Propositional Logic

From Logic

Until now, we've only looked at classical forms of logic: syllogisms. Modern logicians found that the syllogism was too limiting: not every argument could fit into a 3 line syllogism, not every argument could neatly fit into a comparison of categories. So logicians sought to create new forms of symbolic logic.

Propositional logic allows for more complex argument forms than classical syllogisms. In propositional logic, propositions are represented by symbols and connectors, so that the statement's logical form can be assessed for cases of truth and falsity, which in turn allows us to assess the entire argument's form for validity or invalidity. In symbollic, or propositonal logic, a simple statement, containing one proposition, is is referred to as an atomic statement, and is symbollized by one letter, such as p. A compound statement, with more than one proposition holding some relationship to another proposition, is referred to as a molecular statement, which may be symbolized as p v q. The v symbol just used is a connective: Atomic propositions become molecular propositions when they are joined by connectives. The following is a list of the most common connectives:

Contents

Connectives

Connective              Name                    Meaning

~                       Negation                Not, it is not the case, etc.
.                       Conjunction             and, but, however, also
v                       Disjunction             or (inclusive) and/or
>                       Conditional             if, then, only if, given that
=                       Biconditional           if and only if (equivalency)

Now, let's take a look at some examples of how these connectives work to transform atomic statements into more complex statements:

Examples of Atomic and Molecular Statements

Symbolic Notation       Meaning

A                       An atomic statement, on it's own, affirms its own truth.
~A                      A is false (literally negated)
A v B                   either A or B (or both) is/are true
A > B                   If A is true, then B is true
A > ~B                  A unless B
B > A                   A if B (Tricky one)
A > B                   A only if B
B > A                   Only if A, B
B > A                   A is a necessary condition for B (another tricky one)
A >B                    A is a sufficient condition for B (very tricky)
A ≡ B                   A is a necessary and sufficient condition for B
~(A v B)                Neither A nor B
~A v ~ B                Either not A or Not B
~(~A V ~B)              Neither not A nor Not B
~A . ~B                 Both not A and not B
~(A . B)                Not both A and B
A= B                    A if and only if B, this means that you can't have A without B

Connectives allow us to combine not only atomic propositions with one another but also an atomic proposition with a molecular one and molecular propositions with each other. Remember to use ( ) when you combine molecular statements to other molecular statements: (A v B) v ~ C.

Now that we have a new way of representing arguments freed from pre-specified forms and moods, we have a slight problem: without reference to forms and moods, how do we determine whether our arguments are valid? The new way of making this determination is through the use of Truth Tables. We set up a Truth Table for an argument by showing each and every possible permutation of truth and falsity for every proposition in the argument, and then check for the existence of any case where there are all true premises and a false conclusion - i.e an invalid form.

But before we actually look at truth tables for argument forms, let's take a look at truth tables for common propositions in arguments. In other words, let's first understand how these propositions become true or false, based on the truth value of the variables in the statement. Then we'll move on to looking at argument forms, and checking them for validity and invalidity.

Truth Functions and Truth Tables

We use Truth tables for two purposes: first, to examine statements and, second, as we will see later, to determine the validity or invalidity of an argument's form.

Every statement, no matter the specific content, has a particular form that we can examine symbolicaly. To represent a staement symbollicaly, we use variables , (letters like p, q, r,s...) to represent atomic components of the statement and, if necessary, connectives to create molecular statements.

We then create a truth table by plugging in a truth value for every variable in the statement, as well as a truth value for every molecular component. The truth value of a proposition is simply the assignment of truth or falsity to each variable. In a Truth Table . We seek to set up a truth value for every possible permutation of truth or falsity for every variable within a molecular statement. Then we can determine which permuations of values of truth or falsity for the atomic components cause the overall statement to be true or false!

To better understand this process, let's look at the Truth Tables for 2 atomic and 4 molecular propositions used in propositional logic: Negation and Double Negation (the atomic statements), and Conjunction, Disjunction, Conditional and Biconditional. These tables will in turn provide rulesfor calculating the truth value of any molecular proposition built up from atomic propositions by means of the five connectives listed above. Once we understand how these molecular propositions work, we can move on to using Truth Tables to assess arguments forms for their validity.

Negation

In the case of negation only one component exists, so the total number of possibilities for truth and falsity are two: the component could be true or it could be false. So the table will need only two lines.

p
~p
T
F
F
T

Notice how this table allows us to exhaust all the possible combinations, or permutations, of truth and falsity for each and every variable in the proposition. This is why truth tables can serve as proofs. This table shows that whenever the atomic statement P, is true, it's negation will be false, and vice versa.

Double Negation

In the case of double negation, again only one component exists, so the total number of possibilities for truth and falsity are two: the component could be true or it could be false. So the table will need only two lines.


p
~~p
T
T
F
F


This table proves that a double negation is equivalent to the truth value of the original proposition! As trivial as this seems, this will be important later.

Conjunction

Conjunctions involve two components, each of which could be true or could be false, so there are four possible combinations of truth values among the components.


p
q
p & q
T
T
T
F
T
F
T
F
F
F
F
F


The columns for p and q represent all the possible permutations or combinations of truth for BOTH p and q. The column for p & q represents the truth of the conjunction of "p & q" given the particular truth values we have assigned for p and for q. For example, in proposiotional logic, the statement "Both anne and susan are here" (P & q) is true if and only if both Anne and Susan are here. Simple enough, right?

Disjunction

Disjunctions also involve two components, each of which could be true or could be false, so there are four possible combinations of truth values among the components.


p
q
p v q
T
T
T
F
T
T
T
F
T
F
F
F

The columns for p and q represent all the possible permutations or combinations of truth for both p and q. The column for p v q represents the truth of the disjunctive "p v q" given the particular truth values we have assigned for p and for q. For example, in proposiotional logic, the statement "Either Anne or Susan is here" (P v q) is true as long as either one (or both) of them are here.

Conditional (Hypothetical)

Conditionals also involve two components, each of which could be true or could be false, so there are four possible combinations of truth values among the components.


p
q
p É q
T
T
T
F
T
T
T
F
F
F
F
T

The columns for p and q represent all the possible permutations or combinations of truth for both p and q. The column for p ⊃ q represents the truth of the entire conditional p ⊃ q given the particular truth values we have assigned for p and for q. The table indicates that if the antecedent p is true while the consequent q is false, the conditional as a whole will be false, but all other possible combinations of truth value the conditional as a whole will be true. To continue our example, it would be false to hold that Anne is here, given that Susan is here (Susan could be here on her own!) On the other hand, it may seem odd that the statement p ⊃ q is true given that both Anne and Susan are not here. However, since Anne could only be here, given that Susan is here, the fact that Anne and Susan are not here (both p and q are false) would still verify the proposition.

A bit more on conditional statements

According to Copi and Cohen, there are at least four types of conditional statements:

A. If all humans are mortal and Socrates is a human, then Socrates is a mortal. (Logical 
conditional)
B. If Joe is a bachelor, then Joe is unmarried (Defitional or tautological conditional)
C. If this piece of blue litmus paper is placed in acid, then this piece of blue litmus paper will 
turn red. (Scientific, or causal conditional)
D. If the Giants lose this game, I'll eat my hat. ( Decisional conditional)

Copi and Cohen hold that a logical conditional statement includes one element that is true of all the types of relationships given here: they all assert that p & ~q is false. There is also a fifth type of conditional: material implication. . A conditional makes a material implication if it purposely includes a consquent that is known to be false, making the entire statement necessarily false. An example might be: "If George Bush is a good president, then any moron would make a good president."

Biconditional

Biconditionals also involve two components, each of which could be true or could be false, so there are four possible combinations of truth values among the components.


p
q
p º q
T
T
T
F
T
F
T
F
F
F
F
T

The columns for p and q represent all the possible permutations or combinations of truth for both p and q. The column for p v q represents the truth of the biconditional "p º q" given the particular truth values we have assigned for p and for q. The table indicates that when the two statements have the same truth value, the biconditional as a whole will be true, while in all other cases the biconditional will be false.


Tautologies, Self-Contraditions and Contingent Statement Forms

I still have a few more propositional statements for you to examine. I now wish to further explicate on the matter of tautoloy (necessarily true statements) self-contradictions (necessarily false statements) and contingency or contingent propositional forms (statements that can either be true or false).

First, let's look at a tautology

Tautology

A statement form that has only true substitution instances is called a tautologous statement form, or a tautology. We can use a truth table to prove that there are no instances where a tautology is false.


p
~p
p v ~p
T
F
T
F
T
T

We refer to such statements as necessary statements because they are necessarily true, by definition. Any attempt to refute them leads to a self refutation, meaning that any argument that sets up a necessary statement as a false statement is self refuting.

Self (Internal) Contradiction

A statement form that has only false substitution instances is called a self-contradiction. Such a statement is necessarily false.


p
~p
p & ~p
T
F
F
F
T
F

Material Equivalence

Two statements are said to be materially equivalent, or equivalent in truth value when their truth values cannot be the negation of each other: i.e. they must both be true or both be false. We can read material equivalence as p if and only if q.


</TR> </TR>
p
q
p º q
T
T
T
T
F
F
F
T
F
F
F
T


Whenever two statements are materially equivalent, they materially imply each other.

Logical Equivalence

Two statements are logically equivalent when the statement of their material equivalence is a tautology. We can also consider this the principle of double negation.


p
~p
~~p
p º ~~p
T
F
T
T
F
T
F
T


The difference between material equivalence and logical equivalence is important. Two statements are logically equivalent only when it is logically impossible for the two statements to have different truth values. whereas materially equivalent statements merely happen to share the same truth value. You can consider this a difference of correlation and causal connectednesss. For this reason, logically equivalent statements can be considered equivalencies, and substituted for each other, whereas materially equivalent statements may not.

De Morgan' Theorms

captain_morgan.jpg

Captain Morgan not only was a infamous party hound, he also took time out from his busy schedule of pirating and rum running to create some logical theorms. There are two logical equivalancies (logically true bi-conditionals) that express interrelations among conjunction, disjunction and negation. Since the disjunction p v q asserts that at least one (if not both) of the disjuncts are true, asserting the negation of the disjunction p v q is held to be logically equivalent to asserting the conjunction of the negations of p and q. Symbollically: ~( p v q) ≡(~p & ~q). The following truth table is a proof of this claim:


p
q
p v q
~(p v q)
~p
~q
~p & ~q
~(p v q) ≡ (~p & ~q)
T
T
T
F
F
F
F
T
T
F
T
F
F
T
F
T
F
T
T
F
T
F
F
T
F
F
F
T
T
T
T
T

Similarly, since asserting the conjunction of p and q asserts both are true, to contradict this assertion we need only assert that one of them is false. Ergo, using biconditionals, this claim can be stated symobolically, thusly: ~(p & q) ≡ (~p v ~q). Together, these two tautologous binconditionals are known as Spicy Augustus De Morgan's theorms. You can also state them thusly: 1) the negation of the disjunction of two statements is logically equivalent to the conjunction of the negations of the two statements. and 2) the negation of the conjunction of two statements is logically equivalent to the disjunction of the negations of the two statements.

Let's cover two more issues before moving on to examine truth tables for arguments: material implication and the difference between necessary and contingent argument forms.

Earlier, I stated that material implication, a proposition form such as: "p ⊃ ~q", was a fifth type of conditional wherein one purposely set up a conditional with a considered to be impossible, for rhetorical effect. Logically, the statement could be read "It is not the case that p is true while q is false." The definiens, or group of symbols that have the same meaning as this defintion, would be ~(p & ~q), the denial of this conjunction, or, again, "It is not true that p is true and q is false".

Now, by De Morgan's theorem, we now know that any such denial, ~(p & ~q), is logically equivalent to the disjunction of the denials in the conjuncts, so we now know that ~(p & ~q) is logically eqvuilalent to ~p v ~~q. And, this expression, in turn, through the principle of double negation, is logically equivalent to "~p v q". Therefore, since all three phrases are logical equivalents for each other "p ⊃ ~q","~(p & ~q)" and "p v ~q", we can declare that "p ⊃ ~q" is logically equivalent to ~p v q. The reason for this long expostualtion on this matter? We will find that such replacements are quite useful when examining arguments, particuarly this one.

Necessary and Contingent Truth

For every argument there corresponds a conditional statement whose antecedent is the conjunction of the argument's premise and whose consequent is the argument's conclusion. For example, an argument using the form of modus ponens:

P ⊃ q p therefore q

Could be expressed as a conditional statement thusly: "[(p ⊃ q) & p] ⊃ q." We can read this conditional statement this way: "If it is true that 'p is true, then q is true' AND p is in fact true, then in this case, q is true."

Now why go through all this? Because something very interesting happens to a valid deductive argument when stated in a conditional statement: it becomes a tautology. We learn then that a deductive argument is valid, if and only if its expression in the form of a conditional statement is a tautology.

This all goes back to my original expostulation on the nature of deductive arguments: they deal in equivalencies, in equalities, i.e. with a priori truths. So how do we deal with inductive matters, with matters that concern real world phenomena?

Here, we can only deal in contingent truths. If we think back to the deductively invalid forms of affirming the antecedent and denying the consequent, we will recall while there were possible permutations where all true premises led to a false conclusion, we also can recall that there were permutations where this was not the case. Therefore, along with necessarily contradictory conclusions, these deductively invalid forms also led to contingent truths: claims that may in fact be true. As we will see in the section on inductive logic, we can rely on such forms to give us probable or possible truths.

Once you feel comfortable with the concepts in this section, move on to the section where Truth Tables are used to assess arguments for validity.







References

  • Copi, I. M, Cohen, C., (2001), "Introduction to Logic", 11th Edition.
  • Hurely, P. J. (2000) A Concise Introduction to Logic - 7th Edition
  • Wiebe, B. (2006) Wibster's Wizardry, http://sask.usask.ca/~wiebeb.
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