John F. Sowa
Before Peirce and Tarski, the most detailed development
of the model-theoretic approach was by William of Ockham.
Peirce had studied Ockham’s Summa Logicae in depth,
he owned a copy of the first printed edition (1488), and he had
lecture on Ockham’s logic at Harvard in 1869.
Unfortunately, Coffa (1991) was unaware of the contributions
by Ockham and Peirce when he wrote his history of semantics.
To see the continuity, compare
the following quotations from Ockham, Bolzano, Peirce, and Tarski:
- Ockham (1323) showed how to
determine the truth value of compound propositions in terms of the
truth or falsity of their components and to determine the validity
of rules of inference (regulae generales consequentiarum)
in terms of the truth of their antecendents and consequents.
Following are three quotations from Part II and two from Part III
of the Summa Logicae, which Peirce had studied in detail:
- "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."
- "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."
- "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 that the negation of
p∨q is (~p)∧(~q)].
- "From truth, falsity never follows. Therefore, when the
antecedent is true and the consequent is false, the inference
is not valid."
- "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.'"
- Bolzano (1837):
"If we now state that
M, N, O,... are deducible from
A, B, C,... and this in respect of the notions
we are basically saying... the following:
`All ideal contents which in the place of
i, j,... in the propositions
A, B, C,...
M, N, O,... simultaneously verify the propositions
A, B, C,...
have the property of also simultaneously verifying the propositions
M, N, O,...'" [Quoted by Bocheński (1956)]
- Peirce (1869):
"All that the formal logician has to say is, that if
facts capable of expression in such and such forms of words are true,
another fact whose expression is related in a certain way to the
expression of these others is also true.... The proposition `If A,
then B' may conveniently be regarded as equivalent to `Every case
of the truth of A is a case of the truth of B.'".
- Tarski (1936):
"In terms of these concepts [of model],
we can define the concept of logical consequence
as follows: The sentence X follows logically from the sentences
of the class K if and only if every model of the class K is also
a model of the class X."
In addition to the quantifiers and Boolean operators,
Ockham also analyzed truth conditions for tenses and plurals.
(Page numbers refer to the translation by Fredoso and Schuurman.)
- 'A boy will be an old man' is true, and yet 'A boy is an old man'
will never be true. Rather, 'This is an old man' will be true
referring to that person who is now a boy. [p. 106]
- ... 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. [pp. 101-102]
Ockham, William of (1323) Summa Logicae,
Paris: Johannes Higman, 1488; the edition owned by C. S. Peirce.
Also volume 1 of Opera Philosophica, ed. by P. Boehner,
G. Gál, & S. Brown,
St. Bonaventure, NY: Franciscan Institute, 1974.
Ockham, William of (T)
Ockham’s Theory of Terms,
translated by Paul Vincent Spade.
Ockham, William of (T)
Ockham’s Theory of Terms,
translation of Part I of Ockham (1323) by M. J. Loux,
University of Notre Dame Press, Notre Dame, IN, 1974.
Ockham, William of (P) Ockham’s Theory of Propositions,
translation of Part II of Ockham (1323) by A. J. Freddoso &
H. Schuurman, Notre Dame, IN: University of Notre Dame Press, 1980.
Ockham, William of (W) Philosophical Writings, ed. and translated
by Philotheus Boehner, revised by S. F. Brown,
Hackett Publishing Co., Indianapolis. Includes Latin and English
selections from Ockham (1323) and other works.
Spade, Paul Vincent,
Writings about and translations of medieval logic and philosophy.
For other references see the
combined bibliography for this web site.