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General Intelligence :
Term logic, also known as traditional logic, has been in decline since the advent of modern predicate logic proposed by Frege, Russell, and others, in 1880-1910. Recently it is revived by Fred Sommers and his student Englebretsen in an enhanced form known as Term Functor Logic (TFL).
The following summary is highly condensed from [Sommers & Englebretsen 2000], An Invitation to Formal Reasoning. The original book, as the title suggests, is much easier to read.
Propositions start with lowercase letters, eg p,q. They are similar to modern propositional logic:
English |
Propositional Logic |
Term Logic |
| not p | ¬p | -p |
| p and q | p^q | p+q |
| p or q | pvq | pvq |
| if p then q | p→q | p⊃q |
Terms start with uppercase letters, eg X,Y. All terms are either positive or negative. Complex terms are pairs of connected terms.
+ and - are functors that connect terms. The + sign may be omitted when it is obvious (as in arithmatic).
Examples of simple terms:
wise / unwise: +W / -W
happy / unhappy: +H / -H
massive / massless: +M / -M
Connective functors are pairs of + and -, viz, +...+, +...-, -...+, -...-. The first sign indicate the quantity: + for "some" or "at least one", - for "every" or "all". The second sign is the copula ("connector") meaning "is", "are", "was", "isn't", "arn't", etc.
Singular terms, such as "Romeo", are marked with an * after. They denote individuals, and are indifferently + or -, denoted ±.
Parentheses are used to group pairs of connected terms.
Examples:
English |
Term Logic |
Algebraic |
Predicate Logic |
| Socrates is wise. | ±Socrates* + Wise | ±S*+W | W(s) |
| Some philosophers are wise. | +Philosopher + Wise | +P+W | ∃x P(x)^W(x) |
| Romeo loves Juliet. | ±Romeo* +(Loves ±Juliet*) | ±R*+(L±J*) | L(r,j) |
| It is cold and it is wet. | + it_is_cold + it_is_wet | +c+w | c^w |
Statements can be prefixed by a sign of negative judgement, eg, "not a creature was stirring" means "not: some creature was stirring": -(+C+S)
More examples:
English |
Term Logic |
| every X is Y | -X+Y |
| some X is Y | +X+Y |
| some X is non-Y | +X+(-Y) |
| not an X is a Y | -(+X+Y) |
| no X is Y | -(+X+Y) |
| not every X is non-Y | -(-X+(-Y)) |
There is no distinction between nouns and adjectives. For example in "some girls are rich" (+G+R) the first term is a noun and the second term is an adjective; but they are both "terms". A term represents a class of things with a certain property. So, "some girls are rich" (+G+R) is the same as "some rich-things are girls" (+R+G).
Law of commutation:
+X+Y = +Y+X
Law of obversion:
-(+/-X +/-Y) = +(-/+X -/+Y)
The general condition for 2 statements to be equivalent is that they are algebraically equal and covalent. The first means that all the corresponding terms have equal signs. The second means that the statements have equal valence: existential statements ("some") have positive valence; universal statements ("every") has negative valence.
There are a total of 32 (25) possible simple statements:
±(±(±X)±(±Y))
yes/no: some/every X/nonX is/isn't Y/nonY
Compound terms are formed by joining 2 terms by "and" or "or". They are enclosed by <>, eg <+G+S> means "both gentleman and scholar".
Law of association:
+X + <+Y+Z> = +<+X+Y> + Z
eg "some farmer is a gentleman and scholar" = "some farmer and gentleman is a scholar"
The way for doing propositional logic with multiple statements (eg ^, v, ¬) is similar, but the notation follows the +...+ functor style which is somewhat eccentric:
p+q means "p and q"
-p+q means "if p then q"
-(-p)-(-q) means "p or q"
The book explains in detail why the notation is like this.
Notice that p-q is not "p or q", instead it means "p andn't q" similar to "X isn't Y".
Compount statements can be formed by enclosing simple statements with [ ], eg "if Tim is not lying then Tim is crazy" = -[T*-L]+[T*+C]
Again, there are a total of 32 (25) possible ways to combine 2 statements:
±(±(±p)±(±q))
yes/no: both/if p/¬p { and/andn't, then/thenn't } q/¬q
A relation involves 2 or more terms, for example "John envies Mary" in which envy is a diadic relation. In "John kicks Mary the ball" kick is a triadic relation.
"Some girls like coffee" would be +G1+L12+C2 where the subscripts denote how terms are paired with each other. Here "like" is a diadic relation.
"The sailor gives every child a toy" would be S*1+G123-C2+T3 where "gives" is a triadic relation.
{ More examples... }
The rest of Sommers & Englebretsen's book talks about inference in 3 forms: syllogistic, propositional, and those involving relations.
{ More about inference in term logic... }
[Sommers & Englebretsen 2000] An Invitation to Formal Reasoning — The Logic of Terms, Ashgate publishing, UK
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20/Oct/2006 (C) GIRG