What is the basis for the definition of logical connectives? Are they just arbitrary convention? Or does it depend on the meaning of the constituent propositions? Does the individual meaning of two propositions determine or constrain what kind of logical connectives can be formed between them? In other words, are the definition given to logical connectives based on empirical observation and logical necessity or are they just arbitrarily defined concepts and rules(Axioms)?
To give a context for this question, I provide the following statements from Mathematics texts and my commentary of them:
In a discrete Mathematics book, the following definition was given to conjunction and implication.
(I)Conjunction- ''Let P and Q be propositions. The proposition ''P and Q'', is the proposition that is true when both P and Q are true and is false otherwise''.
Comment on the above definition: As per the above definition, I can consider any arbitrary pair of propositions and connect it by the ''Conjunction'', since it does not impose any constraint on the meaning of P and Q. So according to the definition, both of the following two examples seem reasonable. Example
(1). P=This car weighs 50 tons. Q=This car is green in color. P and Q = This car weighs 50 tons and is green in color.
(2). P= This car is green. Q=The milky way galaxy is 100,000 light years wide. P and Q = This car is green and the milky way galaxy is 100,000 light years wide.
As per the definition, both examples are valid cases of Conjunction. But it appears that both the examples do not have the same empirical and\or logical footing. In the former case, it is easy to see that the region of the Venn diagram corresponding to P and Q, is made of points corresponding to individual physical objects(car), that possess two properties: Weighing 2 tons and having a green color. In the latter case, however, the region of the Venn diagram pertaining to ''P and Q''is supposedly made of points, each corresponding to two distinct objects possessing two distinct properties. From an empirical standpoint, the second example looks artificial while the former looks more ''natural'', in the sense that there are arbitrarily many conjunctions that can be formed from any random pair of propositions but only a countable number of conjunctions formable from propositions corresponding to the properties of a Physical object.
Definition of implication from the same book
(II)Implication-''Let P and Q be two propositions. The implication ''If P then Q'' is the proposition that is false when P is true and Q is false, and true otherwise''.
Again this definition also does not impose any condition, based on meaning, on the constituent propositions. So the following two examples seem reasonable.
(1) P= x < 2, Q=x < 6
for all x in the set of real numbers.
If P then Q = If P<2, Q<6 for all x in the set of real numbers.
(2) P= This box is big, Q= Tomorrow it will rain,
If P then Q= If this box is big, it rained yesterday.
Commentary: According to the definition of ''implication'' logical connective, both the examples could be said to constitute valid examples of ''Implication''
Consider, in particular, the second example where the ''If P then Q'' assumes a truth value of F, only if The box is big and it did not rain yesterday(According to the definition). Clearly, this second example is not both logically and empirically on the same footing as the first example. The second example seems to have been constructed '' artificially'', based on the rules specified in the definition. On the other hand, the former example seems to have a logical necessity independent of the definition given to ''Implication''.
I am not completely sure if the second example in both, the case of Conjunction and the case of Implication are ''allowed''. If they are not allowed, what are the rules constraining the formation of logical connectives? It seems that the constraining rules depend on the meaning of the constituent propositions. On the other hand, if they are allowed, is not the arbitrariness (which was illustrated in the examples) troubling?