MHB Mano's question from Facebook on logic

[Jameson]

 

[Evgeny.Makarov]

We have \(p\to q=\neg p\lor q\). I assume \(\neg p\lor q\to r\) means \(((\neg p)\lor q)\to r\), which is the usual convention. So,
\(\begin{align*}\neg p\lor q\to r&=\neg(\neg p\lor q)\lor r\\&=(\neg\neg p\land\neg q)\lor r\\&=(p\land\neg q)\lor r\end{align*}\)

The argument in the second question is valid.

 

[soroban]

Hello, Jameson!

Use truth table to determine the validity of the argument:

. . \begin{array}{c}q \to p \\ p \vee q \\ \hline \therefore\;p \end{array}

. . \begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} p & q & [(q & \to & p) & \wedge & (p & \vee & q)] & \to & p \\ \hline T&T &T&T&T&T & T& T&T&{\color{blue}T}&T \\ T&F &F&T&T&T & T&T&F&{\color{blue}T}&T \\ F&T & T&F&F&F & F&T&T&{\color{blue}T}&F \\ F&F & F&T&F&F & F&F&F&{\color{blue}T}&F \\ \hline && 1&2&1&3&1&2&1&4&1 \end{array}

 

[Deveno]

growing old leads to death.

i am either growing old, or i am dead.

therefore, i am dead.

(strictly tongue-in-cheek, i assure you).

 

[solakis1]

growing old leads to death.

i am either growing old, or i am dead.

therefore, i am dead.

(strictly tongue-in-cheek, i assure you).

Going towards death ia a completely different thing from ,i am already dead

 

[Deveno]

duh! well of COURSE it's not sound logical reasoning (unless, perhaps, i am a zombie...you never know...)

but what is "wrong" is not my identification of "dying" with "dead", but rather a subversion of what "implies" means by substituting an "english" version for a "math" version:

in math, "p implies q" means: if p is true (now!), q is true (now!). if p is not true now, but will be at some point in the future (say we have a function defined on N, and every so often we replace f(k) with f(k+1)), then q must be true "at that same time".

in english "p implies q" means: if p is true (now, or at some point in the future)), q either is, or will be true (we use the same word for an act of deductive and inductive reasoning). this is commonplace: where certain things have a cause-and-effect relationship, we use implication to say the effect results from the cause, even if there is a large time interval involved (the very language of "classical logic" reflects this, we speak of a consequent, which implies a certain time-delay, when the actual relationship is "time-invariant").

for example, we often say things like: if it rains, then the ground will get wet (a classic "if...then" statement). obviously, there is that brief period of time when the first raindrops are falling, but have not yet hit the ground, when this statement isn't true.

logically, the statements:

it is raining, so therefore the ground is wet

and

it is raining so the ground will be wet soon

are distinct, but they are often used in common language interchangeably.

the "inductive" sense of "implies" (which i used as "leads to" to further confuse the issue) can't be turned around to justify a deduction.

for example, look what a simple rephrasing does:

dying (now, or in the future) leads to being dead (in the future).

i am either dying now, or will be dead in the future.

therefore, i will be dead (in the future).

specifying the "time" at which my statements are true changes them slightly, and allows for an "apples to apples" logical comparison.

(even more technically, a statement of causality, in a world where time flows, is *different* than a statement of logical deduction, which occurs at some *fixed* point of time. but they often look similar enough in ordinary language to "blur" the distinction. the "if...then" way of stating implication in natural language reinforces this blurring, but is "extra baggage" from the natural language:

when we say: if x is a multiple of 4,then x is even, we do NOT mean that x is FIRST some multiple of 4 and then at some point later changes to a multiple of 2, x was always a multiple of 2 to begin with).

but...such a re-formulation isn't funny. explaining a joke does tend to ruin it. :(

 

[solakis1]

in math, "p implies q" means: if p is true (now!), q is true (now!).

Wrong.

"p implies q" means : if p true and q true then "p implies q" is true.

Unless you mean "p logicaly implies q"

in english "p implies q" means: if p is true (now, or at some point in the future)), q either is, or will be true (we use the same word for an act of deductive and inductive reasoning)

Again wrong. The implication is the same as that in maths

 

[Deveno]

no, "if p and q" is not the same as "if p then q".

implication in english carries with it a "time-sense", often one will say:

"if (this happens) then (that *will* happen)". note the change of tense.

logical implication has no such "time-sense" bound up within it (in general), logical priority is not temporal. we do not say: 2 is even implies 2 will not be odd, someday.

natural language does NOT express the same ideas as in math: natural language is inherently ambiguous, the whole purpose of logic is to remove (insofar as possible) that ambiguity. the semantics of natural language is much subtler than mathematics, if i say:

"see what i did there?"

there are two obvious meanings:

1) did you perceive the event that occurred
2) did you understand the intent of my action

because the word "see" has a literal and figurative meaning. math (and especially logic) strives to eliminate the figurative meaning (although some of it creeps in in the names we gives to mathematical objects). i can't call a number "prime" just because it fetches a high price in the marketplace, at least not in a proof of number theory.

 

[solakis1]

no, "if p and q" is not the same as "if p then q".

I never said that : "p and q" is the same as "if p then q"

All i said is that:

"p implies q" means : if p true and q true then "p implies q" is true

implication in english carries with it a "time-sense", often one will say:

"if (this happens) then (that *will* happen)". note the change of tense.

logical implication has no such "time-sense" bound up within it (in general), logical priority is not temporal. we do not say: 2 is even implies 2 will not be odd, someday.

natural language does NOT express the same ideas as in math: natural language is inherently ambiguous, the whole purpose of logic is to remove (insofar as possible) that ambiguity. the semantics of natural language is much subtler than mathematics, if i say:

"see what i did there?"

there are two obvious meanings:

1) did you perceive the event that occurred
2) did you understand the intent of my action

because the word "see" has a literal and figurative meaning. math (and especially logic) strives to eliminate the figurative meaning (although some of it creeps in in the names we gives to mathematical objects). i can't call a number "prime" just because it fetches a high price in the marketplace, at least not in a proof of number theory.

Is logical implication ,for you , the same as implication?