Can these propositions be equivalent?
$begingroup$
We all know that:
$P rightarrow Q $
and
$(Not)Q rightarrow (Not)P$
are equivalent.
Is it possible that in specific cases
$P rightarrow Q $
is equivalent with
$Q rightarrow P $
or
$(Not)Q rightarrow P $
or
$P rightarrow (Not)Q $
Can this happen or is that impossible? If it is, how should I attempt to prove it?
logic propositional-calculus
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
$endgroup$
add a comment |
$begingroup$
We all know that:
$P rightarrow Q $
and
$(Not)Q rightarrow (Not)P$
are equivalent.
Is it possible that in specific cases
$P rightarrow Q $
is equivalent with
$Q rightarrow P $
or
$(Not)Q rightarrow P $
or
$P rightarrow (Not)Q $
Can this happen or is that impossible? If it is, how should I attempt to prove it?
logic propositional-calculus
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
$endgroup$
1
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28
add a comment |
$begingroup$
We all know that:
$P rightarrow Q $
and
$(Not)Q rightarrow (Not)P$
are equivalent.
Is it possible that in specific cases
$P rightarrow Q $
is equivalent with
$Q rightarrow P $
or
$(Not)Q rightarrow P $
or
$P rightarrow (Not)Q $
Can this happen or is that impossible? If it is, how should I attempt to prove it?
logic propositional-calculus
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
$endgroup$
We all know that:
$P rightarrow Q $
and
$(Not)Q rightarrow (Not)P$
are equivalent.
Is it possible that in specific cases
$P rightarrow Q $
is equivalent with
$Q rightarrow P $
or
$(Not)Q rightarrow P $
or
$P rightarrow (Not)Q $
Can this happen or is that impossible? If it is, how should I attempt to prove it?
logic propositional-calculus
logic propositional-calculus
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
asked Jan 13 at 11:59
Ayoub RossiAyoub Rossi
11110
11110
1
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28
add a comment |
1
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28
1
1
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
From a truth table $Pto Q$ is equivalent to $Qto P$ iff $P=Q$, to $neg Qto P$ iff $Q$, and to $Ptoneg Q$ iff $Pland Q$. You can easily choose $P,,Q$ to achieve these conditions.
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
$endgroup$
add a comment |
$begingroup$
There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
$endgroup$
add a comment |
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2 Answers
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$begingroup$
From a truth table $Pto Q$ is equivalent to $Qto P$ iff $P=Q$, to $neg Qto P$ iff $Q$, and to $Ptoneg Q$ iff $Pland Q$. You can easily choose $P,,Q$ to achieve these conditions.
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
$endgroup$
add a comment |
$begingroup$
From a truth table $Pto Q$ is equivalent to $Qto P$ iff $P=Q$, to $neg Qto P$ iff $Q$, and to $Ptoneg Q$ iff $Pland Q$. You can easily choose $P,,Q$ to achieve these conditions.
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
$endgroup$
add a comment |
$begingroup$
From a truth table $Pto Q$ is equivalent to $Qto P$ iff $P=Q$, to $neg Qto P$ iff $Q$, and to $Ptoneg Q$ iff $Pland Q$. You can easily choose $P,,Q$ to achieve these conditions.
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
$endgroup$
From a truth table $Pto Q$ is equivalent to $Qto P$ iff $P=Q$, to $neg Qto P$ iff $Q$, and to $Ptoneg Q$ iff $Pland Q$. You can easily choose $P,,Q$ to achieve these conditions.
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
answered Jan 13 at 12:15
J.G.J.G.
33.6k23252
33.6k23252
add a comment |
add a comment |
$begingroup$
There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
$endgroup$
add a comment |
$begingroup$
There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
$endgroup$
add a comment |
$begingroup$
There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
$endgroup$
There is no such thing as two statements being 'sometimes logically equivalent' or 'possibly logically equivalent'. Logical equivalence means always having the same truth-value. If it's not always, it's not equivalence.
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
answered Jan 13 at 16:02
Bram28Bram28
64.6k44793
64.6k44793
add a comment |
add a comment |
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1
$begingroup$
NO; logical equivalence means that "they have the same truth value in every model" (in the case of propostoonal logic : in every truth assignment.
$endgroup$
– Mauro ALLEGRANZA
Jan 13 at 12:28