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Slight edits on April 7 2023 for an entry on academia.edu
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
---
From back in April, 2019 (please see archives for my previous work and reviews):
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.
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.
January 13 2020
Langer mentions that the text shows that for every proposition there is also an analogous one (221). If there is an entity that when multiplied with any term, leaves that term unchanged, then there is also one that can be added to that term without altering it (221).
Langer's theorem:
-(a + b) = -a x -b (221)
She notes that this theorem is the complement of a + b. (221).
The complement is the amount added to something to make it whole. Each entity needs to complete the other is a universe of discourse. (143).
Langer writes (paraphrased) that she is not explicitly explaining her argument here. (222). She states in regards to breaking down the theorems...
'But this is left to the brave and ambitious reader.' (223).
But in philosophical terms, for the sake of logic, her theorems represent the law of duality. (223). The law of duality between + and x. (223). The theorems which explain the relation between sums (+) and products (x) express this law of duality. (223). The relation between addition and multiplication.
Webster
Definition of algebraic sum : the aggregate of two or more numbers or quantities taken with regard to their signs (as + or −) according to the rules of addition in algebra the algebraic sum of −2, 8, and −1 is 5
Study.com
What Is a Product? When speaking mathematically, the term product means the answer to a multiplication problem. For example: 5 * 3 = 15
15 is the product The term product first showed up in England in the 1400s and comes from the Latin word productum, which means 'to produce.'
Philosophical relevance?
Philosopher (and Mathematician) Langer opines that everything she noted about sums is also true of products, (224). I am a philosopher and not a mathematician, but seems to me, she is demonstrating the logic and consistency of symbolic logic within algebra, mathematics and philosophy.
Overall, I reason, symbolic logic has minimal practical use, even within most philosophy. But I appreciate that Langer demonstrates the consistency of logic, and as well that the logical is not necessarily true. But, in my embraced philosophy and theology, the truth is always logical. In other words, the truth always can be made sense of with reasonable premises and conclusions.
l = Logic
t = Truth
l ˜ = t
(Logic does not equal truth, strictly philosophically speaking)
l < t
(Logic is included in truth)
(l < t) ˜ ⊨ (l = t)
(Logic is included in truth, does not entail logic equals truth)
Research and study within four academic degrees and years of academic website writing has shown me that I have read and researched many logically presented premises and conclusions, meaning, many logical arguments, that are not likely true.
BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press.
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
---
From back in April, 2019 (please see archives for my previous work and reviews):
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.
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.
January 13 2020
Langer mentions that the text shows that for every proposition there is also an analogous one (221). If there is an entity that when multiplied with any term, leaves that term unchanged, then there is also one that can be added to that term without altering it (221).
Langer's theorem:
-(a + b) = -a x -b (221)
She notes that this theorem is the complement of a + b. (221).
The complement is the amount added to something to make it whole. Each entity needs to complete the other is a universe of discourse. (143).
Langer writes (paraphrased) that she is not explicitly explaining her argument here. (222). She states in regards to breaking down the theorems...
'But this is left to the brave and ambitious reader.' (223).
But in philosophical terms, for the sake of logic, her theorems represent the law of duality. (223). The law of duality between + and x. (223). The theorems which explain the relation between sums (+) and products (x) express this law of duality. (223). The relation between addition and multiplication.
Webster
Definition of algebraic sum : the aggregate of two or more numbers or quantities taken with regard to their signs (as + or −) according to the rules of addition in algebra the algebraic sum of −2, 8, and −1 is 5
Study.com
What Is a Product? When speaking mathematically, the term product means the answer to a multiplication problem. For example: 5 * 3 = 15
15 is the product The term product first showed up in England in the 1400s and comes from the Latin word productum, which means 'to produce.'
Philosophical relevance?
Philosopher (and Mathematician) Langer opines that everything she noted about sums is also true of products, (224). I am a philosopher and not a mathematician, but seems to me, she is demonstrating the logic and consistency of symbolic logic within algebra, mathematics and philosophy.
Overall, I reason, symbolic logic has minimal practical use, even within most philosophy. But I appreciate that Langer demonstrates the consistency of logic, and as well that the logical is not necessarily true. But, in my embraced philosophy and theology, the truth is always logical. In other words, the truth always can be made sense of with reasonable premises and conclusions.
l = Logic
t = Truth
l ˜ = t
(Logic does not equal truth, strictly philosophically speaking)
l < t
(Logic is included in truth)
(l < t) ˜ ⊨ (l = t)
(Logic is included in truth, does not entail logic equals truth)
Research and study within four academic degrees and years of academic website writing has shown me that I have read and researched many logically presented premises and conclusions, meaning, many logical arguments, that are not likely true.
BLACKBURN, SIMON (1996) Oxford Dictionary of Philosophy, Oxford, Oxford University Press.
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.
PIRIE, MADSEN (2006)(2015) How To Win Every Argument, Bloomsbury, London.
Langer's theorem:
quote
-(a + b) = -a x -b (221)
She notes that this theorem is the complement of a + b. (221).
Further
-a + (a + b) = 1 (221)
-1 + (1 + 1 = 2) = 1 (My add)
-a x ab = 0
-1 x a-b (1-1 = 0) = 0 (With assist from 221)
There is a law of absorption:
a absorbs any sum of itself and any term multiplied with any other product. (217).
a x (a + b) means the common part of a and (a + b) which is just a. (217). a + (a x b) means the class of a or a x b is also just a. (217-218). Not just b. Same with c, d, e, f, g, h, i, etcetera.
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