From:
Weisstein, Eric W. "Implies."
From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/Implies.html
Weisstein writes:
""Implies" is the connective in propositional
calculus which has the meaning "if p, q" In
formal terminology, the term "conditional" is often
used to refer to this connective
(Mendelson 1997, p. 13)."
"The symbol used to denote "implies" is [)]"
(Carnap 1958, p. 8; Mendelson 1997, p. 13), or >."
"The Mathematica command Experimental`ImpliesRealQ[ineqs1, ineqs2]
can be used to determine if the system of real algebraic equations
and inequalities ineqs1 implies the system of real
algebraic equations and inequalities ineqs2."
"In classical logic, p)q is an abbreviation ~pvq, where ~ denotes NOT
and v denotes OR (though this is not the case, for example, in intuitionistic logic)."
vide Dummett.
") is a binary operator that is implemented in Mathematica as
Implies[A, B], and can not be extended to more than
two arguments."
" ) has the following truth table (Carnap 1958, p. 10; Mendelson 1997, p. 13).
T T T
T F F
F T T
F F T
If p ) q and q ) p, then p and q are said to be equivalent, a relationship which is written symbolically as p )( q (Carnap 1958, p. 8)."
SEE ALSO: Connective, Equivalent, Exists, For All, Quantifier
REFERENCES:
Carnap, R. Introduction to Symbolic Logic and Its Applications.
New York: Dover, p.8, 1958.
Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, 1997.
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Wolfram and Weisstein (I refer to them as "Wolfram") as well as Wikipedia are quite valuable resources because they tend to give rather short summaries in ordinary simple English, although sometimes not as simplified as I would do them (of course, one can always boast about such things).
ReplyDeleteAn interesting thing arguably happens when one considers x and y as being continuous truth values of specific propositions between 0 and 1 (including fractions, decimals, square roots, and so on). To avoid confusing them with x and y of my probability posts, I will use a and b for them as "truth-valued propositions". Then truth tables no longer reflect quantities like ( or in alternative notation -->. In Multivalued Logics (ML), we define:
1) (a-->b) = 1 + b - a, where b < = a (Lukaciewicz/Rational Pavelka ML)
2) (a-->b) = b/a where b < = a (Product/Goguen ML)
3) (a-->b) = b where b < = a (Godel ML)
The respectively probabilistic analogs are:
4) P(A-->B) = 1 + P(AB) - P(A)
5) P(B given A) = P(AB)/P(A) for P(A) not 0.
6) P(Godel A-->B) = P(B)
Curiously, in (4), we can DEFINE:
7) (A-->B) = (AB ' ) ' = A ' U B
where A, B are (random) sets/events, A ' is the "complement" of A or the part of the Universe outside A, and U is set/event "union" which is "A and/or B or both". Then taking the probability of (A-->B) and using the standard ("mainstream") laws of probability on (7) yields (4) by actual proof rather than by only definition, with both giving the same results.
Osher Doctorow
I should also mention a second type of probabilistic analog of the P(A-->B) type, namely:
ReplyDelete1) P ' (A-->B) = (definition) 1 + P(B) - P(A) with P(B) < = P(A).
The conditions under which P ' reduces to P or equals P are quite fascinating. They are basically when P(AB) = P(B). Recall in set/event theory that AB = B if and only if B is a subset of A. With random/probabilistic sets/events, P(AB) = P(B) if and only if AB = B except possibly for sets of probability 0, that is to say there may be sets of probability 0 on which AB does not equal B. By the way, this is not true for P(A) = P(B) in general - there is no requirement that A = B except for sets of probability 0 in that case, and A and B can be very different types of set/events.
I have extended this type of definition to conditional probability as well by:
2) P ' (B given A) = (definition) P(B)/P(A), meaning P(B) divided by P(A) if P(A) is not 0.
This is totally unexplored in "mainstream" conditional probability, another example of the danger of focusing on only one theory or sub-theory when small differences in assumptions yield viable alternatives.
Osher Doctorow
Thanks for sharing. Glad to learn of non-mainstream conditional probability, too!
ReplyDeleteIn another source on 'implies' (as used by logicians), an author was saying, "of course, 'implies' can only confuse you. Just stick to 'if'". Will see if I can find the link.
...
Let me see...
This is Dr. Peterson at:
http://mathforum.org/library/drmath/view/72083.html
"In the symbolic logic I am
familiar with, what is commonly
read as "implies" (p ) q) is not,
really, an 'implication.' It should be
read, simply, as "if p, q", and is taken to be true when the truth values of p and q are
such that they do not contradict the claim that q is true whenever p is true. It is important NOT to read into it any
claim of a cause-and-effect connection, or anything of that sort."
-- which IS a good point, but makes you wonder why Grice is still talking of 'logical implication' and entailment in "Presupposition and Conversational Implicature" and of course gives him all the credit for having 'coined' "implicate" to do the duty for 'implies'.
---
I am of the opinion (and I suspect Quine was) that ANY formal arguments which depend on induction and probability are, well, not really formal arguments. So,you start with true premises, and determine whether the conclusion follows (ie is valid). Or at least they are assumed to be true for a certain type of...conversational or informal logic (even say legalistic reasoning, Sherlock Holmes stories, social "science," economics, etc). That holds for supposed "causal" conditionals as well. While people use -> to indicate cause, that's merely a colloquial habit, a type of generalization (ie "splotches on face-> chicken pox"). That sort of generalization might be necessary if not crucial at times (ie a department of medicine would seem to suggest as much...) --but it's not really "logic", any more than the "all swans are white" problems...
ReplyDeleteThere may be a type of inductive logic (ie, Toulminian if you will) but it has little or nothing to do with formal arguments, ie first order logic, but is more like...social science, sampling, data-collection, so forth.
Good points. We may analyse here in a special post, considering induction VERSUS probability and induction versus deduction.
ReplyDeleteI recently received a nice formal treatment of continuous logic in ProofPower from Rob Arthan (who sent it to me because it illustrated some point not specific to continuous logic) and this is the first time I have paid any attention to it. The formal treatment is nice but I don't know what merit the logic has (above being sound!).
ReplyDeleteOf course, inductive inferences (of the kind one may be supposed to need in empirical science, as opposed to mathematical induction) are not deductively sound, but it is nevertheless ("of course" shall I say? I hate that phrase but sometimes can't stop myself) possible to reason soundly (and formally) _about_ probabilities (its just a branch of mathematics and can be formalised in a system like Principia). Also, to be really picky, a formal system does not _have_ to be sound, it does not in that case cease to be a formal system, and some logicians have argued that the inconsistency of a logical system does not necessarily entail its uselessness (though I think he was inhabiting the "propositions as types" alternate universe).
Rob's little paper will eventually (I expect) appear among the "examples" of applications of ProofPower at lemma-one.com, but hasn't arrived there yet.
RBJ
Good. Glad Jones support the use of "probability" in something like a formal system, which "of course" (Note to Roger: If you don't like, as I don't like it either, the expression, 'of course' try to soften it by the introduction of the iota-operator, and say, "off THE course"), or off the course, need not be 'sound'.
ReplyDeleteSound is perhaps one of the most homophonic words in the English language. There is this song I sang for St. Patrick's day,
"There's a depth in my soul never sounded."
----
There is Long Island Sound, which is NOT 'He's a sound boy'.
The etymology of 'sound' in "sound boy" (if you find one) is pretty difficult, and I copy it from an online source:
"sound" -- "uninjured," Old English gesund "sound, safe, healthy," from Proto-Germanic *sundas, from root *swen-to- (cf. Old Saxon gisund, Old Frisian sund, Dutch gezond, Old High German gisunt, German gesund "healthy," source of the post-sneezing interjection gesundheit; also Old English swið "strong," Gothic swinþs "strong," German geschwind "fast, quick"), with connections in Indo-Iranian and Balto-Slavic. Meaning "financially solid or safe" is attested from 1601; of sleep, "undisturbed," from 1548. Sense of "holding accepted opinions" is from 1526. Soundly "completely" is attested from 1577."
Presumably the origin of this use of "sound" is in the way in which one tests wood for rot. You tap it and listen, and if it sounds sound you are OK.
ReplyDeleteIn my youth I had a vac job with "The Coal Board" (now deceased) which included visiting pits to test whether the main axle on the pit head was still sound. This we did with ultrasonics.
RBJ
Should have to confirm that, but it has provoked another etymythological reduction in me, thanks!
ReplyDelete