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As I forgot to stipulate the pretermission of the logical equivalence P → Q ≡ ¬P ∨ Q here, some of the answers presume this and resemble petitio principii, and so are really proofs rather than intuitive explanations. So for this question: exclude, and do NOT use or rely on, any foreknowledge of ¬P ∨ Q as the answer, formal proofs, Substitution Instances or Truth Tables. Before attempting any proofs, how would you conjecture or divine that P → Q:

  1.  ≡  ¬P ∨ Q  and
  2.  not P ∨ ¬Q ?
  • Seems counter-intuitive to use formal notation while excluding truth tables, etc. Perhaps it might help you to write it all out in ordinary language, i.e. write out the supposition of "if a proposition then an other proposition"... and so forth to get at the meat of the implication and equivalence relations you are concerned with? E.g. sense & reference – Mr. Kennedy Nov 13 '16 at 20:20
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    You have already received 5 answer in your previous post here and a couple others on MSE. Why do you think that someone can convince you of the usefulness of this "standardization" ? – Mauro ALLEGRANZA Nov 13 '16 at 20:31
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    In the end, it is enough to consider intuitionsitic logic, where the truth-functional definition of the conncetives is not adopted and thus the above equivalence does not hold. – Mauro ALLEGRANZA Nov 13 '16 at 20:32
  • As it is, this is a pretty much a duplicate of your first question. Unless you are asking for an intuitive reason petitio principi is false. If that is what you are asking, please remove the redundant part of the question. – jobermark Nov 14 '16 at 0:15
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    Possible duplicate of Intuitively, why does (B → S) ≡ (¬B ∨ S)? – jobermark Nov 14 '16 at 0:16
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Unfortunately, you can't put intuition into words for the simple reason that intuition is prelinguistic (although perhaps "sublinguistic" is a more apt description?) That said, I will try and describe the basic idea of "if P, then Q" which is also known by "P implies Q", "P then Q", "P → Q" and such.

Each term [P, Q] can evaluate to either and only (true or false).

We can then rewrite the set of terms [P, Q] as a redundant collection of term conditions: [(true or false), (true or false)].

Note that with (true or false) we are using the logical constant "or" in an exclusive sense, not an inclusive sense. For example, if a parent tells a child they can have either a hamburger or a hot dog for dinner and the child requests both or neither, they have not met the condition established with the parents use of exclusively either a hamburger "or" a hot dog. When the child requests only a hamburger, the dinner choice established by the parent (the condition) is satisfied. When the child requests only a hot dog, the condition is satisfied. Four choices only: two fail (both, neither), two satisfy (hamburger, hot dog).

In the case of the inclusive sense of the logical constant "or" when the parent asks the child if they want peas or carrots with their dinner and the child says neither, they have failed to satisfy the condition established by the parents inclusive use of either peas "or" carrots. When the child requests only peas, only carrots or both, the parents condition is satisfied. Four choices only: one fails (neither), three satisfy (peas, carrots, both).

What is to be made of "if (true or false), then (true or false)"?

What is to be made of "(true or false) implies (true or false)"?

What is to be made of "(true or false), then (true or false)"?

What is to be made of "(true or false) → (true or false)" and such?

For ease of investigation, I will limit the next section to the P → Q notation and we can look at all the possible combinations:

  1. true  →  true.
  2. true  →  false.
  3. false →  true.
  4. false →  false

So here then is my "intuitive" sense of these options.

1) "If true, then true"
We start with a true proposition and we end with a true proposition. Seems pretty straightforward that the entire statement is true. In the expression of "true implies true" it is tempting to imagine that the truth of the former 'true' implies the truth of the latter 'true, however, "implies" is misleading in this senses of "causes the latter to be true" and "causes the supposition to be true." It seems awkward to imagine any positioning of two true propositions that would result in false, so this supposition made of to two 'true's is true simply that we start with true and "...then true." At the very least the statement values are consistent.

2) "If true, then false" We start with true and end up with false. Why would we end up with false if we start with true? That seems confounded. In the expression of "true implies false" this feels almost embarrassing without even imagining that "implies" means "causes." In the sense of "true, then false" it is as if a mistake has been made or something working has broken; an intermittent problem has returned; the lights have gone out in the neighborhood. How could this supposition be other than false?

3) "If false, then true" Hmm... at first I feel like starting out false, it is disingenuous that we end up with true, but there we are: "then true." Can we not start in ignorance and confusion and come to knowledge and analysis by distinction of utterance as true or false? "False implies true" seems counter-intuitive in the incorrect sense of implication as cause, but "false then true" is as if the lights in the neighborhood, the television, refrigerator and all things electrical have started working again. How could this supposition be other than true?

4) "If false, then false" We start with false and end with false and that we end up with what we started with, this feels true. "False implies false" in the incorrect sense of implication is not so troublesome here in that it is true that false is false. "False, then false" is at the very least consistent and it is true that these are both false. See how we have come to truth from falsehood?

And such is my statement of an intuitive sense of supposition and two logical constants. I will leave it to you to consider the remaining forms with these descriptions of my intuitive senses.

What follows will be a more explicit articulation of supposition.

The supposition "if a proposition, then an other proposition" can be analyzed by explicitly articulating conditions which would satisfy the truth or falsity of the supposition.

For the purposes of this examination the previous sentence is equivalent to, "The supposition 'if a proposition, then an other proposition' can be [analyzed; considered; evaluated; computed] by conditions which would satisfy *its* truth or falsity." Please note the former expansion in the brackets and the latter italicized reduction as well the removal of "explicitly articulating". We shall examine the constituent propositions and their relation. For ease of distinction "a proposition" may hereafter be referred to as the "former proposition" and "an other proposition" may hereafter be referred to as the "latter proposition".

The constituent propositions boundaried by the supposition (or, suppositional form) may be either true or false. Within the scope of this consideration I will consider any proposition with a nil truth value to be false. So far we have a supposition constructed from two propositions: a proposition, the former, and, an other proposition, the latter. The two propositions are distinct and not equivalent or identical. Keep this in mind as I shall reduce the propositions to true or false and may not always explicitly state "true proposition" and "false proposition". Lastly, the two propositions may be only either exclusively true or false neither proposition may be both.

In addition to the constituent propositions, the supposition expresses a relationship of the two propositions. This relationship is "if true proposition or false proposition, then true proposition or false proposition" and the aim of this investigation: how then shall we render a truth value from the supposition of two propositions which may be either exclusively true propositions or false propositions? I will also address - tho likely not convincingly - the question of, what does the suppositions truth value mean for the constituent propositions? (Hint: nothing whatsoever) I.e. my aim is also at the question, does the supposition change the truth value of the constituent propositions? In short, no, it does not. Also, in doing so I will comment upon the question, does the supposition cite logical (or physical) cause? The answer here as well is no.

For purposes of this investigation we are only considering a unidirectional relationship such that the resulting truth value of the supposition may sufficiently be said to be derived from the formal positioning (relationship) of the former propositions and the latter proposition. The suppositions truth value has not one iota of relevance to the relationship of the content of either the former or latter proposition. Also, we are not here concerned with whether or not the latter proposition necessarily implies the former proposition, i.e. we are not investigating the suppositionally bidirectional conjunction "if a proposition, then an other proposition, and, if the latter proposition, then the former proposition". Lastly, for the purposes of this investigation, it may also be said that the supposition expresses a relationship between or of or bounded to the two propositions by the suppositional form, but, as stated just prior, we are not investigating every relationship between or of or bounded to these two propositions when placed in proximity within the suppositional speculation. again, it is the suppositional speculation that binds the two propositions together and constructs a boundary within which the suppositional truth value can be rationally assessed. So on with rationally assessing the constituent propositions and then to rationally assessing the supposition.

The former proposition may be either true or false.

The latter proposition may be either true or false.

Is it enough to state that, "if the former is true and the latter is true, then the implication expressed by [derived from] the supposition 'if a proposition, then an other proposition' is also true."? Have we only presumed as much? Are we not simply expressing the conclusion redundantly by repeating the supposition "if something, then some other thing"? It may appear so. Such is a pernicious difficulty with the fundamental and trivial. Note that I do not use trivial in the pejorative sense of pertaining to useless information but in the sense that stating that the statements "the statement that 'Obama is President' is true" and "Obama is President" are true statements, and yet the two statements are distinguishable by the trivium of logic, reason and rhetoric.

Another strategy to avoid conjecturing the suppositional conjecture that "'if true proposition, then true proposition' is true" is to simply state example cases where both propositions which the supposition is constructed with are true, i.e. examples following the form "if true proposition, then true proposition", and then assess if the suppositional statement is true. In stating the shortened form of "if true, then true" we can immediately see that this makes trivial sense (in the pejorative sense). But are we convinced that "if a true proposition, then an other true proposition" is true? Maybe not. So before we conclude that "if true, then true" is true, let's examine a couple examples which fit the form.

  1. if all dividends require financing, then an Euclidian circle is all points upon a plane equidistant from a central point.

  2. if the Sun is ~93 million miles from the Earth, then Angelina has filed for divorce from Brad Pitt.

It is trivially (in the sense of the trivium) worth noting that the first example is axiomatic (the former self-evident by definition and the latter axiomatic) and the second empirical (the former brute and the latter institutional); and, that all four propositions are true.

It is non-trivially worth noting by these examples that a supposition is not citing cause - whether physical or logical. The former does not cause the latter to be true in the way that the cold may cause someone to shiver. That the cold may cause someone to put on a sweater, however, we can see that the relationship which the supposition provides a minimal bound upon results in a rationally assessable, if not trivial, truth value. I will leave it to you to decide upon this most recent sense of "trivial". How then to cite proof that this rational assessment is reasoned by logic and not an imposed rationalizing of the psychopathological varietal? It is said that axiom is only true because we do not allow it to be otherwise, but is logic nothing more than dogma? Let us consider the following instead of the standard "if..., then..." expression.

Here I offer a formulation of a supposition without the logical constants "if" and "then": "I suppose that because [since] the Earth is an oblate spheroid that when an object gets dropped mid-air it will fall to the ground." Note that this supposition does not correlate to cause tho the constituent propositions are true and the content they describe have causes. What of the speculative supposition then? Do not these kinds of statements presume uncertainty? Do they presume uncertainty even when certainty may be achieved (I am hesitant to say "obtained" as tho knowledge of the empirical and axiomatic may be obtained, knowledge is not a dissolutive salve for every skeptics every doubt.) Also, if enough to inspire doubt, is the mere form of supposition enough to convince of a falsity where there is otherwise a truth? These are not questions this investigation seeks to address conclusively, tho I do ask that they be considered inasmuch as the reader is unconvinced of the rationally assessable truth values of suppositional propositions of the form, "if a proposition, then an other proposition". At this point I think we've reached a limit of useful investigation by only considering the former and latter propositions when they are both individually true propositions. What of "if true, then false"? For example:

  • If dividends require financing, then 2+2=5.
  • If Obama is President, then all swans are white.

Whether or not there is logical or physical cause (and there isn't), the truth value of a supposition which concludes a falsehood cannot be true. We simply do not allow it. Note that what we are not saying with this supposition may shed some light on the computational logistics of rendering truth value: we are not saying "if true then true must be false". We are saying that the former proposition is true; that the latter proposition is false; and that the relation expressed by the "if..., then..." supposition boundary containing "the former is true" and "the latter is false" is a false relation. This last point might make more sense in ordinary language expressed in a different order: that the relation expressed by the "if..., then..." supposition boundary containing "the latter is false" and "the former is true" is a false relation. It might not.

What of "if false, then true"?

  • If 2+2=5, then dividends require financing.
  • If all swans are white, then Obama is President.

In both cases that the latter is true the supposition is true. Again, is this merely a reformulation of an "if..., then..." offering circular proof that "if..., then..." is true (because it's true?)? No. It is demonstration. Note here that a suppositional proposition is not an argument and all we are examining here is the relation of the two statements bound in the suppositional form. We are not saying "if false then false must be true." We are not citing cause, i.e. dividends requiring financing does not make 2+2=5. While "imply" in this instance might seem to suggest that "if false, then true" means "false implies true" or that "'if false, then true' is true" means true supposition makes false proposition true. Neither of those implications are implicit to the propositional relationship in a suppositional condition.

What of "if false, then false"?

  • If all bachelors require pedicures, then 2+2=5.
  • If CarrotTop is President of these United States, then all ravens are orange.

How, it must be asked, could it be reasonably asserted that any supposition constructed from false propositions be true? Simply by noting that it is true that both propositions are false.

At this point I will leave it to you to consider the equivalence cases and hope that my answer is enough to convince you that while I appreciate apprehension regarding formalities, their is much to be gained from the expediency of the formal notation, and, something to be gained by laboring their point.

Q.E.D.

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P → Q means: If P is true then Q is true. Implied is further: If P is not true, then I don't say anything about Q.

P ∨ ¬Q means: At least one of the two statements P and ¬Q is true; they are not both false. If one of the statements in an "or" is true, then we have no information, no requirement for the other statement, because the whole "or" statement is already true. But if one of the statements is false, then this means we have a requirement for the other statement; the other statement must be true because they cannot both be false.

So if P is true, then nothing is known about ¬Q. And if ¬Q is true, then nothing is known about P. But if P is not true, or ¬Q is not true (that is Q is true), then we have a requirement for the other statement.

Compare with what I said about P → Q: "If P is not true, then I don't say anything about Q.". But now we have "If P is not true then ¬Q must be true". So one statement we have no requirements if P is not true, the other statement we have a requirement. Both statements cannot be the same.

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One way to view this intuitively is to construct a graph.

The following is a graph of a relationship of P to Q that represents "P → Q". The domain of the graph is the part in the square. P is the beige oval. Q is the green oval. Everything else in the domain is yellow.

Note that the Q oval contains the entire P oval. That is what makes the conditional work.

enter image description here

Consider what happens when the P and Q ovals overlap such that the P oval is not completely contained within the Q oval. Then the conditional fails because there is now a region where we have P and not Q.

enter image description here

To consider what happens when not P ∨ ¬Q look for the regions on the graph that this is true.

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