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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.

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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
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    Possible duplicate of What does the truth-value of a material implication represent? – Mauro ALLEGRANZA May 31 '18 at 17:39
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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.

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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 2 '18 at 5:00
  • @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 2 '18 at 12:38

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