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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 ⊃)."
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)
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.
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).
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])
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)
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 , especially the relation of not being symmetrical.
In the conceptual interpretation, when and denote concepts, the relation signifies that the concept is subsumed under the concept ; that is, it is a species with respect to the genus . From the extensive point of view, it denotes that the class of 's is contained in the class of 's or makes a part of it; or, more concisely, that "All 's are 's". From the comprehensive point of view it means that the concept is contained in the concept or makes a part of it, so that consequently the character implies or involves the character . Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".
In the propositional interpretation, when and denote propositions, the relation signifies that the proposition implies or involves the proposition , which is often expressed by the hypothetical judgement, "If is true, is true"; or by " implies "; or more simply by ", therefore ". 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 and .
Consequently, whenever a 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.
The false claim is: (quote) The standard definition of implication allows us to conclude that, since the sun is made of gas, 3 is a prime number. (end quote)
The standard definition omits restrictions on the antecedent and consequent, however, the proper way to teach introductory logic is together with the fallacies. The claim I quoted is an example of False Cause, and in a well-taught logic course, by the time students are introduced to material implication they have first practised rejecting fallacies.
Thus forming: A = the sun is made of gas B = 3 is a prime number
given input: Evaluate A implies B.
correct output: Reject request for evaluation based on False Cause.
In Friendship, Jennifer —Preceding unsigned comment added by 66.183.47.131 ( talk) 23:52, 15 May 2011 (UTC)
I added a no footnote tag as the article uses only general references. Otr500 ( talk) 05:02, 24 December 2011 (UTC)
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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 ⊃)."
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)
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.
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).
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])
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)
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 , especially the relation of not being symmetrical.
In the conceptual interpretation, when and denote concepts, the relation signifies that the concept is subsumed under the concept ; that is, it is a species with respect to the genus . From the extensive point of view, it denotes that the class of 's is contained in the class of 's or makes a part of it; or, more concisely, that "All 's are 's". From the comprehensive point of view it means that the concept is contained in the concept or makes a part of it, so that consequently the character implies or involves the character . Example: "All men are mortal"; "Man implies mortal"; "Who says man says mortal"; or, simply, "Man, therefore mortal".
In the propositional interpretation, when and denote propositions, the relation signifies that the proposition implies or involves the proposition , which is often expressed by the hypothetical judgement, "If is true, is true"; or by " implies "; or more simply by ", therefore ". 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 and .
Consequently, whenever a 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.
The false claim is: (quote) The standard definition of implication allows us to conclude that, since the sun is made of gas, 3 is a prime number. (end quote)
The standard definition omits restrictions on the antecedent and consequent, however, the proper way to teach introductory logic is together with the fallacies. The claim I quoted is an example of False Cause, and in a well-taught logic course, by the time students are introduced to material implication they have first practised rejecting fallacies.
Thus forming: A = the sun is made of gas B = 3 is a prime number
given input: Evaluate A implies B.
correct output: Reject request for evaluation based on False Cause.
In Friendship, Jennifer —Preceding unsigned comment added by 66.183.47.131 ( talk) 23:52, 15 May 2011 (UTC)
I added a no footnote tag as the article uses only general references. Otr500 ( talk) 05:02, 24 December 2011 (UTC)