Help me understand rules of inference (logic)

[r0bHadz]

My school uses Rosens book which is an awful book, would not recommend. Can anyone please help me solidify if what I am saying is true:

If you are presented an argument, to even take this argument into consideration, you must prove whether the argument is valid or not. After this is done, you can move on to determining if the conclusion is true.

 

[r0bHadz]

Or, since rules of inference are based on tautologies, if we can show an argument is valid, then the conclusion must be true?

 

[Mark44]

If you are presented an argument, to even take this argument into consideration, you must prove whether the argument is valid or not. After this is done, you can move on to determining if the conclusion is true.

I prefer the term logical proposition, which can be true or false.

The only situation in which a logical proposition is false is when the hypothesis (the "if" part) is true but the conclusion (the "then" part) is false.
For example,
If x = 2, then x² = 4 (proposition is true)
When x happens to be equal to 2, the above is a true proposition. If x happens to be any other numeric value, the above is still true.

"All dogs have four legs." This is false. As a counterexample, my neighbors across the street have a dog with only three legs.
The statement above could be rewritten more symbolically, with qualifiers, as $\forall d \in \{\text{dogs}\}, \text{d has four legs}$. This could also be written as an implication: $\text{d is a dog} \Rightarrow \text{d has four legs}$

My counterexample of the dog across the street makes the hypothesis true -- Jackson is indeed a dog -- but the conclusion that Jackson has four legs is false, therefore the proposition is false.

Some valid propositions are true, but meaningless (i.e., fatuous). For example, $x < 0 \text{ and } x > 0 \Rightarrow \text{The sky is polka-dotted}$

Or, since rules of inference are based on tautologies, if we can show an argument is valid, then the conclusion must be true?

If the conclusion is true, the logical proposition is true, regardless of the hypothesis.

 

[r0bHadz]

Concise and well explained. Thank you. You should give Rosen tips on explaining things

 

[Mark44]

Which Rosen textbook are you using? I found "Discrete Mathematics and Its Applications," 7th Ed., on the web, and checked out the first couple of sections of Ch. 1. The book seems OK to me.

 

[r0bHadz]

Which Rosen textbook are you using? I found "Discrete Mathematics and Its Applications," 7th Ed., on the web, and checked out the first couple of sections of Ch. 1. The book seems OK to me.

That's the one I am using! I guess I prefer Grimaldi's (not sure if you've read his discrete math book) style more.

 

[Mark44]

Rosen is no doubt smart enough, having received a PhD from MIT in mathematics. His ratings for an earlier version of the book you're using are pretty good, about 3.8 on a scale of 5.
I'm not familiar with the Grimaldi book, and in fact don't have any discrete math books. I have plenty of math books and CS books, mostly C, C++, but a couple on C#, as well as a bunch on assembly languages.

 

[sysprog]

My school uses Rosens book which is an awful book, would not recommend.

Rosen's Discrete Mathematics and its Applications (7th ed, 2012) (pdf) is very good for fully explaining things like why 367 is the exact minimum number of persons that must be present in the same room for establishment with 100% certainty by the fact alone of that number being present that at least 2 persons in that room have the same birthday (pigeonhole principle), and some considerably more advanced concepts, but that book is too broad in its scope to be able to provide a more expansive and less condensed introduction to logic. Its approach to logic fits well with its general tenor as a textbook for its intended topic areas and anticipated readership.

Can anyone please help me solidify if what I am saying is true:

If you are presented an argument, to even take this argument into consideration, you must prove whether the argument is valid or not. After this is done, you can move on to determining if the conclusion is true.

This is saying that evaluation of the validity of an argument is necessary, prior to being able to be sure of reliability in determining, from learning whether the premises are true, whether the conclusion is true.

In other words, the first question in evaluating an argument is not, "are the premises true", nor is it "is the conclusion true"; rather, it is "do the premises genuinely entail the conclusion" (valid), or put conversely, "is it possible that the premises could be true, while the conclusion is false" (invalid).

A basic form of valid argument is a "sufficient conditions" (modus ponens) argument.

Once we know that such an argument is valid, i.e. that its premises genuinely entail its conclusion, then we need only determine that the premises are true in order to be certain that the conclusion is true; however, if we find that one or more of the premises is not true, we cannot be sure, from learning that alone, that the conclusion is false.

If the premises and conclusion of a valid argument were structured so that the falsity of one or more of the premises would entail a negation of the conclusion, that would be "necessary conditions" (modus tollens) argument.

Or, since rules of inference are based on tautologies, if we can show an argument is valid, then the conclusion must be true?

No. Valid arguments can have conclusions that are either true or false, depending on whether their premises are true or false. A valid argument might have a false conclusion, but only if at least one of its premises is false.

The only arguments that are invalid are those such that it is possible that the premises are true while the conclusion is false. Showing (proving) that an argument is invalid requires point-by-point demonstration that it is possible for the premises to be true while the conclusion is false. A true conclusion cannot be used to demonstrate the invalidity of an argument.

A pair of principles embraced in the discipline of logic are, that a false conclusion is entailed by anything, and, that a false premise entails anything. These principles don't always square well with ordinary inferential understanding. In everyday discourse, we're accustomed to there being a confluence between logical and physical necessity, and some rationally apprehensible relation between the materialities of the premises and the conclusion, while in pure logic, there is not necessarily any causal or other non-logical relation asserted by a statement of a conditional (if-then) nature.

You might find some of the material in the following wikipedia articles, and some of that in others referenced in them, helpful:
https://en.wikipedia.org/wiki/Validity_(logic)
https://en.wikipedia.org/wiki/Logical_consequence
https://en.wikipedia.org/wiki/Material_conditional
https://en.wikipedia.org/wiki/Paradoxes_of_material_implication
https://en.wikipedia.org/wiki/Necessity_and_sufficiency
https://en.wikipedia.org/wiki/Wason_selection_task

For something intended more as a general introduction to Logic, you might look at Suppes' Introduction to Logic (1957) (pdf).

 

[WWGD]

Yet another perspective.Rules of logic are truth preserving so if you start with a true statement ( this being sentence logic) you will arrive at a true conclusion.

 

[WWGD]

Maybe an important issue in Sentence logic, differentiating it from 1st order logic is that in SL , the structure of the sentence does not matter; all that matters is it's truth value. In First order, the structure matters. Most of Mathematics can be expressed in first order logic. Notice there is a difference between syntax and semantics.

 

[sysprog]

Yet another perspective.Rules of logic are truth preserving so if you start with a true statement ( this being sentence logic) you will arrive at a true conclusion.

That alone doesn't tell you whether the premises genuinely entail the conclusion.

 

[sysprog]

Maybe an important issue in Sentence logic, differentiating it from 1st order logic is that in SL , the structure of the sentence does not matter; all that matters is it's truth value. In First order, the structure matters. Most of Mathematics can be expressed in first order logic. Notice there is a difference between syntax and semantics.

Propositional/Sentential Logic and Predicate/Quantificational Logic are both first order logics. Both have syntactical rules. Sentence structure matters in both.

Example:

$P \land Q \lor T$ is just as ambiguous as $Pa \land Qa \lor Ta$, or $(\forall x)[Px \wedge Qx \vee Tx]$,
because
$(P \land Q) \lor T$ is different in meaning from $P \land (Q \lor T)$,
just as
$(Pa \land Qa) \lor Ta$ differs in meaning from $Pa \land (Qa \lor Ta)$,
or
$(\forall x)[(Px \wedge Qx) \vee Tx]$ has different meaning from $(\forall x)[Px \land (Qx \lor Tx)]$.

 

[WWGD]

Propositional/Sentential Logic and Predicate/Quantificational Logic are both first order logics. Both have syntactical rules. Sentence structure matters in both.

Example:

$P \land Q \lor T$ is just as ambiguous as $Pa \land Qa \lor Ta$, or $(\forall x)[Px \wedge Qx \vee Tx]$,
because
$(P \land Q) \lor T$ is different in meaning from $P \land (Q \lor T)$,
just as
$(Pa \land Qa) \lor Ta$ differs in meaning from $Pa \land (Qa \lor Ta)$,
or
$(\forall x)[(Px \wedge Qx) \vee Tx]$ has different meaning from $(\forall x)[Px \land (Qx \lor Tx)]$.

But the truth value of Qa depends on Q, and on a. Does a satisfy preficate Q? For a sentence S we can assign or assume any truth value. And the two are different in important ways: Sentence logic is decidable, through truth tables, preficate logic is not.

 

[WWGD]

That alone doesn't tell you whether the premises genuinely entail the conclusion.

Well, if by entailment you mean necessarily following, then this is true for valid arguments, those in which if the premises are true the conclusion must be true. If the last is not the case then the argument is invalid.

 

[WWGD]

Propositional/Sentential Logic and Predicate/Quantificational Logic are both first order logics. Both have syntactical rules. Sentence structure matters in both.

Example:

$P \land Q \lor T$ is just as ambiguous as $Pa \land Qa \lor Ta$, or $(\forall x)[Px \wedge Qx \vee Tx]$,
because
$(P \land Q) \lor T$ is different in meaning from $P \land (Q \lor T)$,
just as
$(Pa \land Qa) \lor Ta$ differs in meaning from $Pa \land (Qa \lor Ta)$,
or
$(\forall x)[(Px \wedge Qx) \vee Tx]$ has different meaning from $(\forall x)[Px \land (Qx \lor Tx)]$.

But first or the order of the logic refers to quantification. What quantification are you doing in Sentence logic? And, yes, each has its own syntax , I never said otherwise.

 

[sysprog]

But the truth value of Qa depends on Q, and on a. Does a satisfy preficate Q? For a sentence S we can assign or assume any truth value. And the two are different in important ways: Sentence logic is decidable, through truth tables, preficate logic is not.

Predicate Logic without quantifiers is decidable, but does not have the power to bind variables. With quantifiers, it is generally semidecidable, with some restricted versions that are decidable, and some extended versions that are undecidable.

But first or the order of the logic refers to quantification. What quantification are you doing in Sentence logic?

Sometimes, informally, Sentential or Propositional logic is called zeroth order logic. The Wikipedia article begins with the following superficially inconsistent statement:

Zeroth-order logic is first-order logic without variables or quantifiers.​

Some authors, including the luminary Terence Tao, use the term 'zeroth order logic' to refer to a moderate extension of Propositional logic.

From Terence Tao's An Epsilon of Room, II: pages from year three of a mathematical blog (Author's preliminary version made available with permission of the publisher, the American Mathematical Society):

1.4.2. Zeroth-order logic. Propositional logic is far too limited a language to do much mathematics. Let’s make the language a bit more expressive, by adding constants, operations, relations, and (optionally) the equals sign; however, we refrain at this stage from adding variables or quantifiers, making this a zeroth-order logic rather than a first-order one.​

This article can also be found at the author's earlier page entitled The completeness and compactness theorems of first-order logic.

In that earlier version of the article, the term 'zeroth order logic' is linked to the Wikipedia article, which currently references Terence Tao's AMS published article.

While I recognize the utility of allowing zero to be used as an ordinal in some special cases, e.g. exponentiation, sometimes referred to as taking a number to 'the zeroth power', I also suppose that what is meant by 'zeroth order logic' is a subset of first order logic that does not include variables or quantifiers, with the term 'zeroth order logic' used as an expository convenience.

I think that it is more correct English, albeit less expositorily convenient, to say that while all logics are of order ≥ 1, what is ordinarily called sentential or propositional logic is a first-order foundational subset of first-order predicate and quantificational logics.

Historically, Pierce in his 1885 paper, made reference to "first-intentional logic", in which the quantifiers range only over the atomic individuals of the universe of discourse; by which reference he intended to distinguish from his use of the term "second-intentional logic" later in the paper, in which the quantifiers are permitted to range over predicates.

I use the term 'first order logics' in the plural to refer to any logics that are not second or higher order logics; however, I must defer to Prof. Tao in acknowledging that his distinguishing designation and description of 'zeroth order logic' is useful, perspicuous, reasonable, and if only by virtue of his having used it, authoritative.

 

[WWGD]

All that seems too advanced for a mostly intro level post. I thought I was posting some basic results.

 

[sysprog]

All that seems too advanced for a mostly intro level post. I thought I was posting some basic results.

Fair enough, WWGD; however, as you know, the advanced matters are present in the simplicities.

Well, if by entailment you mean necessarily following, then this is true for valid arguments, those in which if the premises are true the conclusion must be true. If the last is not the case then the argument is invalid.

Sometimes, when trying to detect invalidity, we can take a shortcut by looking first at a truth-value assignment in which the conclusion is false, and then checking whether that allows the premises to be true.

The Wason selection task test is susceptible to truth-table analysis; however, test-takers who, instead of using any similarly exhaustive procedure, make a conscious and deliberate mental search for 'all and only' data that could or can falsify the rule being tested, might perform as a group better than those who respond on immediate intuition.

Each card has a number on one side, and a patch of color on the other. Which card or cards must be turned over to test the idea that if a card shows an even number on one face, then its opposite face is red?