honestrosewater said:P: What happened today will happen tomorrow.
Q: The sun rose today.
R: The sun will rise tomorrow.
[itex](P \wedge Q) \Rightarrow R[/itex] is how I would have put the previous propositions together.
I'm being a bit of a stickler for logic here, but stricly speaking, that hypothetical conditional is not deductively true due simply to the truth of its antecedent and its consequent. If you state it in argument form, you get simply P AND R, therefore Q. Stated as such, the truth value of Q is independent of the truth values of P and R. It requires a different formulation of the propositions to produce a valid argument form. So let's start over.
What happened today will happen tomorrow.
The sun rose today.
Therefore, the sun will rise tomorrow.
First we'll restate this as:
For any x, If x happened today, Then x will happen tomorrow.
s happened today.
Therefore, s will happen tomorrow.
Where x is the general propositional variable and s is "the sun rose." We will use H to mean "happened today" and T to mean "will happen tomorrow." We can now translate to:
For any x, If Hx, Then Tx.
Using symbolic connectives, the argument form is:
1. [itex](x)(Hx \Rightarrow Tx)[/itex]
We can then prove the validity of this argument by the following two steps:
3. [itex]Hs \Rightarrow Ts[/itex] From line 1 by Universal Instantiation
4. [itex]Ts[/itex] From lines 3 and 2 by Modus Ponens
This is the only way to capture the inner logical structure of the propositions, by virtue of which the conclusion "The sun will rise tomorrow" becomes deductively valid.