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May 20, 2022April 8, 2023

St. Thomas Aquinas’ Five Proofs for God’s Existence

To answer the question concerning God’s existence, St. Thomas Aquinas presented five ways or proofs in his most notable work, the Summa Theologica. This is also called “Aquinas’ Five Proofs for God’s Existence”. These five arguments draw proof or evidence from man’s experience with the world, which are noticeably influenced by Aristotle and his concept of the four causes.

The first argument that Aquinas formulated is the argument from motion. After observing objects in motion, Aquinas was convinced that whatever is currently in motion were once at rest but had changed states when it was moved by something else. This mover was something once at rest as well but was also moved by something else. This line of thought would go on and on until it forms an almost infinite series of concurrent events where the objects are both movers and moved. But if this series of events needed something to begin the movement, then, it is logical to assume that at the very beginning of this infinite series is the first mover, which starts the movement. Aquinas describes this first mover as the “unmoved mover,” a label which is quite similar to Aristotle’s “prime mover.” As we can see, both see this mover as one that is not caused or moved by anything other than itself. And for Aquinas, this is God.

The second argument is the argument from causation, which builds upon Aristotle’s concept of the efficient cause. The main idea here is that every object, action, or event, according to Aristotle, has an efficient cause or an entity or event responsible for its creation or change. Just like how a baby finds their efficient cause in their parents and their parents in their own parents and so on and so forth, Aquinas uses these examples of dependent relationships to show that every person or object in the world depends on a creator (efficient cause) and that this creator also has its own creator, and this new creator also has its own efficient cause. This cycle, much like the argument from motion, can go on infinitely but, according to Aquinas, it should not be so since in the first place the series would not have begun.

It is then logical to assume that at the very beginning, there is the existence of a “First Maker” or an “Uncaused Cause,” which, as the name suggests, is the efficient cause that is not caused by others or anything but itself. This “Uncaused Cause” is, of course, attributed by Aquinas to the Christian God.

The third argument is the argument from contingency which necessitates the distinction between “necessary” and “possible” beings. “Possible” beings, simply put, are beings that can be created and corrupted or are beings that can exist and not exist. An example of a possible being is man. Man is a possible being because we have the potential to exist (birth) and the same potential to not exist (death). Plants, animals, and structures are among some of the other beings included in this category.

With this in mind, it is then reasonable to think that since most beings in the world are possible beings, then there must have been a time that they had not existed at all, which means that nothing ever existed. And if there truly was a time of pure non-existence, then nothing could currently exist because nothingness can only yield nothingness. The only way that our existence at this very moment could be explained, for Aquinas, is if there was a being that already exists despite the nothingness of the possible beings.

This being is called a necessary being. Necessary beings, on the other hand, are beings that necessarily exist or are beings that cannot be nonexistent. For Aquinas, there must be at least one necessary being to exist at the very beginning for the rest of the beings to be able to exist. This being is, of course, God.

The fourth argument is the argument from degrees of perfection. This argument makes use of man’s knowledge of perfection and his tendency to judge or evaluate whether an object or person is more or less perfect. This action of judging something to be more or less perfect means that there is a standard that is used for the said evaluation. But how could man ever have such standards unless there is a being that is all-perfect to compare it to? Aquinas affirms the existence of such a perfect being and says that if any other being would be compared or evaluated against such perfection, they would always be judged as less perfect. He calls this all-perfect being God.

The fifth and last argument in St. Thomas Aquinas’s five proofs for God’s existence is the argument from final causes or design. Some scholars would also call this as the teleological argument. Aquinas once again drew on the notions of causality as presented by Aristotle to justify this argument. The “final cause,” as described by Aristotle, is the fourth cause and is one that refers to “the end, that for the sake of which a thing is done.” Some scholars would describe it, rather simply, as the cause that refers to the purpose of which a specific object or entity has been created to fulfill.

Humans and most natural beings in the world have been “designed” to have a purpose and we behave or act according to that purpose. For instance, the bird’s wings behave in accordance with its design which allows it to fly. Humans talk using their mouths because this is in accordance with their body’s design which allows them to utilize air and various muscles in their body to create sounds.

For Aquinas, if there is some sort of design that is set in our world, then there must be a designer. This designer cannot possibly just be humans or other natural beings themselves as he describes man as imperfect and not intelligent enough to set such a grand design. Some of the natural beings, Aquinas tells us, are not even capable enough to know what their end is. The design of the world, therefore, must have been set by a being that is vastly more intelligent than humans and knowledgeable enough to guide them towards their end. This, of course, is God.

Aquinas’ five proofs for God’s existence, during, of course, Aquinas’ time, were found to be compelling enough and soon grew to be influential in religious discourses. For some religious denominations, these arguments still remain significant in defense of the Faith up until the 21^st century, where most of them have been incorporated into doctrines and statements. But as groundbreaking as St. Thomas Aquinas’ arguments were and are, there is still room for critique.

St. Thomas Aquinas’ Five Proofs for God’s Existence: A Brief Critique

The main criticism that one can immediately infer from these arguments is the fact that a majority of them remain as assumptions. Though St. Thomas Aquinas did invoke observations from man’s experience with natural phenomena as well as logic to prove his point, there is no concrete way of knowing whether these events do happen in the manner that the theologian-philosopher has described it. In the case of the first proof, there is no concrete explanation as to whether every single movement in this world can be traced back to one single cause nor is there enough proof to determine that an event or an object is necessarily moved or affected by the simultaneous movement of another object or entity. In the case of the fifth argument, it is simply too illogical to immediately assume that just because the bird’s wings are aerodynamic or that humans are capable of speech it automatically suggests the presence of both a grand design and of a grand intelligent mind when, in the same paradigm, the notion of spontaneity and adaption exists.

Interestingly, he did speak of this same point in the Summa Theologica as Objection #2 and his response to this response is as follows: “For all natural things can be reduced to one principle which is nature, and all voluntary things can be reduced to one principle which is human reason. Therefore, there is no need to suppose God’s existence.” Though the reduction does serve the purpose of trying the establish concrete principles where he can root his arguments on, the idea that natural and voluntary things can be reduced into just nature and reason is still an assumption by itself.

It is tempting to think that there is indeed such a connection between the beings in the world, but as far as human knowledge is concerned, these conclusions are merely a product of inference and are not concretely proven.

This then leads to the second point of my criticism. Should a person not be satisfied with the assumptions forwarded by St. Thomas Aquinas and decides to do away with them, then Aquinas’s five proofs will become irrelevant. The arguments would not be able to stand once you remove the assumptions, such as the assumption that the one thing is caused by another or that if the notion of a grand design necessitates the existence of a grand designer, as these are the logical links between his premises. To continue to believe in these arguments without said assumptions, one must somehow either see it in a dogmatic light or ignore contrary logical proof.

Despite these criticisms, St. Thomas Aquinas’ philosophy has withstood time and continues to play a significant role in the development of both the Church and modern theology. By incorporating human experience, logic, and Aristotle in his attempt of proving His existence, he not only formulated five succinct and insightful arguments but he had also brought theology further than what his time had expected.

May 20, 2022May 20, 2022

Tautologies and Contradictions

In these notes, I will briefly discuss tautologies and contradictions in propositional or symbolic logic. But please note that this is just an introductory discussion on tautologies and contradictions as my main intention here is just to make students in logic become familiar with the topic under investigation.

On the one hand, a tautology is defined as a propositional formula that is true under any circumstance. In other words, a propositional expression is a tautology if and only if for all possible assignments of truth values to its variables its truth value is always true.

Thus, a tautology is a proposition that is always true. Consider the following example:

Either the accused is guilty or the accused is not guilty. (p)

Obviously, the proposition is a disjunction; yet both disjuncts can be represented by the variable p. Hence, the proposition is symbolized as follows:

p v ~p

Now, in what sense that this proposition is always true? The truth table below will prove this point.

As we can see in the truth table above, if p is true, then ~p is false; and if p is false, then ~p is true. And if we apply the rules in both inclusive and exclusive disjunction, the result of p v ~p is always true. If we recall our discussion on inclusive and exclusive disjunction, we learned that an inclusive disjunction is true if at least one of the disjuncts is true; and an exclusive disjunction is true if one disjunct is true and the other is false, or one disjunct is false and the other is true.

Hence, there is no way that p v ~p will become false. Indeed, the propositional form p v ~p is always true.

On the other hand, a contradiction is defined as a propositional formula that is always false under any circumstance. In other words, a propositional expression is a contradiction if and only if for all possible assignments of truth values to its variables its truth value is always false. Thus, again, a contradiction is a proposition that is always false. Let us consider the examples below.

Man is both mortal and immortal. (p)

Obviously, the proposition is a conjunction; yet both conjuncts can be represented by the variable p. Hence, the proposition is symbolized as follows:

p • ~p

Now, in what sense that this proposition is always false? The truth table below will prove this point.

As we can see in the truth table above, if p is true, then ~p is false; and if p is false, then ~p is true. And if we apply the rule in conjunction here, which says that “A conjunction is true if and only if both conjuncts are true,” then surely there is no way that the proposition “Man is both mortal and immortal” or p • ~p will become true. Indeed, the propositional form p • ~p is always false.

May 20, 2022May 20, 2022

Propositional Logic: Indirect Truth Table Method and Validity of Arguments

In these notes, I will discuss the indirect truth table method in determining the validity of an argument in symbolic logic.

In my other notes (look for “Propositional Logic: Truth Table and Validity of Arguments” in Studypool search engine), I discussed the truth table method in determining the validity of an argument in symbolic logic. But the problem of the truth table method is that it can hardly be used in determining the validity of longer arguments.

Consider the example below.

If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had powerful weapon only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)

As we can see, the argument contains 8 constants, namely, A, D, W, G, E, S, B, and P. If we employ the truth table method in determining the validity of this argument, then this means that we need to construct a truth table that contains 256 rows. Needless to say, that’s going to be a long and arduous process. It is for this obvious reason that logicians invented a shorter, more efficient method of determining the validity of arguments, namely, the indirect truth table method.

Let us determine the validity of the argument above using the indirect truth table method.

First, let us symbolize the argument above proposition by proposition or sentence by sentence to avoid confusion. In case one does not know yet how to symbolize arguments in logic, please refer to my previous post titled “Truth Table and Validity of Arguments”, http://philonotes.com/index.php/2018/03/26/truth-table-and-validity-of-arguments/. See also “Symbolizing Propositions in Symbolic Logic”, http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-logic/.

Proposition 1:

[(A ⊃ D) • (D ⊃ W)] ⊃ (G v E)

Proposition 2:

The airship Albatros had powerful weapon only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground.

(A ⊃ S) • (S ⊃ D)

Proposition 3:

It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax.

(D ⊃ W) v (B ⊃ P)

Proposition 4:

It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax.

~ (B ⊃ P)

Proposition 5:

Furthermore, the captain’s motives for controlling the world were not evil.

~ E

Conclusion:

Therefore, his motives were good.

/∴ G

In the end, the argument above is symbolized as follows:

Now, in determining the validity of the argument above using the indirect truth table method, what we need to do is try to make the conclusion false and all the premises true. This is because if we recall our discussion on the rule in determining the validity of an argument in symbolic logic, we learned that an argument is invalid if the conclusion is false and all the premises are true. Thus, in using the indirect truth table method in determining the validity of an argument, we aim to make the argument invalid. If it is possible for us to make the argument invalid, then obviously the argument is invalid. If it is impossible for us to make the argument invalid, then obviously the argument is valid.

But how do we make the argument invalid?

First, let’s write the premises and the conclusion in a horizontal manner for convenience’s sake.

And second, assign truth-values to the conclusion and the premises that would result in the form “false conclusion and all true premises”. In doing so, start with the conclusion and assign the value “false”, and then go back to the premises and try to make all of them true. In making the premises true, always start with the first premise in order to avoid confusion and, of course, save time. Consider the example below.

As we can see, the argument above is valid because, although the conclusion is false, we cannot make all of the premises true. No matter what we do, Premise #5 cannot be true. Let me explain this further.

As said, let us always start with the conclusion and assign a false value to it. Please note that we should not assign a true value to the conclusion because if we do so, then we are defeating the purpose. This is obviously because if we assign the value true for the conclusion, then the argument will already appear valid. So, again, we should assign the value false for the conclusion. Look at the illustration below.

Please note that in this example, we can easily make the conclusion false by assigning the truth-value “false” to it because the conclusion above is a simple proposition. But even if it’s a compound proposition, we can still easily make it false if we have mastered the rules in compound propositions in symbolic logic. For example, if the conclusion is p ⊃ q, then we just need to assign the value true for p and false for q to make the proposition (conclusion) false. If we recall, the conditional proposition is false if the antecedent is true and the consequent false.

Now, since we have made the conclusion false, let’s go back to the premises and try to make all of them true. And let’s start with the first premise. Look at the illustration below.

As we can see, the first premise is a conditional proposition whose antecedent is [(A ⊃ D) • (D ⊃ W)] and the consequent is (G v E). Because this is a conditional proposition, and since our goal here is to make this premise true, then all we need to do is assign truth-values that would make the consequent (G v E) true. Please note that we need not assign any values to the antecedent [(A ⊃ D) • (D ⊃ W)] because whatever its truth-value, the premise is already true since the consequent is true. Again, the only instance wherein the conditional proposition becomes false is when the antecedent is true and the consequent false. Thus, whenever the antecedent is false or the consequent is true, the conditional proposition becomes automatically true.

So, how do we make the consequent (G v E) true?

Since the consequent is an exclusive disjunctive, then we need to see to it that (G v E) should not have the same truth-value in order for it to become true. If we recall our discussion in exclusive disjunction, an exclusive disjunction is true if one disjunct is false and other is true, and vice versa. And since we already have the value false for G in the conclusion, then we cannot make it true in the premise. Please note that in indirect truth table method, once the variable or constant has a fixed truth-value, then we cannot change it. Thus, if we change the truth-value of one variable or constant, then we need to change the truth-value of the same variable or constant in the entire indirect truth table. Since G is false, then we are forced to assign the truth-value “true” for E to make (G v E) true. So, since G is false and E is true, then the exclusive disjunction (G v E) is now true. And since the proposition is a conditional one, and because the consequent (G v E) is true, then Premise #1 is now true.

Let’s proceed to the second premise and try to make it true. Look at the illustration below.

Premise #2 is a conjunctive statement whose conjuncts are both conditional propositions. If we recall our discussion on conjunctive statements, we learned that a conjunctive statement is true if both conjuncts are true. Hence, if one conjunct or both are false, then the conjunctive statement is false. Since our goal here is to make Premise #2 true, then we have to see to it that both conjuncts must be true. In other words, (A ⊃ S) and (S ⊃ D) must be true.

How do we make (A ⊃ S) and (S ⊃ D) true?

Let’s start with (A ⊃ S). Since we don’t have a value for A and S yet, then we are free to assign whatever value that will make (A ⊃ S) true. So, if we assign a true value for both A and S, then (A ⊃ S) is true. Hence, as you can see in the diagram above, A is true and S is true.

The second conjunct is (S ⊃ D). Please note that we already have a value for S, which is true. Hence, we cannot assign a false value to D because it will make the proposition false. So, we are forced to assign the value true for D. Since S and D are now true, then the proposition (S ⊃ D) is true.

And since the conjuncts (A ⊃ S) and (S ⊃ D) are now true, then Premise #2 is now true (see diagram above).

Let’s proceed to Premise #3. Look at the illustration below.

Premise #3 is an inclusive disjunction whose disjuncts are both conditional propositions. If we recall our discussion on inclusive disjunction, we learned that an inclusive disjunction is true if at least one of the disjuncts is true. So, in Premise #3, we just need to make either of the disjuncts true in order for it to become true. And in the illustration above, we just made the first disjunct (D ⊃ W) true.

How do we make the first disjunct (D ⊃ W) true?

Since we already have the value true for D (see Premise #2), and since the proposition is conditional, that is, (D ⊃ W), then we cannot assign a value false for W; otherwise, we are making the proposition false. Hence, we are forced to assign the value true for W. Now, Since D is true and W is true, then the first disjunct (D ⊃ W) is true. If we look at the illustration above, we do not assign a value to the second disjunct (B ⊃ P). Of course, we are free to assign a value for (B ⊃ P), but that is not necessary because whatever value we have for (B ⊃ P), the premise (D ⊃ W) v (B ⊃ P) is already true because the first disjunct is true.

Let’s proceed to Premise #4. Look at the illustration below.

Premise #4 is a conditional proposition, but it is completely negated. In this case, it is relatively easy for us to make this premise true. All we need to do is make B ⊃ P false; so that if B ⊃ P is false, then ~ (B ⊃ P) is true.

How do we make B ⊃ P false?

Because B ⊃ P is a conditional proposition, there is only one way to make it false, that is, assign a true value to the antecedent B and false to consequent P. If we recall our discussion on conditional propositions, we learned that a conditional is false if the antecedent is true and the consequent false. Hence, if B is true and P is false (see illustration above), then B ⊃ P false. Again, since B ⊃ P false, then ~ (B ⊃ P) is true (see illustration above).

Lastly, let us make Premise #5 true. Look at the illustration below.

Premise #5, as we can see, is just a simple proposition. So, it is very easy for us to make this premise true. Since the premise is ~ E, all we need to do is assign the value false for E. This is because if E is false, then ~ E is true.

However, if we go back to Premise #1, we notice that we have assigned the value true for E. And since in indirect truth table method we are not allowed to change the value of a variable or constant, then we are forced to use the value true for E in Premise #5. So, if E is true, then ~ E is false (see illustration above).

At the end of it all, it’s impossible for us to make the argument above invalid. Therefore, the argument is absolutely valid.

Just a final note. There are, of course, several ways of making a conclusion false and all the premises true in indirect truth table method. For example, we can make Premise #1 in the argument above by making the antecedent [(A ⊃ D) • (D ⊃ W)] false. But I have exhausted all the ways in making the argument above invalid but to no avail. The argument, therefore, is valid.

May 20, 2022May 20, 2022

Propositional Logic: Truth Table and Validity of Arguments

In these notes, I will discuss the topic truth table and validity of arguments, that is, I will discuss how to determine the validity of an argument in propositional logic using the truth table method.

However, it must be noted that there are two basic methods in determining the validity of an argument in symbolic logic, namely, truth table and partial truth table method. Again, in this post, I will only discuss the truth table method, thus the topic “truth table and validity of arguments”. I will discuss the partial truth table method in my next post.

Validity and Invalidity of Arguments

How do we know whether an argument is valid or invalid?

On the one hand, a truth-functionally valid argument form is an argument that is composed of propositions that have truth-functional forms such that it is impossible for its premises to be all true and its conclusion false. In other words, an argument is valid if it does not contain the form “all true premises and false conclusion”.

On the other hand, a truth-functionally invalid argument form is an argument that is composed of propositions that have truth-functional forms such that it is possible for its premises to be all true and its conclusion false. In other words, an argument is invalid if all of its premises are true and its conclusion false.

Let’s consider the example below.

1. If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights. (S, C, F)

So, how do we determine the validity of the argument above?

Before we can apply the truth table method in determining the validity of the argument above, we need to symbolize the argument first. After symbolizing the argument, we will construct a truth table for the argument, and then apply the rule in determining the validity of arguments in symbolic logic.

But how do we symbolize the argument above?

In case one does not know how to symbolize arguments in symbolic logic, please refer to my previous post titled “Symbolizing Propositions in Symbolic Logic”, http://philonotes.com/index.php/2018/02/14/symbolizing-propositions-in-symbolic-logic/.

In symbolizing arguments in symbolic logic, we just need to apply the techniques that we employed in symbolizing propositions. Hence, we symbolize arguments in symbolic logic proposition by proposition or sentence by sentence.

Now, if we look at the argument above, the first proposition is “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights.” And then we see the constants “S, C, and F” at the end of the argument.

If we recall my discussion on symbolizing propositions, we learned that the variables or constants provided after the proposition (argument in this case) represent the propositions in the entire proposition (argument in this case) respectively. Hence, the constant S stands for “The squatters settle here”, C for “The cattlemen will be angry”, and F for “There will be a fight for water rights”. Thus, the first proposition “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights” is symbolized as follows:

S ⊃ (C • F)

The next proposition in the argument above says “The squatters are going to settle here”. As we notice, this proposition is just a repeat of the proposition in the previous statement, and this proposition is symbolized by the constant S. Hence, the second proposition “The squatters are going to settle here” is symbolized as follows:

The third and last proposition is obviously the conclusion because of the signifier “therefore”. This proposition is also a repeat of the proposition in the first sentence, which is symbolized by the constant F. Hence, the conclusion “Therefore, there will be a fight for water rights” is symbolized as follows:

At the end of it all, the argument “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights. (S, C, F)” is symbolized as follows:

S ⊃ (C • F)
S /∴ F

Please note that in the symbolized form of the argument above, S ⊃ (C • F) is the first premise, S is the second premise, and F is the conclusion.

Now, how do we construct a truth table for this argument?

First, we need to construct a truth table that contains columns for the variables or constants and columns for the premises and the conclusion. In order to do this, we will use the formula 2 raised to the power n (2ⁿ), where 2 is constant and n is a variable.

The n in the formula 2ⁿ represents the number of variables or constants used in the argument. Since the argument above contains 3 constants, namely, S, C, and F, then the formula now reads:

2³

So, 2³ = 8. This means that we need to construct a truth table that contains 8 rows. But first we have to draw columns for the constants and the premises and the conclusion, which will look like this:

After drawing the columns for the constants and the premises and conclusion, we will now draw 8 rows. With this, the truth table will now look like this:

Now that we have constructed the truth table that contains the columns for the constants and the premises and conclusion, let us provide the truth values of the variables S, C, and F. We need to do this because the truth values of the premises and the conclusion will be based on the truth values of the variables or constants.

But how do we do this?

First, it must be noted that the product 8 above (2³ = 8) also represents 4 true values and 4 false values for the variable or constant (S in the case of the example above). And the rule here is that we write the true values first and then the false values. So, the truth table will now look like this:

For the next column (that is, the column for C), we need to divide the 4 true and false values by 2. Thus, we will have 2 true values and 2 false values. The rule is we will write 2 true values first and then 2 false values. For the remaining rows, we will write 2 true values and 2 false values alternately. So, the truth table will now look like this:

For the next column (that is, the column for F), we need to divide the 2 true and false values by 2. Thus, we will have 1 true and 1 false value. The rule is we will write 1 true value first and then 1 false value. So, the truth table will now look like this:

For the remaining rows, we will write 1 true value and 1 false value alternately. So, the truth table will now look like this:

Since we already have the truth values of the constants S, C, F, we can now determine the truth values of the premises and the conclusion. Please note that we need to provide the truth values of all the premises and the conclusion before we can apply the rule in determining the validity of an argument. So, let’s provide the truth values of the premise.

The first premise is S ⊃ (C • F). As we can see, the first premise is a conditional proposition whose consequent is a conjunction. We also need to remember that in determining the truth value of this premise, we need to apply the rules in conditional and conjunctive propositions. If one does not know yet the rule in conditional propositions, one may visit our previous post titled “Conditional Propositions”, http://philonotes.com/index.php/2018/02/11/conditional-propositions/. And for the rule in conjunctive statements, please see “Conjunctive Statement”, http://philonotes.com/index.php/2018/02/03/conjunctive-statements/.

Now, before we can apply the rule in conditional proposition in the premise S ⊃ (C • F), we need to simplify first the consequent (C • F). This can be done my determining its truth value using the rule in conjunction. Since the rule in conjunction says “A conjunction is true if both conjuncts are true”, then the truth table will now partially look like this:

Please note that in the truth table above, I temporarily removed the column for S to avoid confusion, that is, in order to show that we are just using the values for the columns C and F.

Since we have already simplified (C • F), then we can now proceed to determining the final truth values of the first premise S ⊃ (C • F). So, the truth table will now look like this:

Please note that the final truth values of the first premise S ⊃ (C • F) are the ones in bold red. Let me illustrate how I arrived at those values. But let me just illustrate the first two rows. In the first row, S is true, C is true and F is true. So,

In the second row, S is true, C is true, and F is false. So,

For the truth values of the second premise which is S, we just need to copy the truth values of S in the first column. This is obviously because the second premise is a simple proposition. So, the truth table will now look like this:

For the truth values of the conclusion which is F, we just need to copy the truth values of F in the third column. This is obviously because the conclusion is a simple proposition. So, the truth table will now look like this:

As we can see, the truth table is now complete. So, we may now apply the rules in determining the validity of arguments in symbolic logic. But before we proceed to that, let us remove the columns for the variables in order to avoid confusion. It must be remembered that the rule talks about the premises and the conclusion only, and so we may now drop them. Of course, as we can see, the columns for the variables/constants are needed only in determining the truth values of the premises and conclusion. So, the truth table of the argument above will finally look like this:

If we recall, the rule in determining the validity of an argument in symbolic logic says that an argument is valid if it does not contain the form “all true premises and false conclusion” and an argument is invalid if “all of its premises are true and its conclusion false”. Please note that in applying the rule, we need to consider all rows in the truth table.

Now, the easiest and most convenient way to do it is to look for an invalid form in each row, that is, a row that contains all true premises and a false conclusion. Thus, if we cannot find one, then the argument is obviously valid.

If we look at the final truth table of the argument above, we cannot find at least one row that contains the form all true premises and a false conclusion. Therefore, at the end of the day, the argument “If the squatters settle here, then the cattlemen will be angry and that there will be a fight for water rights. The squatters are going to settle here. Therefore, there will be a fight for water rights” is absolutely valid.

Finally, let me give an example of an invalid argument so we can fully understand why the argument above is valid. Consider the example below.

2. If Marco had been a poor businessman, then he would have had to undertake extensive lecture hours. He did undertake extensive lecture hours. Hence, he must be a poor businessman. (p, q)

I need not explain again here why we have come up with the truth table above. The discussion above is enough for us to know how to construct a truth table.

Now, although not necessary, let’s remove the columns for the variables in the truth table above to avoid confusion. So, the truth table will now look like this:

If we recall, the rule in determining the validity of an argument in symbolic logic says that an argument is valid if it does not contain the form “all true premises and false conclusion” and an argument is invalid if “all of its premises are true and its conclusion false”. Please note that in applying the rule, we need to consider all rows in the truth table.

The easiest and most convenient way to do it is to look for an invalid form in each row, that is, a row that contains all true premises and a false conclusion. Thus, if we cannot find one, then the argument is obviously valid.

Now, if we look at the final truth table of the argument above, we can indeed find one row (row 3) that contains the form all true premises and a false conclusion. Thus, at the end of the day, the argument “If Marco had been a poor businessman, then he would have had to undertake extensive lecture hours. He did undertake extensive lecture hours. Hence, he must be a poor businessman. (p, q)” is absolutely invalid.

May 20, 2022May 20, 2022

Symbolize the following arguments

If either Algebra is required or Geometry is required, then all students will study mathematics. Algebra is required and trigonometry is required. Therefore, all students will study mathematics. (A, G, S, T)

Either Smith attended the meeting or Smith was not invited to the meeting. If the directors wanted Smith at the meeting, then Smith was invited to the meeting. Smith did not attend the meeting. If the directors did not want Smith at the meeting and Smith was not invited to the meeting, then Smith is on his way out of the company. Therefore, Smith was on his way out of the company. (A, I, D, W)

If a scarcity of commodities develops, then prices rise. If there is a change of administration, then fiscal controls will not be continued. If the threat of inflation persists, then fiscal controls will be continued. If there is overproduction, then prices do not rise. Either there is overproduction or there is a change of administration. Therefore, either scarcity of commodities does not develop or the threat of inflation does not persist. (S, P, C, F, I, O)

If the investigation continues, then new evidence is brought to light. If new evidence is brought to light, then several leading citizens are implicated. If several leading citizens are implicated, then newspapers stop publicizing the case. If continuation of the investigation implies that the newspapers stop publicizing the case, then the bringing to light of the new evidence implies that the investigation continues. The investigation does not continue. Therefore, new evidence is not brought to light. (C, N, I, S)

If the king does not castle and the pawn advances, then either the bishop is blocked or the rook is pinned. If the king does not castle, then if the bishop is blocked, then the game is a draw. Either the king castles or if the rook is pinned, then the exchange is lost. The king does not castle and the pawn advances. Therefore, either the game is a draw or the exchange is lost. (K, P, B, R, D, E)

If Andrews is present, then brown is present, and if Brown is present, then Cohen is not present. If Cohen is present, then Davis is not present. If Brown is present, then Emerson is present. If Davis is not present, then Farley is present. Either Emerson is not present or Farley is not present. Therefore, either Andrews is not present or Cohen is not present. (A, B, C, D, E, F)

Democracy can survive only if people can be taught to work for the public good rather than for their own self-interest. People can be taught to work for public good rather than for their own self-interest only if the theory of psychological egoism is false. Therefore, democracy can survive only if the theory of psychological egoism is false. (D, P, T)

If either George enrolls or Harry enrolls, then Ira does not enroll. Either Ira enrolls or Harry enrolls. If either Harry enrolls or George does not enroll, then Jim enrolls. George enrolls. Therefore, either Jim enrolls or Harry does not enroll. (G, H, I, J)

If Tom received the message, then Tom took the plane, but if Tom did not take the plane, then Tom missed the meeting. If Tom missed the meeting, then Dave was elected to the board, but if Dave was elected to the board, then Tom received the message. If either Tom did not miss the meeting or Tom did not receive the message, then either Tom did not take the plane or Dave was not elected to the board. Tom did not miss the meeting. Therefore, either Tom did not receive the message or Tom did not miss the meeting. (R, P, M, D)

If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had powerful weapon only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)

Dialectical materialism is true if and only if economic determinism is true. If dialectical materialism is true if and only if economic determinism is true, then both nature as a whole and economic conditions contain opposing forces. If nature as a whole and economic conditions contain opposing forces, these forces can be expressed as laws. Given that such forces can be expressed as laws, their formulation may be discovered by observing history. Their formulation may be discovered by observing history only if there is progress in history and enough time has elapsed for the pattern to be observable. Hence, Marx claimed, there is progress in history and enough time has elapsed for the pattern to be evident. (D, E, N, C, L, F, P, T)

If Prof. Sparks is right, then this is the best of all possible worlds. Of course, this is the best of all possible worlds only if the evils it contains are necessary evils. If the evils it contains are necessary evils, this implies the truth of the Principle of Sufficient Reason ⎯ i.e., there is good reason for everything being as it is and not otherwise. If there is good reason for everything being as it is and not otherwise, then there must be good reason for Louie having been kicked out by the baron and for the earthquake having destroyed Lisbon. And if either Louie had not been involved in the Inquisition or had not desired lady Cunnie, then either he would not have wandered over America on foot or he would not have lost the sheep from Eldorado. If he had not wandered over America on foot or had not lost the sheep from Eldorado, then some principle other than the Principle of Sufficient Reason would be required to explain what happens in this world. And no principle other than the Principle of Sufficient Reason is required to explain what happens in this world. If it were not true that either Louie had not been involved in the Inquisition or that he had not desired lady Cunnie, then it would be true that Prof. Sparks is right. Therefore, there must be a good reason for Louie having been kicked out by the baron and for the earthquake having destroyed Lisbon. (P, B, E, R, C, L, I, D, W, S, O)

May 20, 2022May 20, 2022

Symbolizing Statements in Propositional Logic

In these notes, I will be discussing the topic “symbolizing statements in propositional (or symbolic) logic.” This is very important because, as I have already said in my earlier post before we can determine the validity of an argument in symbolic logic by applying a specific rule, we need to symbolize the argument first. So, how do we symbolize propositions in symbolic logic?

First, we need to identify the major connective. This is because once we have identified the major connective, we will be able to punctuate the proposition properly.

Second, we have to keep in mind that the variables or constants, such p and q or Y and Z, stand for the entire proposition, and not for the words within the proposition itself.

Third and last, we need to put proper punctuation and negation if necessary.

Let us consider the examples below.

If the squatters settle here, then the cattlemen will be angry and there will be a fight for water rights. (p, q, r)

As we can see, this example is a combination of a conditional proposition and a conjunctive proposition. However, if we analyze the proposition, it becomes clear to us that it is a conditional proposition whose consequent is a conjunctive proposition. Thus, the major connective in this proposition is “then.” Hence, when we symbolize the proposition, we need to punctuate the consequent. So, if we let p stand for “The squatters settle here,” q for “The cattlemen will be angry,” and r for “There will be a fight for water rights,” then the proposition is symbolized as follows:

p ⊃ (q • r)

If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth. (p, q, r, s)

This example is indeed a complicated one. But it can be easily symbolized.

If we analyze the proposition, it becomes clear that it is a conjunctive proposition whose conjuncts are both conditional propositions with a component inclusive disjunction and conjunction respectively.

Now, if we let

p stands for “The butler is telling the truth”
q for “The maid is telling the truth”
r for “The job was an inside one” and
s for “The lie detector is accurate”

then we initially come up with the following symbol: p v q ⊃ r • s ⊃ p • q

The symbol above, however, is not yet complete. In fact, it remains very complicated. So, we have to punctuate it.

Since the major connective of the proposition is “however,” then we have to punctuate the component conjuncts. Thus, we initially come up with the following symbol:

[p v q ⊃ r] • [s ⊃ p • q]

However, the symbolized form of the proposition remains complicated because the component conjuncts have not been properly punctuated. As already said, there should only be one major connective in a proposition. So, let us punctuate the first conjunct.

Since it is stated in the first conjunct that the proposition is a conditional proposition whose antecedent is an inclusive disjunction, then we have to punctuate p v q. Thus, we initially come up with the following symbol:

[(p v q) ⊃ r] • [s ⊃ p • q]

And then let us punctuate the second conjunct. Since it is stated in the second conjunct that the proposition is a conditional proposition whose consequent is a conjunctive proposition, then we have to punctuate p • q. Thus, we come up with the following symbol:

[(p v q) ⊃ r] • [s ⊃ (p • q)]

Now, the symbol appears to be complete. Thus, the final symbol of the proposition “If either the butler or the maid is telling the truth, then the job was an inside one; however, if the lie detector is accurate, then both the butler and the maid are telling the truth” is as follows:

[(p v q) ⊃ r] • [s ⊃ (p • q)]

Neither Lucas is hard-working nor is he intelligent. (p, q)

This example is obviously an inclusive disjunction; hence, we may initially symbolize the proposition as p v q. However, the words “Neither…nor” is a signifier of a negation, and these words suggest that the entire proposition is negated. Thus, we finally symbolize the proposition “Neither Lucas is hard-working nor is he intelligent” as follows:

~ (p v q)

Please note that ~ (p v q) is not the same with ~ p v ~ q. And ~ p v ~ q is not the proper symbol of example #3 because the words “Neither…nor” suggest that the proposition has to be completely negated. As we learned in my previous post titled “Punctuating Propositions in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/11/punctuating-propositions-in-symbolic-logic/), when the proposition is completely negated, then the entire proposition has to be punctuated.

But let me explain why ~ (p v q) is not the same with ~ p v ~ q. If we recall, the rules in inclusive disjunction say “The inclusive disjunction is true if at least one of the disjuncts is true.” With this, let us determine the truth value of ~ (p v q) and ~ p v ~ q in order to prove that they are not the same.

Let us assign the truth value “true” for p and “false” for q.

symbolizing propositions in symbolic logic

It is not the case that the manager will resign if she does not receive a salary increase. (p, q)

Please note that since the negation sign “It is not the case” precedes the entire proposition, then the entire proposition has to be negated. Thus, we need to punctuate the entire proposition and put the negation sign outside of it.

As I discussed in one of my previous posts, we learned that 1) the variables provided after the proposition represent the propositions in the entire proposition respectively, and 2) since in the example above the antecedent is written after the consequent, then q must be our antecedent and p our consequent. Hence, we initially come up with the following symbol: ~q ⊃ p. Please note that q is negated because it is clearly specified in the proposition. In other words, the proposition contains a negation sign “not.”

Now, since the negation sign “It is not the case” precedes the entire proposition, then, again, the entire proposition must be negated. Thus, we finally symbolize the proposition “It is not the case that the manager will resign if she does not receive a salary increase” as follows:

~ (~q ⊃ p)

If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence. (p, q, r)

In this example, since the negation sign “It is not the case” does not precede the entire proposition, then we do not negate the entire proposition. We only negate the proposition where the negation sign immediately precedes. Thus, the negation sign in the example above only negates the proposition “The professor will take a leave of absence if and only if the administration allows him to.” It does not clearly negate the proposition “There must be another good reason why the professor will take a leave of absence.”

Now, if we analyze the proposition, we notice that:

p stand for “The professor will take a leave of absence”
q for “The administration allows him to” and
r for “There must be another good reason why the professor will take a leave of absence.”

Please note that we do not repeat the variable “p” for the proposition “There must be another good reason why the professor will take a leave of absence” because the thought of the proposition is completely changed. This is because of the addition of the idea “There must be another good reason.” Thus, instead of repeating the variable “p,” we use the variable “r” to represent the proposition “There must be another good reason why the professor will take a leave of absence.”

So, we symbolize the proposition “If it is not the case that the professor will take a leave of absence if and only if the administration allows him to, then there must be another good reason why the professor will take a leave of absence” as follows:

~ (p ≡ q) ⊃ r

May 20, 2022May 20, 2022

How to Symbolize Arguments in Propositional Logic?

In these notes, I will discuss how to symbolize arguments in propositional or symbolic logic, which uses all the basic symbols, especially the use of parentheses. As I have mentioned in my other notes, symbolizing arguments in logic is important because before we can determine the validity of an argument in symbolic logic, we need to symbolize the argument first.

In symbolizing arguments in symbolic logic, we need to do the following:

First, we need to symbolize the argument sentence by sentence.

Second, we have to identify the major connectives in each sentence of the argument. This is important because once we have identified the major connective we will be able to punctuate the sentence or proposition properly.

Third, we need to remember that the variables or constants, such p and q or Y and Z, stand for the entire sentence or proposition, and not for the words within the sentence or proposition itself.

Lastly, we need to put proper punctuations and negation signs if necessary.

Let us consider the example below.

If the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth, then the captain either had good motives for controlling the world or his motives were evil. The airship Albatros had a powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground. It is either the case that if the Albatros could destroy objects on the ground its captain could enforce his will all over the earth, or it is the case that if he attempted to blow up the British vessel then his passengers would recognize the hoax. It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax. Furthermore, the captain’s motives for controlling the world were not evil. Therefore, his motives were good. (A, D, W, G, E, S, B, P)

As we can see, the argument above is quite long and indeed complicated. But again, we can easily symbolize this argument because, as I already mentioned, we will symbolize this argument sentence by sentence (or proposition by proposition). So, let’s start with the first sentence.

Sentence 1

If we analyze this sentence, it is clear that the major connective is “if…then” or just “then”. Hence, it is a conditional proposition. Now, in symbolizing this sentence, we need to punctuate the antecedent and the consequent.

If we look at the antecedent, we notice that it is a compound proposition whose conjuncts are both conditional propositions. Because there are several connectives in the sentence, then we also need to punctuate the antecedent. Hence, the antecedent (which reads: the fact that the airship Albatros had powerful weapon meant it could destroy objects on the ground, and its capability of destroying objects on the ground meant that the captain could enforce his will all over the earth) is symbolized as follows:

(A ⊃ D) • (D ⊃ W)

As we can see, the consequent of the proposition above is an exclusive disjunction. Thus, we need to underscore the wedge to differentiate it from an inclusive disjunction. The consequent (which reads: the captain either had good motives for controlling the world or his motives were evil) is symbolized as follows: G v E.

Please note that the constants provided at the end of the argument above represent the propositions in the entire argument respectively. Thus, in the first proposition, the constant A stands for “the airship Albatros had powerful weapon”, D stands for “it could destroy objects on the ground”, W stands for “the captain could enforce his will all over the earth”, G stands for “the captain either had good motives for controlling the world”, and E stands for “his motives were evil”.

Now, when we symbolize the entire proposition, we need to punctuate both the antecedent and the consequent because, as the rule says, there should only be one major connective in each proposition. Thus, the proposition above is symbolized as follows:

[(A ⊃ D) • (D ⊃ W)] ⊃ (G v E)

Note: Please apply the principles discussed above in symbolizing the rest of the sentences below. If you have questions or clarifications, please leave a comment below. The PHILO-notes team is happy to respond to them.

Sentence 2

The airship Albatros had powerful weapon if and only if its captain had more advanced scientific knowledge than his contemporaries; and if the captain had more advanced scientific knowledge than his contemporaries, then Albatros could destroy objects on the ground.

(A ≡ S) • (S ⊃ D)

Sentence 3

(D ⊃ W) v (B ⊃ P)

Sentence 4

It is not the case that his attempt to Blow up the British vessel resulted in his passengers’ recognizing the hoax.

~ (B ⊃ P)

Sentence 5

Furthermore, the captain’s motives for controlling the world were not evil.

~ E

Sentence 6 (which is the conclusion)

Therefore, his motives were good.

In the end, the argument above is symbolized as follows:

How to Symbolize Arguments in Symbolic Logic

May 20, 2022May 20, 2022

Punctuating Statements in Propositional Logic

In these notes, I will briefly discuss the topic “punctuating statements in propositional (or symbolic) logic.”

But why do we need to punctuate propositions in symbolic logic? This is because, in many instances, propositions in symbolic contain more than one connective; but in symbolic logic, all propositions should only have one major connective.

Thus, if there are two or more connectives, then we have to punctuate the proposition accordingly so that the major connective will become clear.

Symbolic logic uses parentheses ( ), brackets [ ], and braces { } as punctuation symbols.

Let us consider the example below.

If the road is wet, then either it rains today or the fire truck spills water on the road. (p, q, r)

As we can see, the example contains three propositions, namely:

1) The road is wet,

2) It rains today, and

3) The fire truck spills water on the road.

And as I already discussed in my previous posts, we learned that the variables provided after the proposition represent the propositions in the entire proposition respectively. Thus, in the example above, p stands for “The road is wet,” q for “It rains today,” and r for “The fire truck spills water on the road.” Hence, initially, the proposition is symbolized as follows:

p ⊃ q v r

However, the symbol above is not yet complete because, at this point, it is not yet clear what type of proposition it is. This is the reason why we need to punctuate the proposition. Please see my previous discussion on “Propositions and Symbols Used in Symbolic Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/) for some idea on how to symbolize a proposition in symbolic logic.

Now, if we analyze the proposition, it would become clear that it is a conditional proposition whose consequent is an inclusive disjunction. For this reason, we need to punctuate the consequent.

Thus, the proposition “If the road is wet, then either it rains today or the fire truck spills water on the road” is symbolized as follows:

p ⊃ (q v r)

I will discuss more about this when I go to the discussion on “symbolizing propositions” in symbolic logic.

Meantime, let me give examples of a punctuated proposition just to show that statements in propositional or symbolic logic that contain two or more connectives have to be punctuated accordingly. Please see examples below then.

punctuating propositions in symbolic logic

May 20, 2022May 20, 2022

Negation of Statements in Propositional Logic

In my other notes titled “Propositions and Symbols Used in Propositional (or Symbolic) Logic” (see http://philonotes.com/index.php/2018/02/02/symbolic-logic/), I discussed the two basic types of a proposition as well as the symbols used in symbolic logic. I have also briefly discussed how propositions can be symbolized using a variable or a constant.

In these notes, I will discuss the topic “negation of statements in propositional (or symbolic) logic” or the way in which propositions or statements in symbolic logic are negated.

To begin with, we have to note that any statement used in symbolic logic can be negated. And as I have already mentioned in the previous discussion, symbolic logic uses ~ (tilde) to symbolize a negative proposition.

But how do we know that the statement is negative?

A statement is negative if it contains at least one of the following signifiers:

Not

It is false

It is not the case

It is not true

For example, let us consider the following statements:

Either no students are interested in the party or it is not the case that the

administration requires the students to attend the party.

If the company does not increase the salary of the workers, then the union will go on strike to press its various demands.
The professor will not be absent if and only if he is not sick.

As we notice, example #1 is a compound statement, and both component statements contain the negation signs “no” and “it is not the case.” For this reason, when we symbolize the entire statement, then both component statements should be negated. Hence, if we let p stand for “No students are interested in the party” and q for “It is not the case that the administration requires the students to attend the party,” then the statement “Either no students are interested in the party or it is not the case that the administration requires the students to attend the party” can be symbolized as follows:

~ p v ~ q

In example #2, only the first component statement contains the negation sign “not.” Hence, only the first statement should be negated. Thus, if we let p stand for “The company does not increase the salary of the workers” and q for “The union will go on strike to press its various demands,” then the statement “If the company does not increase the salary of the workers, then the union will go on strike to press its various demands,” is symbolized as follows:

~ p ⊃ q

In example #3, both component statements contain a negation sign “not.” Thus, when symbolized, both component statements have to be negated. Hence, if we let p stand for “The professor will not be absent” and q for “He is not sick,” then the statement “The professor will not be absent if and only if he is not sick” is symbolized as follows:

~ p ≡ ~ q

Now, sometimes a statement can be double (or even triple) negated. In other words, the statement contains two or more negation signs. If this happens, then the statement has to be symbolized accordingly. Consider this example: “It is not true that the professor is not sick.” If we let p stand for the entire statement, then it is symbolized as follows:

~~ p

However, since a double negation implies affirmation, then the statement can also be symbolized as follows:

In some cases, contradictory words, such as “kind and unkind” and “mortal and immortal,” may signify negation if and only if it is clearly specified in the statement; otherwise, the statement should not be negated. Consider the following examples:

Lulu is generous, while Lili is unkind.
Either George is kind or Bert is unkind.

In example #1, the word “unkind” does not clearly signify negation. Thus, the statement “Lili is unkind” is not a negative statement. Let us symbolize example #1. If we let p stand for “Lulu is generous” and q for “Lili is unkind,” then the proposition “Lulu is generous, while Lili is unkind” is symbolized as follows:

p • q

However, the word “unkind” in example #2 above clearly signifies negation because of the presence of the contradictory words “kind and unkind” in the statement. Now, if we let p stand for “George is kind” and q for “Bert is unkind,” then the statement “Either George is kind or Bert is unkind” is symbolized as follows:

p v ~q

This is because the statement “Either George is kind or Bert is unkind” can also be stated in this manner: “Either George is kind or Bert is not kind.”

Rule in Negation

The negation of a true statement is false; while the negation of a false statement is true.

Obviously, the rule in negation says that if a particular statement is true, then it becomes false when negated. And if a particular statement is false, then it becomes true when negated. The truth table below illustrates this point.

Let us determine the truth-value of a negative statement by applying the rule in negation.

Consider the example below.

It is not the case that the administration requires the students to attend the party.

Again, if we let p stand for the statement “The administration requires the students to attend the party,” then the statement is symbolized as p. However, since the statement contains a negation sign “It is not the case,” then the statement is negative. Thus, the statement has to be symbolized as follows:

~ p

Now, if we assume that the statement “The administration requires the students to attend the party” is true, that is, the administration did indeed require the students to attend the party, then the statement “It is not the case that the administration requires the students to attend the party” is absolute false. To illustrate:

Please note that when we assign a truth-value to a statement, we assign it to the statement without the negation sign. Thus, if we have the statement ~ p, and if we assign, for example, True value to the statement, we assign it to p and not to ~ p.

May 19, 2022May 19, 2022

Biconditional Statements in Propositional Logic

Biconditional statements are compound propositions connected by the words “if and only if.”

The symbol for “if and only if” is a ≡ (triple bar). Let’s consider the example below.

I will take a leave of absence if and only the administration allows me to. (p, q)

If we let p stand for “I will take a leave of absence” and q for “The administration allows me to,” then the biconditional proposition “I will take a leave of absence if and only if the administration allows me to” is symbolized as follows:

p ≡ q

Please note that the connective “if and only if” should not be confused with “only if.” The connective “only if” is a connective of a conditional proposition. Let’s take the example below:

I will take a leave of absence only if the administration allows me to. (p, q)

We have to take note that the proposition that comes after the connective “only if” is a consequent. Thus, if we let p stand for “I will take a leave of absence” and q for “The administration allows me to,” then the proposition is symbolized as follows: p ⊃ q.

Rules in Biconditional Propositions

A biconditional proposition is true if both components have the same truth value.
Thus, if one is true and the other is false, or if one is false and the other true, then the biconditional proposition is false.

As we can see, the rules in biconditional propositions say that the only instance wherein the biconditional proposition becomes true is when both component propositions have the same truth value. This is because, in biconditional propositions, both component propositions imply each other. Thus, the example above, that is, “I will take a leave of absence if and only if the administration allows me to” can be restated as follows:

If I will take a leave of absence, then the administration allows me to; and if the administration allows me to, then I will take a leave of absence.

Thus, the symbol p ≡ q means p is equal to q, and q is equal to p.

The truth table below illustrates this point.

The truth table above says:

If p is true and q is true, then p ≡ q is true.
If p is true and q is false, then p ≡ q is false.
If p is false and q is true, then p ≡ q is false.
If p is false and q is false, then p ≡ q is true.

Now, suppose we have the example ~p ≡ q. How do we determine its truth value if p is true and q is false?

Let me illustrate.

The illustration says that p is true and q is false. Now, before we apply the rules in biconditional in the statement ~p ≡ q, we need to simplify ~p first because the truth value “true” is assigned to p and not to ~p. If we recall our discussion on the rule in negation, we learned that the negation of true is false. So, if p is true, then ~p is false. Thus, at the end of it all, ~p ≡ q is true.