In one of the letters written to the Infeld group in Warsaw Einstein wrote:
"A new manner of thinking is essential if humankind is to survive."

  
      IMPLICATION (Exact 2D Propositional Calculus)


      Let's consider two statements 
      
      p: "it has been raining over the street"
      q: "the street is wet"
      
      NOTE: By p and q we mean that: 
      K. The rain was sufficient to wet the street. 
      L. It's been raining recently and the street had no time to dry, 
      M. It's the same street in p and q.

      Implication in metalanguage:
      "If it has been raining over the street then the street is wet" 
      or "it has been raining over the street implies that the street is wet"
      
      Implication in Calculus:
      
      case. imp(pq)
      1:....1...11
      2:....0...10
      3:....1...01
      4:....1...00
      
      Let imp(pq) axiom of theory T.
      
      We shall discuss: 
      A. Application of T 
       AA.Deductive 
       AB. Inductive 
      B. Research on T
      
      A. APPLICATION OF T (Application is based upon belief that imp(pq) holds).
      -------------------

      AA. Deductive 
      -------------
      Meteo forcasts rain in concerned area. We deduce from case 1:
      
      case. imp(pq)
      1:....1...11
      
      that the street will soon be wet and act accordingly, eg. clean the gutters.

      AB. Inductive
      -------------
      Meteo stated that it has been raining in concerned area, but we state that our 
      street is dry. Upon our belief in imp(pq) holding, we induce from case 4:
      
      case. imp(pq)
      4:....1...00
      
      that p=0, i. e. that it has not been raining in concerned area and we inform 
      meteo system that it has a bug.
      
      On the other hand, if we state that our street is wet, we induce from case 1:
      
      case. imp(pq)
      1:....1...11 
      
      that meteo was right and keep happy and quiet.
      
      RESEARCH ON T
      -------------

      We gather FACTUAL INFORMATION, i. e. OBSERVATIONS of p and q to see how they 
      fit pertinent cases of the axiom: "imp(pq)":
      
      case. imp(pq)
      1:....1...11
      2:....0...10
      3:....1...01
      4:....1...00 
      
      Let's note that T CAN BE DISPROVED by a single observation fitting case 2.
      
      On the contrary, it cannot be PROVEN. Indeed, no matter how many observations 
      may fit cases 1,3,4, they don't prove that some day we will not observe the 
      case 2.
      
      The more observations fit 1,3,4, and the stronger gets our PRAGMATIC BELIEF in 
      imp(pq) holding, but no matter how strong our belief, it is not a PROOF.
      
      Billions of observations per minute fit the "gravity theory", 
      making us believe so strongly in gravity, that we take it for obvious and 
      granted. Still, gravity is not PROVEN and while we believe that it will be 
      there to morrow, there is no logical reason to be 100% certain that it will 
      not cease in next second.
      
      We encounter here a central premise in the philosophy of science, the Principle 
      of Falsifiability, first formally discussed by Karl Popper. This principle 
      states that in order to be useful (or even scientific at all), a scientific 
      statement ('fact', theory, 'law', 'principle', etc) must be falsifiable, i. e. 
      able to be proven wrong. Without this property, it would be difficult (if not 
      impossible) to test a scientific statement against the evidence.
      
      It's surprising to find that it took 2000 years to formulate this Principle, 
      when it is obviously inherent to Implication which has been known to 
      Aristoteles.

      EXERCISE
      --------

      Let: p: "it has been raining over the
      street" q: "the street is wet"
      
      NOTE: By p we mean additionally that: 

      K. The rain was sufficient to wet the street.
      
      L. It's been raining recently and the street
      had no time to dry,
      
      M. The street in p is the same as in q.
      
      Let imp(pq) axiom of theory T.
      
      case. imp(pq)
      1:....1...11
      2:....0...10
      3:....1...01
      4:....1...00 
      
      1. Try to find factual examples for case 3. and explain why they do not refute T.
      
      2. Explain why case 1. does not prove T.
      
      3. Try to refute T.