# Why don't two equivalent propositions contribute to the same semantics?

We often have 2 propositions that have the same truth table, in that they are true and false given the same conditions.

Nevertheless, we still feel as though there different semantics (i.e meaning..), lurking in the background.

Example : the law of non-contradiction: ~(P ∧ ~P), and the proposition if P then P, (P → P) have the same truth table:

╔═══╦═══════════╦═════════╗
║ P ║ ~(P · ~P) ║ (P → P) ║
╠═══╬═══════════╬═════════╣
║ F ║     T     ║    T    ║
╠═══╬═══════════╬═════════╣
║ T ║     T     ║    T    ║
╚═══╩═══════════╩═════════╝

But when translated into English, we can see that a semantic layer emerges, which is not contained in the logical form. Mainly, these two propositions seem to have a different meaning:

Law of non-contradiction: It is not the case that a cat is white and not-white.

If P then P: If a cat is white then it is white.

The second proposition, does not say anything about white being not-not-white as the first proposition implies.

The law of non-contradiction seems to determine what not- stands for, in that nothing can be and not-be.

While the second proposition does not imply this meaning of non-contradiction.

In our example, the first proposition implies that a cat cannot be white and not-white at the same time, while the second only asserts that if it is a white then it is white (without saying anything about the not- case)

Although both have the same truth table.

Why is that? Are there any philosophical resources to study these propositions on a semantic level? And how does meaning emerge from propositions with the same terms (i.e: letters like P), and the same truth table?

Thanks

Note I am personally inclined to say that the different semantics are due to the meaning already encoded in the propositional symbols (· , → , ~ ) and the parentheses '(' and ')', but I need some resources to look into this in further detail.

• Is your question essentially: what is there more to semantic meaning than truth conditions? If so, the answer is: a lot, and you may want to take a look here: plato.stanford.edu/entries/meaning/#SemThe. – Eliran Mar 3 at 17:22
• The truth-functional propositional logic is a very very very simplified model of language... and it is very very very useful. But it is a model. – Mauro ALLEGRANZA Mar 3 at 17:28
• The basic assumption of truth-functional propositional logic is exactly that "meaning" is equated with truth value. – Mauro ALLEGRANZA Mar 3 at 17:28
• But still, the two are "similar" : both instances of the same tautology express a "trivial" truth. In classical logic we have double negation law; thus "If a cat is white then it is white" is the same as "If a cat is white then it is not the case that it is not-white". And this, in turn, is quite similar to "It is not the case that a cat is white and not-white." – Mauro ALLEGRANZA Mar 3 at 17:31
• @MauroALLEGRANZA I agree ... Don't you think that the propositional symbols are the factor that contributes to this difference in meaning? Because, if the truth table is the same, the terms are the same, then the only source of difference would be in if...then, and , not- ...etc. – SmootQ Mar 3 at 17:41