Ramble, Part 57: Digging Deeper
Aug. 31st, 2008 06:00 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowThe symbolic logic devised by George Boole and his successors was both powerful and flexible, a major step forward. Still, it was limited; the logic used by Aristotle and extended by the Scholastics of the Middle Ages was capable of feats beyond the reach of Boolean logic. It was Charles Sanders Pierce, again, who found the way forward, unifying the two kinds of logic.
The syllogistic of Aristotle could draw the conclusion "This is a striped mammal" from the premises "Tigers are striped mammals" and "This is a tiger". To Boolean logic, these statements are opaque; none of them can be constructed from simpler statements by the use of the operators AND, OR, NOT, and IF...THEN.... If there is complexity to these statements, it is of a new kind.
If simple statements are regarded as atoms and complex ones (built with Boolean operators) as molecules, Pierce's work may be viewed as a search for subatomic particles - for pieces smaller than entire statements, which nonetheless can be handled with logical tools. Four types of such smaller pieces were eventually discerned.
First, there are terms. A term can be regarded as a noun phrase - but not just any noun phrase. A term must designate a single definite individual: a specific person, a precise number, a particular concept. It could be a single noun ("stoutfellow", "five", "democracy") or a complex phrase ("the person currently third in line to the British throne", "the positive zero of x5+x-29", "the prohibition of ex post facto laws"). It must not be indefinite ("one of my sisters"), general ("every positive integer"), or even plural ("two members of the Connecticut delegation"). (There's more going on than this, including some apparent exceptions, but let's not get bogged down in details.)
Second, there is the identity relation "=", which indicates that the terms (and they must be terms) on either side designate the same individual: "2+2=4". "2+2" is a term; so is "4", and they are two names for the same individual.
Third, there are predicates. If terms are regarded as noun phrases, predicates are verb phrases - more or less. More precisely, a predicate is a template for statements, which can be completed by the insertion of terms. "x is a tiger" is a one-place predicate: inserting an appropriate term into the x slot produces a statement, which may be true or false. ("Appropriate" is a bit of a weasel-word; I'll discuss that later on.) "x is taller than y" is a two-place predicate; "x gave y to z", a three-place predicate, and so on. (Statements themselves may be regarded as zero-place predicates.)
Finally, there are quantifiers. Ordinary language has many different quantifiers ("a", "exactly one", "three or four", "many", "most", "all", and so on), but mathematical logic makes do with only two, the existential and universal quantifiers. A quantifier can be attached to a predicate, or more specifically to one place of a predicate, producing a new predicate with one less place. The existential quantifier asserts that there is at least one value that can be attached to that place which makes it true; the universal, that every appropriate value, attached to that place, makes it true. Thus, if you consider the one-place predicate "x is a star" (where "star" refers to entertainers), attaching the existential quantifier produces the statement "There is some x which is a star", or more succinctly "Stars exist"; attaching the universal quantifier results in "Everybody is a star". Again, using the two-place predicate "x is taller than y", you may attach an existential quantifer to the x place, producing the one-place predicate "Somebody is taller than y"; attaching the universal quantifier to the y place produces "x is taller than everybody". (Here, again, I'm assuming that the places in this predicate refer to people, for specificity's sake.)
Using these elements, in combination with the operators of Boolean logic, it becomes possible to analyze the syllogism with which we began. "Tigers are striped mammals" can be unpacked as "For all x, IF x is a tiger THEN x is a striped mammal". (Note that this is where Pierce's IF...THEN... shines: this statement says that inserting a value into the x slot will produce a true statement, whether that value is a striped tiger, a non-striped non-tiger, or a striped non-tiger.) "This" is a term (requiring a pointing finger or a gesturing hand to make it specific, but we do that all the time), which can be inserted into either of the predicates "x is a tiger" and "x is a striped mammal", and there are laws of inference, describing what conclusions can be drawn from universal or existential statements, which can be applied to show that this syllogism is a valid one.
I won't go into the details of those laws of inference, any more than I did for Boolean logic. The point is that they can be laid out, and they subsume Boolean logic and traditional syllogistic into a single framework. I do, however, want to look more closely at that word "appropriate", but that will require some more preliminary work, in the next Ramble.
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The syllogistic of Aristotle could draw the conclusion "This is a striped mammal" from the premises "Tigers are striped mammals" and "This is a tiger". To Boolean logic, these statements are opaque; none of them can be constructed from simpler statements by the use of the operators AND, OR, NOT, and IF...THEN.... If there is complexity to these statements, it is of a new kind.
If simple statements are regarded as atoms and complex ones (built with Boolean operators) as molecules, Pierce's work may be viewed as a search for subatomic particles - for pieces smaller than entire statements, which nonetheless can be handled with logical tools. Four types of such smaller pieces were eventually discerned.
First, there are terms. A term can be regarded as a noun phrase - but not just any noun phrase. A term must designate a single definite individual: a specific person, a precise number, a particular concept. It could be a single noun ("stoutfellow", "five", "democracy") or a complex phrase ("the person currently third in line to the British throne", "the positive zero of x5+x-29", "the prohibition of ex post facto laws"). It must not be indefinite ("one of my sisters"), general ("every positive integer"), or even plural ("two members of the Connecticut delegation"). (There's more going on than this, including some apparent exceptions, but let's not get bogged down in details.)
Second, there is the identity relation "=", which indicates that the terms (and they must be terms) on either side designate the same individual: "2+2=4". "2+2" is a term; so is "4", and they are two names for the same individual.
Third, there are predicates. If terms are regarded as noun phrases, predicates are verb phrases - more or less. More precisely, a predicate is a template for statements, which can be completed by the insertion of terms. "x is a tiger" is a one-place predicate: inserting an appropriate term into the x slot produces a statement, which may be true or false. ("Appropriate" is a bit of a weasel-word; I'll discuss that later on.) "x is taller than y" is a two-place predicate; "x gave y to z", a three-place predicate, and so on. (Statements themselves may be regarded as zero-place predicates.)
Finally, there are quantifiers. Ordinary language has many different quantifiers ("a", "exactly one", "three or four", "many", "most", "all", and so on), but mathematical logic makes do with only two, the existential and universal quantifiers. A quantifier can be attached to a predicate, or more specifically to one place of a predicate, producing a new predicate with one less place. The existential quantifier asserts that there is at least one value that can be attached to that place which makes it true; the universal, that every appropriate value, attached to that place, makes it true. Thus, if you consider the one-place predicate "x is a star" (where "star" refers to entertainers), attaching the existential quantifier produces the statement "There is some x which is a star", or more succinctly "Stars exist"; attaching the universal quantifier results in "Everybody is a star". Again, using the two-place predicate "x is taller than y", you may attach an existential quantifer to the x place, producing the one-place predicate "Somebody is taller than y"; attaching the universal quantifier to the y place produces "x is taller than everybody". (Here, again, I'm assuming that the places in this predicate refer to people, for specificity's sake.)
Using these elements, in combination with the operators of Boolean logic, it becomes possible to analyze the syllogism with which we began. "Tigers are striped mammals" can be unpacked as "For all x, IF x is a tiger THEN x is a striped mammal". (Note that this is where Pierce's IF...THEN... shines: this statement says that inserting a value into the x slot will produce a true statement, whether that value is a striped tiger, a non-striped non-tiger, or a striped non-tiger.) "This" is a term (requiring a pointing finger or a gesturing hand to make it specific, but we do that all the time), which can be inserted into either of the predicates "x is a tiger" and "x is a striped mammal", and there are laws of inference, describing what conclusions can be drawn from universal or existential statements, which can be applied to show that this syllogism is a valid one.
I won't go into the details of those laws of inference, any more than I did for Boolean logic. The point is that they can be laid out, and they subsume Boolean logic and traditional syllogistic into a single framework. I do, however, want to look more closely at that word "appropriate", but that will require some more preliminary work, in the next Ramble.
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