The subject line is a reference to N.D. Belnap (Jr.), 'Tonk Plonk and
Plink' repr. in Sir Peter F. Strawson (ed.) Philosophical Logic, Oxford
University Press, Oxford -- Oxford readings in philosophy, ed. by Oxford's
Vice-Chancelor (and Grice's friend), Sir Geoffrey Warnock.
In a message dated 12/2/2015 8:55:40 A.M. Eastern Standard Time,
donalmcevoyuk@xxxxxxxxxxx quotes from Wikipedia
"If an argument form is valid, the conjunction of the premises will be
logically equivalent to the conclusion and this can be clearly seen in a truth
table; it is displayed. The concept of TAUTOLOGY is thus central to
Wittgenstein's Tractarian account of logical consequence, which is strictly
deductive"
and notes:
"This notion of valid argument may be attacked,"
Indeed. Validly, I hope (Geary uses 'attack' literally, "on the field,"
as he puts it).
"as may the notion of logical equivalence and how these relate to
deduction."
Indeed. For one, a philosopher (who craves for generality, like both
Witters and Grice did) needs a notion of valid reasoning that applies to BOTH
deduction and the rest of types of reasoning -- inductive, abductive, what
have you.
McEvoy:
"There may be valid deductive arguments where the conclusion will not be
logically equivalent to the conjunction of the premises. That is, conclusion
A may follow from premise B, C, D even though A is neither logically
equivalent to B or C or D nor logically equivalent (in any useful sense) to
their conjunction."
Well, I suppose the Wikipedia author is dealing with a conceptual analysis
of 'valid deductive argument' for which the above would NOT count as
'valid deductive argument'.
My favoured view is not to use 'logical equivalence', but the method of
the 'associated conditional', as someone called it.
Wikipedia (I expect a different author?) prefers 'corresponding'
conditional:
https://en.wikipedia.org/wiki/Corresponding_conditional
In logic, the corresponding conditional of an argument (or derivation) is
the material 'horse-shoe' whose antecedent is the conjunction of the
argument's (or derivation's) premises and whose consequent is the argument's
conclusion.
An argument is valid, by definition, iff its corresponding or thus
associated conditional is a logical truth.
It follows that an argument is valid iff the negation of its corresponding
conditional is a contradiction.
The construction of a corresponding or associated conditional therefore
provides a useful technique or algorithm for determining the validity of an
argument.
Consider the argument A:
i. P1: Either it is hot or it is cold.
P2: It is not hot.
-------------
C: ∴ Therefore it is cold.
This argument is of the form:
ii. P1: Either P or Q
P2: Not P
------
C: ∴ Therefore Q
or (using standard symbols of the propositional calculus):
iii. P1: p v q
P2: ~p
-----
C: ∴ q
The corresponding or associated conditional is:
((P or Q) and not P) ⊃ Q
or (using standard symbols):
((p v q) & ~p) ⊃ q
and the argument is valid iff the associated conditional is a necessary
truth.
If the associated conditional is a necessary truth then ~C entails Falsity
(The False).
Thus, any argument is valid iff if the denial of its associated ⊃
-statement leads to a contradiction.
If we construct a truth table for the associated ⊃-statement, we will find
that it comes out T (true) on every row (and of course if we construct a
truth table for the negation of C it will come out F (false) in every row.
These results confirm the validity of the argument A.
Some arguments need first-order predicate logic to reveal their forms and
they cannot be tested properly by truth tables forms.
Consider the argument A1:
iii. Some mortals are not Greeks
Some Greeks are not men
Not every man is a logician
------------------
Therefore Some mortals are not logicians
To test this argument for validity, construct the corresponding associated
⊃-statement-1 (you will need first-order predicate logic), negate it, and
see if you can derive a contradiction from it.
If you succeed then the argument is valid.
Instead of attempting to derive the conclusion from the premises one may
proceed as follows.
To test the validity of an argument
(a) one may translate, as necessary, each premise and the conclusion into
sentential or predicate logic sentences
(b) one may then construct from these the negation of the corresponding
conditional
(c) one may check if from it a contradiction can be derived (or if
feasible construct a truth table for it and see if it comes out false on every
row.)
Alternatively one may construct a truth tree and see if every branch is
closed.
Success proves the validity of the original argument.
In case of difficulty trying to derive a contradiction one may proceed as
follows.
From the negation of the corresponding conditional derive a theorem in
conjunctive normal form in the methodical fashions described in text books.
Iff the original argument was valid will the theorem in conjunctive normal
form be a contradiction, and if it is then that it is will be apparent.
But Grice and Witters knew that!
In "Aspects of reason", Grice is into providing a GENERAL (typical of him
-- "philosophers crave for generality") of 'valid' as applied to at least
alethic and practical reasoning -- and he succeeds. He finds the common
element in the idea of satisfactoriness.
The fun part of this is that he gave those lectures TWICE: once as the Kant
lectures at Stanford (they revere Kant at Stanford) and another as the
Locke lectures at Oxford (they revere Locke at Oxford).
Cheers,
Speranza
References
First-order Logic: An Introduction, By Leigh S. Cauman Published by Walter
de Gruyter.
The Cambridge Companion to Mill, By John Skorupski Published by Cambridge
University Press.
The Languages of Logic: An Introduction to Formal Logic, By Samuel D.
Guttenplan Published by Blackwell Publishing.
The Value of Knowledge and the Pursuit of Understanding, By Jonathan L.
Kvanvig Published by Cambridge University Press,
Logic
By Paul Tomassi Published by Routledge.
---
Cfr.
Corresponding conditional from the Free On-line Dictionary of Computing
http://books.google.co.uk/books?id=TQlvJJgUiVoC&pg=PA19&lpg=PA19&dq=Correspo
nding+conditional&source=web&ots=V0GmWFcKsg&sig=JXjvWnQJpOKjU_-Nr-e3vE6s8PE&
hl=en&sa=X&oi=book_result&resnum=3&ct=result
http://books.google.co.uk/books?id=BVHwg_qNxosC&pg=PA40&lpg=PA40&dq=Correspo
nding+conditional&source=web&ots=MHRGHboBUd&sig=ha4gxQrKdKsINVcSOWBfrpvNQ00&
hl=en&sa=X&oi=book_result&resnum=6&ct=result
http://www.earlham.edu/~peters/courses/log/terms2.htm
http://www.csus.edu/indiv/n/nogalesp/SymbolicLogicGustason/SymbolicLogicOver
heads/Phil60GusCh2TruthTablesSemanticMethods/TT
ValidityCorrespondingConditional.doc
http://books.google.co.uk/books?id=xfOdpyj1bSIC&pg=PA90&lpg=PA90&dq=Correspo
nding+conditional&source=web&ots=PNBSh6fukg&sig=7BEBKbCD5Qhq9TOIBri9Oa5Zah4&
hl=en&sa=X&oi=book_result&resnum=6&ct=result
http://books.google.co.uk/books?id=OxXopc5AjQ0C&pg=PA175&lpg=PA175&dq=Corres
ponding+conditional&source=web&ots=FCFY5L4_HB&sig=7pkTUrJ87AtojCVRzeej5eHgqn
A&hl=en&sa=X&oi=book_result&resnum=2&ct=result
http://books.google.co.uk/books?id=tb6bxjyrFJ4C&pg=PA153&dq=Corresponding+co
nditional+logic
Categories: Conditionals Statements
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