1 Logic
1.1 Logical vs. Material equivalence:
1.1.1 “Two wffs (statements) are logically equivalent if they
necessarily have the same truth-value. The are materially
equivalent if they have the same truth-value.” (Susan Haack,
philosophy of logics 246).
1.2 Logical truth: “A wff is logically true in L iff it is true in all
interpretations of L.” (Haack, p.248)
1.3 Contrary: “Wffs (statements) A and B are contrary if they
can’t be both true but may be both false.” (Haack 244).
Extension/intension “Reference (extension) versus sense
(intension) of an expression. For a singular term, the
extension is its reference, for a predicate, the set of things it
is true of, for a sentence, its truth-value. Two expressions
with the same extension are co0-extensive. Related
terminology: Dedeutung (- extenson) versues Sinn (intenson) of an expression (Frege); denotation versus
connotation (Mill); extensional versus intentional contexts.
(Haack 246). See the same page for a good definition of
extensional/intensional contexts.
1.3.1 Richard Kirkham: “The extension of an expression is the object
or set of objects referred to, pointed to, or indicated by, the
expression…The extension of a predicate is the set of all
objects to which the predicate truly applies….The attempt to
produce such a noncircular description of the set of all trueh
things (the extension of the prediate ‘is true’) is what I shall call
the extensional project.” (Kirkam, Theories of Truth, p. 4)
1.3.2 “Imagine someone in the process of learning English. He has
learned the meaningof, and hence the extension of, ‘vertebrate
with a liver’, but he does not yet know the meaing of ‘vertebrate
with a heart’. By informing him tha the extension of ‘vertebrate
with a heart’ is identical to the extension of ‘vertebrate with a
liver’, we have effectively specified for him the extension of the
former. (We shall no, however, have completely explained the
meaning of ‘verbrate with a heart’ until we have accounted for
the other component of its meaning, its intension.) [This
demonstrates that meaning has two components, sense and
reference].” Kirkham, 5.
1.3.3 “Put in a very general way, the intension of an expression is the
informational content of the expression, as distinct from the set
of objects deonted by the expression…the intension of ‘the
morning star’ can be worded as ‘the star visible around dawn’,
while the intension of the ‘the evening star’ is ‘the star visible
around sunset’. But being visible around dawn is not the same
things as being visible around sunset, so whle ‘the morning
star’ and ‘the evning star’ are extensionally equivalent, they are
not intensionally equivalent. They convey different information.
And since intension is one component of meaning, the two
expressions do not mean the same thing, despite the fact that
they both anme the same planet. (9) …¶ One implication of the
intensional equivalence of x and y is that we can substitute y
for x in any English sentence in which the latter appears
without changing the meaning or, hence, the truth value, of the
sentence. This principle does not hold good in cases of mere
extensional equivalence. [Are ‘heart’ and ‘blood-pumping
organ’ intensionally equivalent. I can see one would say no,
since the latter focuses the mind on more fine-grained
properties of the heart. Certainly, they are extensionally
equivalent expressions. Modes of representation may be in
play here as well. How is ‘heart’ presenting itself to me, as a
blood-pumping organ? No, just a 3-dimensional shape that is
red in color.] (Kirkham, Theories of Truth)
1.3.4 George Bealer “Fine-grained type-free intensionality”
1.3.4.1 (178) “An item x is an extensional entity if fit satisfied all
relevant principles of extensionality. Principles of extensionality
come in two forms. First, let ‘R’ be any copula (‘is’, ‘is an
instance of’, ‘is related by’, ‘is a member of’ ‘is a part of’, ‘has’,
‘falls under’, etc.) and let the item x be in the same ontological
category as some item in the range of R. Then for all y in the
same ontological category as x: (z)(zRx ↔ zRy)  x = y.
Second, let ‘T’ be any verifier (‘is true’, ‘is so’, ‘obtains’, ‘holds’,
‘happens’, ‘occurs’, etc.) and let the iterm x be in the same
category as some item that is an instance of T. Then for all y in
the same cateogory as x: (Tx ↔ Ty)  x = y. With this
definition of extensional entity we may define intensional entity:
an item x is intensional if it is not extensional. That is, an item x
is intensional if it fails to satisfy one or more of these
extensionality principles.
1.4 See p. 247 for definition of material vs. strict implication.
1.5 Oratio Obliqua “Indirect (reported) speech, as ‘s said that p’
(Haack 249).
1.6 Port Royal Grammar/Logic
1.6.1 Chomsky, Language and Mind, (16):
1.6.1.1 It seems that one of the innovations of the Port-Royal
Grammarof 1660—the work that initiated the tradition of
philosophical grammar—was its recognition of the importance
of the notion of the phrase as a grammatical unit. Earlier
grammar had been largely a grammar of word classes and
inflections. In the Cartersian theory of Port-Royal, a phrase
corresponds to a complex idea and a sentence is subdivided into
consectutive phrases, which are further subdivided into phrases,
and so on, until the level of the word is reached. In this way we
derived what might be called the “surface structure” of the
sentence in question. To use what became a standard example,
the sentences “Invisible God created the visible world” contains
the subject “invisible God” and the predicate “created the visible
world,” the latter contains the complex idea “the visible world”
and verb “created,” and so on. But it is interesting that although
the Port-Royal Grammar is apparently the first to rely oin a
fairly systematic way on analysis into surface structure, it also
recognized the inadequacy of such analysis. According to the
Port-Royal theory, surface structure correspons only to sound—
to the corporeal aspect of language; but when the signal is
produced, with its surface structure, there takes place a
corresponding mental analysis into what we may call the deep
structure, a formal structure that related directly not to the
sound but t o the meaning. In the example just given, “Invisible
God created the visibile world,” the deep (17) structure consists
of a system of three propositions, “that God is invisible,” “that
he created the world,” “that the world is visible.” The
propositions that interrelate to form the deep structure are not,
of course, asserted when the sentence is used to make a
statement; if I say that a wise man is honest, I am not asserting
that men are wise or honest, even though in the Port-Royal
theory the propositions “a man is wise” and “a man is honest”
enter into the deep structure. Rather, these propositions enter
into the complex ideas that are present to the mind, though
rarely articulated in the signal, when the sentence is uttered.”
(Chomsky, Language and Mind)
1.7 Intensional Logic
1.7.1 Bealer and Mönnich, “Property Theories”.
1.7.1.1 (138) “It appears, therefore, that failure of substitutivity is
indeed the best criterion to use in characterizing intensional
logic.”
1.7.2 Bealer “The Logical Status of Mind”
1.7.2.1 (234) ¶ “there have been two prevalent conceptions of
intensional entities in the history of logic and philosophy. On the
first conception, intensional entities are taken to be identical if
and only if they are necessarily equivalent. … ¶ The second
conception of intesniaonal entities imposes far stricter conditions
on what it takes for a definition to be correct. On this
conception, when an intensional entity is dfined completely, the
result is unique and noncircular. … ¶ The first conception
underlies the currently popular possible-worlds treatment of
intensional entites, and it also underlies Alonzo ?church’s
“Alternaitve (2)” formulation of Frege’s theory of sense. This
conception is parituclarly well suited to the treatment of the
modalities (Necessity, possibility, etc.), probability statements,
counterfactuals, and so forth. The second conception has a
rather livelier history. It underlies Leibniz’s doctrine of simple
and complex properties and russell’s doctrine of logical atomism.
And when intensional entities are identified with ideas (concepts,
thoughts), we see that this conception was adopted at least
implicityly by neary all modern philosophers from Descartes and
Locke thorugh Kant. … (237) ¶ … Though the concept of being
a triangle and the concept of being a trilateral are distinct, they
correspond to the same quality of things in the world. The
reason is that qualities, unlike concepts, are identical if there is
no possibility of an object haiving one of them but not the other.
Qualities (and connections) are what fix the actual conditions in
the world, and they do not exhibit distinctions finer than
necessary equivalents. Concpets, on the other hand, pertain to
thinking and to reason; it is here that finer intensional
distinctions show up. Thus, we arrive at the accompanying
picture of the types of intensional entities.