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Are the following four statements always true -

  1. If Proposition B is the Logical consequence of proposition A, then B is material conditionally connected with A.
  2. If Proposition E is material conditionally connected with C, then it is NOT necessarily the case that E is a logical consequence of C.
  3. If proposition H corresponds to an observed fact H* which is supposed to be causally related to observed fact G* (say H* causes G*), then the proposition G corresponding to the observed fact G* is material conditionally connected with Proposition H.
  4. If proposition X is material conditionally connected with Proposition Y, then it is NOT necessarily the case that X and Y are causally related.

I am a student of Physics, researching on quantum logic. I am confused by the above statements which I hypothesized regarding classical logic. Any help would be much appreciated.

Definitions taken from the comments:

Material conditional: Proposition A is material conditionally connected with Proposition B, denoted as "If A then B" when the proposition "If A then B" has a truth value of false only if A is true and B false. For all other combinations of truth values assigned to A and B, "If A then B" is true.

Logical Necessity: Proposition A is a logical consequence of proposition B, if the truth of proposition B (along with maybe other auxiliary axioms), necessitates the truth of Proposition A. Example: The fact that an equilateral triangle has all three sides equal necessitates that all of the three angles are each 60 degrees.

Frank Hubeny
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Varun Immanuel
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  • It is not clear what "logical" or "material conditionally" means here, your source probably has some very specific definitions. In a formalized theory some consequences will be formal (if A then A), and some only material (if red then not green), i.e. specific to the matter of the theory. If "logical" means formal and "material conditionally" means generically valid then 1 and 2 are true. As for 3, 4, they will depend on what role causality plays in the theory, one can imagine theories with material postulates that do not involve causation, e.g. the red does not *cause* the non-green. – Conifold May 31 '18 at 16:34
  • These are the definitions of Logical necessity and material conditional. – Varun Immanuel May 31 '18 at 16:41
  • These are the definitions of Logical necessity and material conditional. Material conditional: Proposition A is material conditionally connected with Proposition B, denoted as ''If A then B'' when the proposition ''If A then B'' has a truth value of False only if A is true and B false. For all other combinations of truth values assigned to A and B,'' If A then B'' is true. – Varun Immanuel May 31 '18 at 16:48
  • Logical Necessity : Proposition A is a logical consequence of proposition B, if the truth of proposition B (along with maybe other auxilary axioms), neccessitates the truth of Proposition A. Example The fact that an equilateral triangle has all three sides equal necessitates that all the three angle are each 60 degree – Varun Immanuel May 31 '18 at 16:51
  • You are using the terms in the wrong context. You seem to want to say this material conditional is a grammar rule for all English sentences in the real world. This is not TRUE. A conditional does not have to have a necessary component, a causation or a sufficent component. If you are go g down that road please stop! In mathematics your context may fit, but YOU have to indicate the CONTEXT and not leave it to readers. There is no universal rule when using If . . . THEN sentence structure. Context makes the difference. – Logikal May 31 '18 at 17:10
  • These definitions only work for mathematics, where 1 is trivially true, 2 is trivially false, and 3,4 make no sense at all. Beyond that the second one is circular, necessity is "defined" in terms of "necessitating". If you wish to use the concepts of modal logic, like necessity, beyond mathematics you need [possible worlds](https://en.wikipedia.org/wiki/Possible_world), and then the first definition becomes useless also, truth values depend on the world. – Conifold May 31 '18 at 17:13
  • I thought of all the above statements within the context of mathematics and physics. I did not refer to the natural language. – Varun Immanuel May 31 '18 at 17:18
  • Material conditional?? There is no such thing! You may mean a material implication if you are in the context of Philosophy. Material implication has a specific context which is not equal to how people speak English ordinarily. Usually material implication is equivalent to an If . . . THEN sentence structure that also expresses a disjunction. So symbolically (s-->p) is rewritten as (~s V p). This is not how many people speak ordinarily. In the Philosophy context that is what people express by material implication. A conditional sentence is an If . . . THEN structure sentence. – Logikal May 31 '18 at 17:22
  • @Conifold. You said that statement 2 is trivially false in the context of mathematics. But consider the following two formulas A:x+y=2 and B:m+k=6, for (x,y)=(1,1) and (m,k)=(2,4). Now if we form a material conditional connective between A and B, we have a truth value of 'T' for the material conditional 'If A then B', by definiton of material conditional. But clearly, B is not a logical consequence of A. Does this not illustrate the truth of the second statment in the question? – Varun Immanuel May 31 '18 at 17:25
  • @Logikal . See this https://en.wikipedia.org/wiki/Material_conditional – Varun Immanuel May 31 '18 at 17:26
  • In mathematics with classical semantics any true sentence is a logical consequence of any other true sentence, logical consequence and material conditional are equivalent by definition. You can have them come apart by using something more elaborate, which is why I asked for definitions. For example, you can distinguish consequences "by logic alone" (formal) from those that use material postulates (of arithmetic, say), as I suggested earlier, but classical "definitions" are not very useful for that. – Conifold May 31 '18 at 17:34
  • I don't need a wiki page to know what I know. There were times wiki did not exist. I wanted to be clear that YOU did not specify math or physics the way your question is worded. I wanted to make clear normal English does not always amount to wiki. – Logikal May 31 '18 at 17:35
  • Standard questions whose answer are available on Formal Logic textbook and on line. See e.g. [Logical consequence](https://en.wikipedia.org/wiki/Logical_consequence) and [Tautological consequence](https://en.wikipedia.org/wiki/Tautological_consequence). – Mauro ALLEGRANZA May 31 '18 at 17:36
  • See also [Tautology](https://en.wikipedia.org/wiki/Tautology_(logic)) and [Tautological implication](https://en.wikipedia.org/wiki/Tautology_(logic)#Tautological_implication). – Mauro ALLEGRANZA May 31 '18 at 17:36
  • See also [Material conditional](https://en.wikipedia.org/wiki/Material_conditional) for the *propositional connective* also called *conditional* and *implication* : the [truth-functional](https://en.wikipedia.org/wiki/Material_conditional#Truth_table) translation of "if..., then...". – Mauro ALLEGRANZA May 31 '18 at 17:38
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    Possible duplicate of [What does the truth-value of a material implication represent?](https://philosophy.stackexchange.com/questions/36117/what-does-the-truth-value-of-a-material-implication-represent) – Mauro ALLEGRANZA May 31 '18 at 17:39
  • You can see also many similar qeustions (with detailed answer) on the twin site of MSE: see at least [Implication and entailment](https://math.stackexchange.com/questions/361735/implication-and-entailment) and [Whats the difference between logical consequence (entailment) and simple implication?](https://math.stackexchange.com/questions/1489150/whats-the-difference-between-logical-consequence-entailment-and-simple-implica) – Mauro ALLEGRANZA May 31 '18 at 17:54
  • In light of this extended discussion in the comments, I recommend that the question be completely revised and resubmitted. – Mark Andrews May 31 '18 at 18:08
  • See also [Varieties of Modality](https://plato.stanford.edu/entries/modality-varieties/) for *necessity* and [Counterfactual Theories of Causation](https://plato.stanford.edu/entries/causation-counterfactual/). – Mauro ALLEGRANZA May 31 '18 at 18:28

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On material implication/conditional: A => B means only that it is not the case that both A is true and B is false. Or equivalently, A is false and/or B is true. Nothing more. No connection (causal or otherwise) is assumed between the propositions A and B.

Note: While the above is often stated as a definition of material implication, it can also be derived as a theorem from simpler, self-evident properties of logical connectives including '=>' itself.

See my recent blog posting, "Material Implication: If Pigs Could Fly". There, I attempt to formally justify each entry of truth table and other well known properties of material implication.

Dan Christensen
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  • So, if for example, the first statement is;'' I am 6ft tall'', is true and the second statement; ''The universe is 4 billion years'' is true, then does the following statement: ''I am six foot tall, and therefore the universe is 4 billion years'', qualify as a valid material conditional which is true? – Varun Immanuel Jun 02 '18 at 05:00
  • @VarunImmanuel Yes, keeping in mind that there is not necessarily any causal or other link between the two statements. See my formal proof of A & B => [A => B] (from the usual first line of the truth table for A => B) at the above mentioned blog posting. – Dan Christensen Jun 02 '18 at 12:38