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We have described our official notation; however, we shall often use an unofficial notation. For example, we shall often use 'x', 'y', 'z', etc. for variables, while officially we should use 'x1', 'x2', etc. A similar remark applies to predicates, constants, and function letters. We shall also adopt the following unofficial abbreviations:

 

(A ∨ B) for (~A ⊃ B);

 

(A ∧ B) for ~(A ⊃ ~B);

 

(A ≡ B) for ((A ⊃ B) ∧ (B ⊃ A));

 

(∃xi) A for ~(xi) ~A.

We have described our official notation; however, we shall often use an unofficial notation. For example, we shall often use 'x', 'y', 'z', etc. for variables, while officially we should use 'x1', 'x2', etc. A similar remark applies to predicates, constants, and function letters. We shall also adopt the following unofficial abbreviations:

(A ∨ B) for (~A ⊃ B);

(A ∧ B) for ~(A ⊃ ~B);

(A ≡ B) for ((A ⊃ B) ∧ (B ⊃ A));

(∃xi) A for ~(xi) ~A.

Aristotle agrees with modern logic that conjunctions, for example, are not atomic formula. They are compound formulas and as such they consist of subformulas. Really, the introduction of connectives comes from our natural use of words such as "and" and "or". We defined "and" as trying to represent what we logically mean when we say "It is raining AND it is cloudy". What does the "and" mean in that sentence? Well it means that both of those atomic formula, the assertions "it is raining" and "it is cloudy" are true at the same time. Therefore we structured the logical "and" to reflect a common connective we use when dealing with propositions, or assertions, in natural language. Due to the fact that we can construct all of our connectives out of just a negation and one other connective, the choice of which connectives we use is arbitrary.

Aristotle agrees with modern logic that conjunctions, for example, are not atomic formulae. They are compound formulas and as such they consist of subformulas. Really, the introduction of connectives comes from our natural use of words such as "and" and "or". We defined "and" as trying to represent what we logically mean when we say "It is raining AND it is cloudy". What does the "and" mean in that sentence? Well it means that both of those atomic formula, the assertions "it is raining" and "it is cloudy" are true at the same time. Therefore we structured the logical "and" to reflect a common connective we use when dealing with propositions, or assertions, in natural language. Due to the fact that we can construct all of our connectives out of just a negation and one other connective, the choice of which connectives we use is arbitrary.

In this excerpt "⊃" is the symbol for implication which is the symbol he chose to define the logic with, "≡" is the symbol for biconditional, "~" is the symbol for negation and (x) is the universal quantifier. The numbers next to the variables in the beginning as well as the i next to the universal quantifier are supposed to be subscripts, however the philosophy.SE sadly does not allow for mathjax typesetting.

In this excerpt "⊃" is the symbol for implication which is the connective he chose to define the logic with, "≡" is the symbol for biconditional, "~" is the symbol for negation and (x) is the universal quantifier. The numbers next to the variables in the beginning as well as the i next to the universal quantifier are supposed to be subscripts, however the philosophy.SE sadly does not allow for mathjax typesetting.