expressed in symbols in general terms and their relations in general it is in algebra. (236). The classes provided through general propositions is genuine algebra. (236).

The principles of logical proof...

Importantly, philosopher Langer explains that there is no guarantee that there is truth in a logical system. (189). Logic does not necessarily promote a fact, rather 'it stands for the conceptual possibility of a system'. (189). Logic documents with the deduction of premises. It stands for 'the consistency of all propositions'. (189). It is standing for logical validity. (189), not factual certainty or truth. (189). This is standard from philosophy, logic, texts. Certainly not something Langer or I manufactured as original.

In many cases when a person states that a premise or argument is logical, the person means that it is true. But a premise or argument can be logical and false. Therefore, it would be more accurate in many cases to claim that a premise or argument is true and or reasonable.

Stanford Encyclopedia of Philosophy

Cited

On standard views, logic has as one of its goals to characterize (and give us practical means to tell apart) a peculiar set of truths, the logical truths...

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).

Also logical, but a true premise:

n= Napoleon
d= Discover
a= America

n ˜ (d+a)

Napoleon did not discover America.