Talk:Indicative conditional

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that conditional statement example is pretty weak in my opinion, that all girls becoming women proves all boys become men. it's probably the logical equivalent, but still pretty far from a law-like truth.

Does anyone know why certain symbols don't appear right on my computer? They are just blank squares. Ex. "often separate symbols (such as ⇒ and ⊃)."

Same here. I would like to know what those symbols are...

Can someone tell me whether or not the discussion of concepts and sets really belongs here? It seems particularly incoherent, but I can't tell if that's entirely because it is, or if it has something to do with me not being familiar with the ideas of a "conditional" as applied in the case of non-propositions. I know there's some kind of common isomorphism with boolean algebra in the background here, but I can't see how to use this knowledge to make this page coherent. --Ryguasu 09:13, 27 Feb 2004 (UTC)

It does need a fairly tough copy edit. Probably it could stay here; but another way would be to define a new page inclusion (sets) that took the strain off this one.

Charles Matthews 11:57, 27 Feb 2004 (UTC)

I think this content should be moved to Material conditional, and Logical conditional should redirect to Conditional. Anyone agree? KSchutte 4 July 2005 11:23 (UTC)

Section 3 should probably be its own article, Indicative conditional. KSchutte 4 July 2005 11:25 (UTC)

[edit] “Material implication”, “contraposition”, “reductio ad absurdum”, and “self-contradiction”

Self-contradiction in reductio ad absurdum argument (“reduction to absurdity” --- in its strictest form, “reduction to self-contradiction” [please refer to Nicholas Rescher, “Reductio ad Absurdum” in Stanford Encyclopedia of Philosophy @Internet]) is inherent with the very definition of material implication --- with P true and ~P --> Q as well as ~P --> ~Q being both true at the same time so that ~P --> P (by contraposition and the transitive property of material implication) or with P false and P --> Q as well as P --> ~Q being both true at the same time so that P --> ~P (by contraposition and the transitive property of material implication). Contraposition (~Q --> ~P) is definitionally equivalent to material implication (P --> Q) --- their truth tables are identical. Moreover, contraposition checks infinite regress of reasoning — that is, one needs to justify P in P --> Q with O --> P, O with N --> O, N with M --> N, and so on ad infinitum but contraposition prevents the necessity for this infinite justifications so contraposition must be a “first principle” (not merely a “theorem”) just like the first principles of identity (P --> P), excluded middle (P OR ~P), and non-contradiction [~(P AND ~P)] (Aristotle’s 3 “laws of thought”) all of which are in fact embodied in the very definition of truth-functional logic (that is, Boolean or 2-valued logic wherein the truth-value of a compound formula is determined by the truth-values of its prime constituents).

A reductio ad absurdum (“reduction to self-contradiction”) proof goes either (~P --> P) --> P or (P --> ~P) --> ~P.

The first reductio ad absurdum tautologous scheme simply says that if one assumes P to be false and establish by logical reasoning (that is, some valid argument) that in fact P is true then P is actually true. By the very definition of material implication, the assumption that P is false means P --> ~P is true so that together with the “proven” antecedent ~P --> P they are equivalent to P <--> ~P which is equivalent to P AND ~P (a self-contradiction) from which (being false) any conclusion immediately follows (again, from the very definition of material implication).

The second reductio ad absurdum tautologous scheme simply says that if one assumes P to be true and establish by logical reasoning (some valid argument) that in fact P is false then P is actually false. By the very definition of material implication, the assumption that P is true means ~P --> P is true so that together with the “proven” antecedent P --> ~P they are equivalent to ~P <--> P which is equivalent to ~P AND P (a self-contradiction) from which (being false) any conclusion immediately follows (again, from the very definition of material implication).

What must be emphasized here is that ---

(1) reductio ad absurdum (“reduction to self-contradiction”) is a self-contradictory argument [that is, it is absurd to derive a self-contradictory proposition P inasmuch as this does not truly establish the truth or falsity of proposition P];

(2) P is a self-contradictory proposition or formula scheme [that is, P is not only false or true it is also true or false, respectively, at the same time]; and (3) P AND ~P is a contradiction [that is, it is always false for any truth-value for P] from which, used as an antedent, any consequent follows.

It is reiterated: in a self-contradiction --- P AND ~P --- it is the conjunction which is false while the proposition P itself is both true and false which violates the very definition of a proposition in truth-functional logic as being single-valued.

In plain words, a reductio ad absurdum (“reduction to self-contradiction”) argument with material implication and contraposition as defined in truth-functional logic is self-contradictory reasoning. Thus, non-classical logics like relevance logic (that is, where it is required that premises be relevant to the conclusions drawn from them, and that the antecedents of true conditionals are likewise relevant to the consequents) had been developed to avoid from the beginning the self-contradictions. With relevance logic, a reductio ad absurdum [should actually be reductio ad falsum (“reduction to falsehood or contradiction”) or reductio ad impossibile (“reduction to impossible”) or reductio ad ridiculum (“reduction to implausibility”) or reductio ad incommodum (“reduction to anomaly”)] argument makes sense because it pre-emptively disallows, or they do not involve, self-contradiction. With the statement calculus and predicate calculus of first-order mathematical logic, the self-contradictions are barred ab initio by agreeing that Aristotle’s 3 “laws of thought” (the 3 are definitionally equivalent) as well as contraposition (which is definitioanlly equivalent to material implication that is typically used, together with negation, as the base statement connectives of first-order theories) are to be “first principles” --- that is, they are over and above all other axioms of any first order theory — in particular, the first principle of non-contradiction which prohibits from the beginning the consideration of a self-contradiction (that is, invoking a logical formula and its negation at the same time in the same respect).

What this means is simply that reductio ad absurdum (“reduction to self-contradiction”) is ridiculous and absurd while the other cases of reductio (that is, not factually “reduction to self-contradiction”) may well be “valid reasoning” if: (1) in the case of the first reductio ad absurdum tautologous scheme, with the assumption that P is false, it is not _factually_ true that P --> Q and P --> ~Q (for any proposition or formula scheme Q) are both true at the same time; or, (2) in the case of the second reductio ad absurdum tautologous scheme, with the assumption that P is true, it is not _factually_ true that ~P --> Q and ~P --> ~Q (for any proposition or formula scheme Q) are both true at the same time.

First, consider the widely accepted “Euclid’s” proof of the proposition P about the infinitude of the prime natural numbers. An assumption is made, ~P, that there are only finitely many prime natural numbers. Now, there is no proposition Q such that both P --> Q and P --> ~Q are factually true at the same time --- thus, this is actually not a true reductio ad absurdum (reduction to “self-contreadiction” ) argument because there is no “self-contradiction” involved here. The initial supposition ~P (which references “infinity”) simply need not be stated --- this is called finitary argument: it simply asserts that given any finite list of prime natural numbers one could always find another prime natural number that is not in the list (this “no last element” scenario is to be taken as the meaning of “infinite”).

Next, consider the standard argument proving that 1/0 is not an element of the field of real numbers. An assumption is made, P, that 1/0 = c is a real number and it is argued that this leads to 1 = 0 [since 0 ٠ c = 0 for all real number c (this is easily derived from the field axioms)]. There is no “reduction to self-contradiction” here but only “reduction to falsehood” or “reduction to contradiction” [that is, the argument is merely reduced to the result 1 = 0 which is a contradiction --- it is false] so this argument is valid.

Next, consider Cantor’s diagonal argument “proving” the “uncountability” of the real numbers by first assuming that all the fractional real numbers are countable and they could be row-listed in the standard enumeration form x1, x2, x3, … from which a fractional real number not in the row-listing could be formed from the anti-diagonal digits. Now, this so-called “proof” is replete with so many self-contradictions and, thus, is an untenable reasoning ---

(1) the assumption that all the fractional real numbers could be row-listed uniquely and exhaustively in the standard enumeration form x1, x2, x3, … presupposes some list inclusion and imposition of order condition; but any of this specification is tantamount to the prescription that the nth-row number must have some particular digit at its nth-column position --- that is, the row-listing could be specified as such that the diagonal digits are all 0s, or all 1s, or alternating 0s and 1s, etc. Clearly, with the adoption of a prefixed fractional expansion point before the diagonal digits, the “real number” thus formed by the diagonal digits must be an “irrational number” because one can easily find an excluded fractional real number from the row-listing if it was a “rational number” (that is, one with a discernible pattern in its digits); hence, the anti-diagonal number must also be an “irrational number” that could not possibly satisfy ab initio its own row-list inclusion and imposition of order condition being different digit-for-digit to it;

(2) the diagonal-digits-with-prefixed-fractional-expansion-point “real number” is a variable --- it is not a true real number which is a constant;

(3) the row-listed fractional real numbers are mostly _intervals_ and not true real number _points_ (so Georg Cantor himself had to posit ordinal numbers --- in particular, omega as the first transfinite number, followed by omega + 1, omega + 2, and so on);

(4) the standard enumeration form x1, x2, x3, … and the fact that each fractional real number is an infinite sequence of place-value base digits (at least 2 for binary system) is a self-contradiction --- the non-standard enumeration (still countable!) form a1, a2, a3, …, b1, b2, b3, … is more appropriate for the row-listing;

(5) etc., etc., etc. . . .

Likewise, Kurt Godel’s argument which invokes the self-contradiction “This assertion cannot be proved”, Alan Turing’s argument which invokes the self-contradiction “a computer program halts if and only if it does not halt”, and many others that involve self-contradictions are untenable “proofs”.

Please read my Wikipedia discussion notes on “Cantor’s diagonal argument”, “Cantor’s theorem”, “Cantor’s first uncountability proof”, “Ackermann’s function”, “Boolean satisfiability problem”, “Entscheidungsproblem”, “Definable number”, and “Computable number”. (BenCawaling@Yahoo.com [14 December 2005])

[edit] Long overdue

In the next day or two I'm going to carry out the suggestion I made above (half a year ago), splitting the content of this page between Indicative conditional and Material conditional.

Here is my strategy for doing it, in order to preserve the page histories: I will move this page (Logical conditional) to Indicative conditional (i.e., change the title of this page). I will then move most of the content from the page to Material conditional (which is currently a redirect). Finally, I will change the redirect that will be left here after moving it to point to Conditional instead. This will bring wiki's nomenclature into consistency with contemporary professional use. KSchutte 03:30, 14 December 2005 (UTC)

I will then, of course, update Conditional to include Indicative conditional, along with a small description of the differences between each kind of conditional listed on that page. KSchutte 03:35, 14 December 2005 (UTC)

[edit] Removed.

Connection with other concepts

The logical conditional, and particularly the material conditional, is closely related to inclusion (for sets), subsumption (for concepts), or implication (for propositions). It also has formal properties analogous to those of the mathematical relation less than or more exactly $\leq$ , especially the relation of not being symmetrical.

In the conceptual interpretation, when $a$ and $b$ denote concepts, the relation $a \in b$ signifies that the concept $a$ is subsumed under the concept $b$ ; that is, it is a species with respect to the genus $b$ . From the extensive point of view, it denotes that the class of $a$ 's is contained in the class of $b$ 's or makes a part of it; or, more concisely, that "All $a$ 's are $b$ 's". From the comprehensive point of view it means that the concept $b$ is contained in the concept $a$ or makes a part of it, so that consequently the character $a$ implies or involves the character $b$ . Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".

In the propositional interpretation, when $a$ and $b$ denote propositions, the relation $a \Rightarrow b$ signifies that the proposition $a$ implies or involves the proposition $b$ , which is often expressed by the hypothetical judgement, "If $a$ is true, $b$ is true"; or by " $a$ implies $b$ "; or more simply by " $a$ , therefore $b$ ". We see that in both interpretations the relation may be translated approximately by "therefore".

Remark. -- Such a relation is a proposition, whatever may be the interpretation of the terms $a$ and $b$ .

Consequently, whenever a $\Rightarrow$ relation has two like relations (or even only one) for its members, it can receive only the propositional interpretation, that is to say, it can only denote an implication.

A relation whose members are simple terms (letters) is called a primary proposition; a relation whose members are primary propositions is called a secondary proposition, and so on.

From this it may be seen at once that the propositional interpretation is more homogeneous than the conceptual, since it alone makes it possible to give the same meaning to the copula in both primary and secondary propositions.

I took this out after dividing the pages and I don't know what to do with it. It seems like a math student's take on logic rather than a logician's explanation. Anyone who sees fit to re-add it would be encouraged to give it some context. KSchutte 17:58, 19 December 2005 (UTC)