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[The following is summary. My commentary is in brackets.]
Introduction to Logic
Ch. 3 Symbolizing Everyday Language
A term is an expression that either (a) names or describes some object, or (b) generates a name or a description for some object whenever we replace the expression’s variables with names or with descriptions. Thus “x + y” both contains terms, namely, x and y, because these variables can be replaced with specific numerals, and also, the whole expression “x + y” itself is a term, because after its variables are substituted, it expresses the value 5.
Suppes will define “term” first by considering an important kind of term, namely, a variable. A variable is often a letter, sometimes with a subscript. In everyday language, they are similar to pronouns and common nouns. So the sentence “"
Everything is either red or not red
can be written with variables as
For every x, x is red or x is not red.
Here, the variable ‘x’ corresponds to the use of “thing” in the prior sentence (44). And we could have used any letter instead of x (44).
Proper nouns are often written as letters from the beginning of the alphabet. So let a = Isaac Newton. Now take this sentence:
Isaac Newton is the greatest mathematician of the last three centuries.
We can translate it to:
a is the greatest mathematician of the last three centuries.
We then make this abbreviation:
b = the greatest mathematician of the last three centuries
. And we now have:
a is b
And since “is” means identity, we can even write it as
a = b
Names and descriptions of objects are called constants, and thus the a and b above are constants (45).
Suppes then defines “term”.
DEFINITION. A TERM is an expression which either names or describes some object, or results in a name or description of an object when the variables in the expression are replaced by names or descriptions.
[I might have the next idea wrong. The idea seems to be the following. 3 is a term, because it names some object. ‘x’ is a term, because it is a variable that can be replaced by the name ‘3’. But the whole expression ‘x + y’ is a term, because, as the definition says, “when the variables in the expression are replaced by names or descriptions” it “results in a name or description of an object”. So we can replace x with ‘2’ and y with ‘3’ to get ‘2+3’, which designates 5. It could also be that the complex expression ‘2+3’ is not itself a term, but just contains them. Let me quote:]
Thus ‘x’ is a term according to the definition because when we replace ‘x’ by the Arabic numeral ‘3’ we obtain an expression which designates the number three. The definition also classifies as terms such expressions as ‘x + y’, for when we replace ‘x’ by ‘2’ and ‘y’ by ‘3’, we obtain ‘2 + 3’, an expression which designates (i.e., names) the number 5. Some other examples of terms are:
x + 3
x2 + y2 - 1
the fattest man in township x.
Suppes, Patrick. Introduction to Logic. New York: Van Nostrand Reinhold / Litton Educational, 1957.