Very Basic Terms of Logic

Very Basic Terms of Logic
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Posted Mar 6, 2004 - 7:08 AM:

We have tried to compile a list of general terms in logic. Of course, this is, quite simplify a generalized introduction to certain Kripke and other systems. Please feel free to add to this thread with relevant information. However, debate and superogatory posts should be avoided; we would like to keep this strictly as an informative thread.

Naturally, this thread doesn't include nearly enough terms to sufficiently understand logic, but it should clarify some of the common philosophical terms, as well as a very brief introduction to symbolic (formal) and modal logic. When I was starting out on logic (and that wasn't too long ago), I would have thought a thread like this to be ideal. Hopefully, it will be a help to others.

Very Basic Terms of Logic

Logic might be minimally defined as:

The study of the principles of reasoning, especially of the structure of propositions as distinguished from their content and of method and validity in deductive reasoning.
http://dictionary.reference.com/search?q=logic

Axiom

An axiom, in its original meaning, signifies something that is self-evident. An axiom, however, has varying meanings depending upon the system. Mathematics (an axiomatic deductive system) does not necessarily propose that axioms are self-evident, but rather, it uses them as a starting point; some "common ground". That is, an accepted truth without proof.

Peano's induction axiom: every number has a successor. In Epistemology, an axiom conforms to the former definition of it - being a self-evident truth. Axioms, particularly in logic, are sometimes called postulates or premises.
Also see http://en.wikipedia.org/wiki/Axiomatization

Deductive argument

In a deductive argument, the truth of the premises assures the truth of the conclusion, and the falsity of it is impossible.. Mathematics is a deductive axiomatic system, since it is based on axioms. If we take the commonly used example, then we can see what we mean:

1. All footballs are balls.
2. All balls are physical objects.
3. Therefore, all footballs are physical objects.

We might review this argument as:

1. All As are Bs
2. All Bs are Cs
3. Therefore, all As are Cs

Now, we might take (1) and (2) as axioms. In this case, (3) logically follows from (1) and (2). If (1) and (2) are true, then it must be the case that (3) is also true. And, that is all that a deductive argument proposes.

Deduction & Induction -- William M.K. Trochim.
Confirmation by Contrapositive Instances -- Rick Garlikov


Inductive argument

An inductive argument is one in which the premises support the conclusion, but do not necessitate it. In a good inductive argument, the truth is merely probable. An argument with inductive reasoning makes the milder claim. For example, an inductive argument:

1. I have seen four million swans, and they have all been white.
2. Therefore, all swans are white.

As we can see, simply because one has not seen a black swan, it does not follow that they do not exist. It has been debated in the past whether it is sufficient to call inductive reasoning valid, because it doesn't reach its conclusion by laws of logic or inference. However, some arguments take that part. A binding, formal logical argument is generally created by deductive methods. Scientific evidence tends to be inductive in nature.

Characteristics of Inductive Reasoning
Inductive Logic -- Branden Fitelson


Truth, Soundness and Validity

There is a necessary distinction between soundness and validity in deductive argument. An argument is valid if all the inferences (and the conclusion) follow logically from the axioms, or, from one another. That is, if an argument is valid, the truth of its premises will ensure the truth of its conclusion. If an argument is created, and the inferences as well as the conclusion logically follows from the axioms, then it is valid.

An argument is sound if (and only if) the reasoning in an argument is logically valid and all its premises are true. If this is the case, then the argument is sound or true. Let's take an example:

Premises:
1. Robert Pires plays football professionally.
2. If one plays football professionally, then they are a footballer.

Conclusion:
3. Robert Pires is a footballer. -- Substitution from 1 and 2.

(1) and (2) here are axioms. The axioms in this argument are true, and the conclusion (3), logically follows from one and two by substitution. So therefore, this argument is both valid and sound. An example of an unsound, yet valid argument:

Premises:
1. All Shakespeare books are blue.
2. All blue books are a good read.

Conclusion:
3. Shakespeare is a good book. -- Substitution from 1 and 2.

Again, in this case (1) and (2) are axioms. This argument is valid because the conclusion (3), once again, follows from the axioms. However, it is unsound since we do not accept that all of Shakespeare's books are blue. Neither might we accept that all blue books are a good read. If one were to agree with (1) and (2), then the conclusion (and the whole argument) would be true for them.

Valid and Sound -- PROF. A. SOBLE
Truth of Statements, Validity of Reasoning -- Peter Suber


Fallacy

A fallacious argument is one that is based on an invalid or false inference. But generally, a fallacy is used to refer to a typical fault of argumentation. Fallacies in a deductive argument are easy to spot, since the inferences will not logically follow from the premises. An example:

1. If God created the world, then it would have order.
2. The Universe has order,
3. Therefore, God created the Universe.

This argument is faulty and invalid, at least, because of premise (3) is reached fallaciously. We have assumed too much in reaching our conclusion. We might say that the Universe has order because of the probability of it doing so. The name given to such a fallacy is affirming the consequent, but more about that later.

There are various types of logical fallacies of formal debate. A decent introduction to these can be found at these links:

Logical Fallacy -- Wikipedia
List of Logical Fallacies -- The Atheism Web
Logical Fallacies -- The Nizkor Project
Stephen Downes' guide to Logical Fallacies


Symbolic Logic

Presenting an argument in formal logic often makes it easier to view. Partly because fallacies become apparent quite quickly. Anyways, the basic symbols:

We'll take "::" to mean "therefore". Ideally, it should be a triangular formation of three dots (wink modal operators. For example:

<>p

This means that p is possibly true. According to p, there exists at least one possible world in OmniSim where p is true. An example of a possible (but not necessary) statement would be that Abraham Lincoln was not president of the United States. This is indeed possible, but not necessary. There exists a world in OmniSim (even if it is not our own) where Abraham Lincoln was never such a President, because such a world is possible. However, there also exists at least one possible world where Abraham Lincoln was in fact the President of the U.S. (e.g. our own). Therefore, the statement is possible even if it is not true in every possible world. A statement that's true in some possible worlds but also false in some other possible worlds is called a contingent statement (because it can be either true or false). Another example:

[]p

This means that p is necessarily true. That is, it is true in all possible worlds. Using the supercomputer analogy, statement p would be true in all of OmniSim's simulated worlds. Examples of a necessary truth would be 2 + 2 = 4, the law of noncontradiction (this law states that for any specified proposition p, it is impossible for both p and not p to be true; e.g. it is impossible for me to exist and to not exist at the same time), and the law of excluded middle. All of these things are true in all possible worlds. Note that a possible statement (one that's true in at least one possible world) can also be necessary (true in all possible worlds), and all necessary statements are possible. However, a contingent statement (one that's true in some possible worlds, and false in some others) cannot be a necessary statement, and no necessary statements are contingent. A brief summary of the terms that describe statements can be found below. All three of those terms belong to the category of modal status, because they describe statements as being necessarily true/false, possibly true/false, or contingently true/false. The table after those definitions is a summary of the basic modal symbolic stuff we've got so far."
http://www.angelfire.com/mn2/tisthammerw/rlgnphil

So basically,

[]p = Necessarily p –– p is true in all possible worlds
<>p = Possibly p –– p is true in some or one possible world(s)
~[]~p = Note that this is equivalent to <>p –– "p is true" is not impossible.
p = p is true in actually.

Contingency

A contingent statement is something that might be true or false. Something contingent exists in some possible worlds, and not in other possible worlds. For example, one could imagine a world where David Beckham did not exist. Therefore, David Beckham's existence is contingent.

More on Modal Logic:
Modal Logic -- James Garson at StanfordA discussion of modal logic by John McCarthy

Logical Formulas


Modus Tollens and Modus Ponens
Section by Libertarian

A modus tollens is a logical formula or rule. It, along with the modus ponens, comprise two very common ways to draw inferences. Suppose you have two propositions, P and Q. P -> Q means that Q is a logical consequence of P. You can read it several ways: P implies Q; Q follows from P; If P is true than Q is true; and so on.

P -> Q is called a conditional implication. That is, the truth of Q is conditional upon the truth of P. As an example, let's take as given that Bob wears a red shirt only on Sunday. We can form an implication this way:

P = "Bob is wearing a red shirt"
Q = "Today is Sunday"

P -> Q
Bob is wearing a red shirt -> Today is Sunday.

That means "If Bob is wearing a red shirt, then today is Sunday". We have made a reasonable assertion because we know that Bob wears a red shirt only on Sunday. For convenience, let's call the P side of the implication the antecedant. And let's call the Q side of the implication the consequent.

"Bob is wearing a red shirt" is the antecedant. And "Today is Sunday" is the consequent.

The rule of modus ponens states that if the antecedant of an implication is true, then so is the consequent. Antecedant implies consequent. Antecedant is true. Therefore, consequent is true. Or...

P -> Q; P; ::Q

So, the above says, "P implies Q; P is true; therefore, Q is true." Using our example:

If Bob is wearing a red shirt, then today is Sunday; Bob is indeed wearing a red shirt; therefore, today is indeed Sunday. That is a modus ponens.

The rule of modus tollens states that if the consequent of an implication is false, then the antecedant is false. That is, if the Q side is false, then the P side is also false. Antecedant implies consequent. Consequent is false. Therefore, antecedant is false. Or...

P -> Q; ~Q; ::~P

So, the above says, "P implies Q; Q is false; therefore, P is false." Using our example:

If Bob is wearing a red shirt, then today is Sunday; Today is indeed not Sunday; therefore, Bob is indeed not wearing a red shirt. That is a modus tollens.

Now, it happens that there are two common logical fallacies that correspond to the modus ponens and the modus tollens. The fallacy that corresponds to the modus ponens is called denial of the antecedant. The fallacy that corresponds to the modus tollens is called affirmation of the consequent.

A denial of the antecedant fallacy takes this form:

P -> Q; ~P; ::~Q

Compare the fallacy above to the rule below:

P -> Q; P; ::Q

Thus, the following is an invalid inference:

If Bob is wearing a red shirt, then today is Sunday; Bob is indeed not wearing a red shirt; therefore, today is indeed not Sunday.

In denying the antecedant, we have assumed too much. All we know is that Bob wears a red shirt only on Sunday. But we do not know anything about what is true when Bob does not wear a red shirt. He might wear a green shirt or a blue shirt on Sunday. It's just that he never wears a red shirt on any other day.

An affirmation of the consequent fallacy takes this form:

P -> Q; Q; ::P

Compare the fallacy above to the rule below:

P -> Q; ~Q; ::~P

Thus, the following is an invalid inference:

If Bob is wearing a red shirt, then today is Sunday. Today is indeed Sunday, therefore, Bob is indeed wearing a red shirt.

In affirming the consequent, we have again assumed too much. Sunday is indeed the only day on which Bob wears a red shirt, but he might also wear a green shirt or a blue shirt on Sunday. We can't say that Bob is wearing a red shirt just because today is Sunday. Interestingly, you will very often spot these two fallacies in arguments made by theists and atheists. Affirmation of the consequent is a very common fallacy among theists. And denial of the antecedant is a very common fallacy among atheists.

Consider this argument by Joe Theist:

If God created the universe, then we would expect the universe to be organized according to rules. And in fact, the universe is indeed organized according to rules. Therefore, God created the universe.

P = "God created the universe"
Q = "The universe is organized"

P -> Q; Q; ::P — fallacy!

Now, consider this argument by Sam Atheist:

If God were to appear to me personally, then that would prove that He exists. But God has not appeared to me personally. Therefore, God does not exist.

P = "God appears to me personally"
Q = "God exists"

P -> Q; ~P; ::~Q — fallacy!


Substitution

In simplistic symbolic language, substitution takes these forms:

P & Q, Q -> Z :: P & Z
P v Q, Q -> Z :: P v Z

Disjunctive Syllogism

Disjunctive Syllogism takes this form:

P v Q; ~P :: Q

If we start again with the above example of Bob and his jumper, we'll get:

Either Bob is wearing a red shirt, or it is Sunday. Bob is not wearing a red shirt; therefore, it is Sunday.

Disjunctive Syllogism –bucks.edu
here.

Basic Laws and Principles

Some of the main "laws" of predicate calculus; and, of course, generally considered as some of the fundamental laws of logic. Below is Principle of Excluded Middle (PEM), the Law of Non-Contradiction (PNC) (or the "Law of Contradiction"), the Principle of Bivalence (PB) and the Principle of Exclusive Disjunction for Contradictories (PEDC). All of these are often confused between each other, so consideration is requiring when using them.

Principle of Excluded Middle

The law or principle of excluded middle (Latin: Tertium non datur -- "the third is not permitted") is among the most famed of logical laws, partly because of what it proposes and partly because of its practical uses. It states that for any given proposition, p, either p or not-p must be true. That is:

p v ~p

This can be applied to practically anything; for example:
  • I am a footballer or it is not the case that I am a footballer.
    The chair is blue or it is not the case that the chair is blue.
    That man is a thief or it is not the case that that man is a thief.


Principle of Non-Contradiction

The law/principle of Non-Contradiction suggests that contradictions do not occur; that two propositions that are contradictory cannot both be true. It was famously stated by Aristotle: "One cannot say of something that it is and that it is not in the same respect and at the same time." That is, it states "not-both":

~(p & ~p)

An example:
It cannot be the case that I am a human and that I am not a human.

Principle of Bivalence

The Principle of Bivalence is alike to the PEM. However, instead of suggesting that a proposition is either true or not-true, it goes one step further to suggest that a proposition is either true or it is false. That is, that every proposition must have a truth value. This perhaps is one of the most rejected of principles in more modern systems of logic; it no longer holds in intuitionistic logic. A common disproof of the law of bivalence is easily constructed in intuitionistic logic:

Define ¬A as ( A → contradiction ) i.e., a false statement is one from which one can derive a contradiction. This is the standard intuitionistic definition of what it is for a statement to be false.

So using this definition, if we have ( A ∧ ¬A ) this can be written as ( A ∧ ( A → contradiction ) ) → contradiction

So ( A ∧ ¬A ) → contradiction

So ¬ ( A ∧ ¬A )
http://en.wikipedia.org/wiki/Principle_of_bivalence

Principle of Exclusive Disjunction for contradictoriness

This is a principle that has often been assumed in Logic, but has not been given a name recently; until Suber. He defines it as the law that proffers that of two propositions, p and ~p, "exactly one is true, exactly one is false". More can be read on it here:
http://www.earlham.edu/~peters/courses/logsys/pnc</A]

Links and sources:
Non-Contradiction and Excluded Middle[/url]
Principle of Bivalence – Wikipedia
Law of Excluded Middle – Wikipedia
Law of Excluded Middle
Intuitionism: the significance of bivalence


Other - Links

A Priori and A Posteriori -- Jason S. Baehr at IEP
Excellent Introduction to Logic with Exercises -- Stefan Waner and Steven R. Costenoble
Logic - A recommended reading list
Modal Logic - A recommended reading list 1, 2.
Logical Symbols -- Philosophypages
Peter Suber, "Symbolic Logic"
Study guide to Philosophy -- Philosophypages
Guide to Philosophy on the Internet -- Compilation by Suber
The Internet Encyclopedia of Philosophy
Columbia Encyclopedia, Sixth Edition -- Bartleby
Introduction to the Hegelian thesis/antithesis/synthesis -- Tobias
19th Century Logic between Philosophy and Mathematics -- Volker Peckhaus
A vast range of links on the Internet to Philosophy -- Thomas Ryan Stone
History of Logic -- Encyclopædia Britannica
Merriam Webster Online Dictionary
Stanford Encyclopedia of Philosophy
Deontic Logic, Mally's -- Gert-Jan Lokhorst at Stanford
Temporal Logic -- Antony Galton at Stanford
Foundations of Temporal Logic -- Per Hasle and Peter Øhrstrøm
Notes on Symbolic Logic -- Peter Williams

Edited by dreamweaver on Sep 22, 2006 - 5:05 AM. Reason: Fixed URL tags
Gassendi1
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Posted Mar 6, 2004 - 7:48 AM:

Dreamweaver:

In response to your invitation for additions to the useful summary you have posted, I would suggest the addition of the idea of necessary and sufficient conditions as a kind of supplement to modus ponens and modus tollens.

For example take, "Oxygen is a necessary (but not, sufficient) condition of combustion." Which means that unless there is oxygen, combustion cannot occur. But does not mean, that if there is oxygen, then combustion will occur. So, the existence of oxygen is a necessary, but not a sufficient condition of combustion.

On the other hand take the following example: Being decapitated is a sufficient condition of death, but it is not a necessary condition of death. Which is to say, although it is true that if a person is decapitated he will die, it is not true that only if he is decaptitated, will a person die. (There are other causes of death than decapitation.) So, although decapitation is a sufficient condition for death, it is not a necessary condition for death.

As you can see, the notion of sufficient and necessary conditions parallels the notions of modus ponens and modus tollens, respectively.

And, equally, the fallacy of affirming the consequent is just the confusion of a necessary condition with a sufficient condition, while the fallacy of denying the antecedent is just the confusion of a sufficient condition with a necessary condition.
muxol
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Posted Apr 10, 2004 - 1:37 AM:

why is the section on modal logic not about modal logic in general but kripkean possible world semantics specifically? there are a great number of modal logics that have nothing to do with necessity and possibility.

i think it useful when introducing modal logic to substitute talk of possible worlds with talk of (nonspatio-temporal) localities. deontic logic, for example, has nothing to do with possibility/necessity and so expressing deontic statements shouldn't be done with possible worlds jargon. it's also beneficial to talk about the accessibility relation on worlds as a the neutral R-relation on localities.
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Posted Apr 10, 2004 - 5:16 AM:

Good point muxol; you're right. Temporal logics too, isn't understood completely with the use of possible world semantics. I guess I dived right in on the deep end; into S5 and the like. The more "famed" ones (at least on these forums). Hey, anyone can add to this post. I tried to start of with some of the basics, but I tried to encourage all others to contribute if they could. If you could provide any addition to this it would be greatly appreciated, as always. Anyhow, I'll try to edit that part as soon as I get some more time on my hands.
MathematicalPhysics Wizard
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Posted May 5, 2004 - 11:21 AM:

perhaps someone may find this helpful:
www.earlham.edu/~peters/cou...s/nonstbib.htm#categorical
/
|
A Bibliography of Non-Standard Logics.

dreamweaver if you have already included this link then i will delete this post of mine.
AKG
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Posted May 8, 2004 - 9:38 PM:

Here's a good link for logic symbol GIF's:

http://www.earlham.edu/~peters/writing/logicsym.htm
Timothy
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Posted May 26, 2005 - 6:18 PM:

I've found this site that some migh consider useful. It features a tableau program that you can download for free.

http://logic.philosophy.ox.ac.uk.
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Posted Sep 15, 2006 - 8:43 AM:

A free one-semester course in Logic

http://www.cmu.edu/oli/courses/enter_logic.html
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Posted Sep 28, 2009 - 8:47 AM:

This is a fantastic page!
Thank you Dreamweaver for making this page!

-Stefan
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Posted Jun 13, 2010 - 10:56 AM:

Truth tables

Truth tables are how the traditional predicate logic operators are defined. Here are some Truth tables corosponding to the standard logical operators.

1. p==>q
P Q p==>q
1 1 1
1 0 0
0 1 1
0 0 1

This is the most controversal truth table that will be listed. The controversal part of it is the last two lines of the table. Indeed, using this operator, and a few tautologies, it's possible to take true statements, convert them into logic, and then derive statements that, when translated back into natural languague, are false.

2. p v q

This is what most people call and/or

p q p v q
1 1 1
1 0 1
0 1 1
0 0 0

3. p and (sorry I don't know how to get the proper symbol) q

p q p and q
1 1 1
1 0 0
0 1 0
0 0 0


Now, using these definitions, it's possible to derive their corosponding negations.


1 ~(p=>q)

looking at the truth table, we see that the implication is only false when p=1 and q=0 or p and ~q

2. ~(pvq)

looking at the truth table definition for this one, we see that the disjunction is only false when both p and q are; p=0, q=0 or ~p and ~q

3. ~ (p and q)

Again, looking at the truth table, we see that the conjunction is false whenever either p or q or both are false.

So, we have (~p) v (~q)


It's important to understand that these truth tables are the definitions of these logical operators. Therefore, it's possible to show that two logical expressions are identical by showing that their truth tables, which you can deduce from their comonent statements and the definitions to them, are identical.

One of the interesting things about this particular system of logic is that it's actually possible to have a complete system without both v and and operators. If your're interested you should show this as an exercise. Here is a useful hint.

p=>q is logicaly equilivent to (~p) v q. How can you write this without the disjunction? Next, show how p v q can be written in terms of and operators, implications (or their equilivencies) and negations, and so on.

Have fun.
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