predicates & quantifiers
How to express these in terms of predicates & quantifiers :
- Some properties are tautologies
- The negation of a contradiction is a tautology
- The dis junction of two contingencies can be a tautology.
- The conjunction of two tautologies is a tautology.
I could find the answer from the answer key in this sequence as:
- $\exists xT(x)$
- $\forall x(C(x)\rightarrow T(\neg x)) $
- $\exists x\exists y(\neg T(x)\wedge \neg C(x) \wedge \neg T(y) \wedge \neg C(y) \wedge T(x\vee y)) $
- $\forall x\forall y((T(x) \wedge T(y)) \rightarrow T(x\wedge y))$
From Rosen 5th edition
And not at all able to know how did he arrive at this answer
Can anyone help ? !!
Thanks in advance
$\endgroup$ 22 Answers
$\begingroup$The variables stand for properties (or propositions). The predicate $T$ is for tautologies, i.e., $T(x)$ means that the property $x$ is a tautology. $C(x)$ means that $x$ is a contradiction.
Now things should be rather straight forward:
$\exists x T(x)$ means "there is a property $x$ such that $x$ is a tautology".
The second line is "for all propositions $x$ such that $x$ is a contradiction, the negation $\neg x$ is a tautology".
The third line is more interesting. I believe the last $\wedge$ should be $\vee$ for disjunction. Then the line can be explained as follows:
What is a contingency? A property that is neither a tautology nor a contradiction. So this line says "there are $x$ and $y$ such that $x$ and $y$ are contingencies and $x\vee y$ is a tautology".
The last line also has a typo, I think. The comma between $x$ and $y$ should be $\wedge$.
$\endgroup$ 2 $\begingroup$@Stefan Geschke explains your typos and what the intended answers seem to be.
But there is something quite horribly confusing going on here (in the model answers). Suppose we take the seemingly intended fourth answer $\forall x\forall y((T(x) \wedge T(y)) \rightarrow T(x \wedge y))$.
Then the first '$\wedge$' is the truth-functional connective, which when used to conjoin two sentences (open or closed) produces a sentence. But what about the second use of the '$\wedge$'? '$T$' is a predicate which applies to terms (constants, variables, or expressions built up from them by the use of functions applying to terms). So if '$T(x \land y)$' is to be well-formed, here $\land$ must be expressing not a sentence-connective but a function which applies to two terms to produce another term (i.e. '$\land$' will be here a function expression which combined with two terms denoting propositions produces a term denoting the conjunction of those proposition). Note the fundamental type-distinction between wffs which express propositions and terms which might denote them.
To deploy these two differently typed uses of '$\wedge$' in an unexplained way in what is supposed to be an elementary exercise is Very Bad if not Logically Wicked!
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