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Re: [ontolog-forum] Two

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: "John F. Sowa" <sowa@xxxxxxxxxxx>
Date: Thu, 21 Jun 2007 21:53:34 -0400
Message-id: <467B2B9E.3030301@xxxxxxxxxxx>
Chris,    (01)

The short answer is no:    (02)

CP> Minor historical point, pre 1870 wasn't Aristotelian Logic
 > accepted as some final word on the matter.    (03)

Some people who weren't familiar with the history of logic,
such as Immanuel Kant, erroneously thought so.    (04)

But the Stoic logicians had a version of propositional
logic (expressed in stylized Greek) in which they had
introduced what is now called material implication.    (05)

The so-called De Morgan laws were familiar to the medieval
scholastics, and Leibniz had proved the following theorem,
which he called the Praeclarum Theorema (Splendid Theorem):    (06)

    ((p -> r) & (q -> s)) -> ((p & q) -> (r & s))    (07)

Boole's first book on the "Laws of Thought" was in 1847, and
quite a few people such as De Morgan, Hamilton, and Peirce were
extending Boolean algebra in the 1860s.  The most important of
the early extensions was Peirce's relational algebra of 1870,
which went beyond monadic relations.   Peirce and others had
experimented with rudimentary kinds of quantifiers then, but
full-blown FOL wasn't invented until Frege 1879 and Peirce's
papers of 1880 and 1885.    (08)

And Ockham had written an early model-theoretic foundation for
Latin in his _Summa Logicae_ of 1323 (excerpts below).    (09)

________________________________________________________________    (010)

Ockham showed how to determine the truth value of compound
propositions in terms of the truth values of their components
and to determine the validity of rules of inference (_regulae
generales consequentiarum_) in terms of the truth of their
antecedents and consequents.  The following quotations are from
his _Summa Logicae_:    (011)

    "We must posit certain rules which are common to the signs
    'every', 'any', 'each', and others like them, if there are
    any others.  These rules are also common to many propositions
    which are equivalent to hypothetical propositions, e.g.
    'Every man is an animal', 'Every white thing is running', etc....
    It should be noted that for the truth of such a universal
    proposition it is not required that the subject and the predicate
    be in reality the same thing.  Rather, it is required that the
    predicate supposit for all those things that the subject
    supposits for, so that it is truly predicated of them."    (012)

    "it should be noted that when the sign 'all' is taken in the plural,
    it can have either a collective or a distributive sense...
    For example, by means of 'All the apostles of God are twelve'...
    if 'all' is understood collectively, then it is not asserted that
    the predicate agrees with each thing of which the subject 'apostles'
    is truly predicated.  Rather, it is asserted that the predicate
    belongs to all the things -- taken at once -- of which the subject
    is truly predicated.  Hence, it is asserted that these apostles,
    referring to all the apostles, are twelve."    (013)

    "A conjunctive proposition is one which is composed of two or more
    categoricals joined by the conjunction 'and' or by some particle
    equivalent to such a conjunction.  For example, this is a conjunctive
    proposition:  'Socrates is running and Plato is debating'....  Now
    for the truth of a conjunctive proposition, it is required that both
    parts be true.  Therefore, if any part of a conjunctive proposition
    is false, then the conjunctive proposition itself is false."    (014)

    "A disjunctive proposition is one which is composed of two or more
    categoricals joined by the disjunction 'or' or by some equivalent.
    For example, this is a disjunctive proposition:  'You are a man or
    a donkey.'  Likewise, this is a disjunctive proposition:  'You are
    a man or Socrates is debating.'  Now for the truth of a disjunctive
    proposition, it is required that some part be true....  It should
    be noted that the contradictory opposite of a disjunctive proposition
    is a conjunctive proposition composed of the contradictories of the
    parts of the disjunctive proposition."  [Note that this is Ockham's
    version of DeMorgan's law.    (015)

    "From truth, falsity never follows.  Therefore, when the antecedent
    is true and the consequent is false, the inference is not valid."    (016)

    "From a false proposition, a true proposition may follow.  Hence
    this inference does not hold:  'The antecedent is false; therefore,
    the consequent is false.'  But the following inference holds: 'The
    consequent is false; therefore, so is the antecedent.'"    (017)

Although Ockham's version was not as formal as Tarski's, he covered
much more ground, since he also analyzed the truth conditions for
temporal, modal, and causal statements.    (018)

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