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Imagine this statement

If there is sun then it will be a bright day

which is of the form $p \to q$. If the sun was there and it was not bright, then the truth value of this statement would be false. So why did I make this statement instead of keeping quiet? As far as common sense and wisdom go, one would prefer to avoid making such statements if it is not always going to be true.

I think I am still not clear on the usage of such propositions. Is this a bad example from "spoken language" to understand why we need this?

In Mathematics, you start with a statement whose truth value is unknown and then enumerate all the possible scenarios to determine the truth value of $p \to q$ as opposed to "spoken language" where you actually (implicitly) mean it is true without a doubt (if I had doubt, I wouldn't say it)? The key point being I start with an unknown trying to establish the truth value?

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    $\begingroup$ Is this a bad example from "spoken language" to understand why we need this? Yes. as opposed to "spoken language" where you actually mean it is true? Not sure what you mean by that. Something like "I'll buy you lunch if that happens" is a fairly common "spoken language" construct which does not presume the truth value of its antecedent. $\endgroup$ – dxiv Mar 13 '18 at 6:37
  • $\begingroup$ @dxiv, Thank You. When I said, as opposed to "spoken language" where you actually mean it is true?, I meant that you say such a statement because you implicitly believe it is true. If it was false, such a sentence wouldn't exist :-) $\endgroup$ – Nishant Mar 13 '18 at 7:15
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    $\begingroup$ You might be interested in reading about 'fuzzy logic'. The wikipedia page is a good intro. $\endgroup$ – Prince M Mar 15 '18 at 22:19
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First of all, it seems to me that your comments are not restricted to conditionals: typically, any claim I make is something that I believe is true. So, you might as well ask: why do we check all possible values of any proposition?

well: I may believe certain things to be true, but whether they are actually true is a whole other thing. That's why we have such things as science, reason, and logic. Indeed, other than observation, one of the main ways we try to establish the truth of claims is through arguments ... but our arguments aren't always valid either. That is, when I make an argument, I once again believe that some conclusion follows from some premises, but that may not actually be the case. And, how do we demonstrate using logic that some piece of reasoning is valid or invalid? We do this by considering all possible truth-conditions.

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In Mathematics, you start with a statement whose truth value is unknown and then enumerate all the possible scenarios ...

NO.

In mathematics we start from axioms and already proved theorems: call $T$ the theory made by the collection of axioms and theorems.

Call $P$ a certain proposition in the subject matter of the theory (i.e. expressed in the corresponding language).

Our aim is to prove $P$.

If we succeed in proving $S \to P$, where $S$ is a (finite) subset of $T$, i.e. a certain sub-collection of the initial set of axioms and of the already availbale theorems, we are entaitled to conclude that:

proposition $P$ is a new theorem of our theory.

In everyday life, we do not (usually) try to prove that "it is a bright day": we (usually) open the window and look outside.

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In mathematics (and in natural language, it would seem), if we have logical propositions $A$ and $B$ that are known to be either true or false, and if we accept certain "common sense," self-evident properties of logical implication (see my recent posting Deriving the Truth Table for Material Implication ) then we can prove:

  • If $A$ is true and $B$ is true, then $A\implies B$ must be true

  • If $A$ is true and $B$ is false, then $A\implies B$ must be false

  • Regardless of whether $B$ is true or false, if $A$ is false, then $A\implies B$ must be true

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