# 3 no namers – logical implications

(P=>Q) => ((Q => R) =>(P => R))

This is actually the hypothetical syllogism in another form. For by considering (P Q) as a proposition S, (Q R) as a proposition T, and (P R) as a proposition U in the hypothetical syllogism above, and then by applying the “exportation” from the identities, this is obtained

((P => Q) ^ (R => S)) => ((P ^R) => (Q ^ S))

For example, if the statements “If the wind blows hard, the beach erodes.” and “If it rains heavily, the streets get flooded.” are true, then the statement “If the wind blows hard and it rains heavily, then the beach erodes and the streets get flooded.” is also true.

((P <=> Q) ^( Q <=> R)) => (P <=> R) This is saying that you can infer that P is equal to R when ((P <=> Q) ^( Q <=> R)).

This just says that the logical equivalence is transitive, that is, if P and Q are equivalent, and if Q and R are also equivalent, then P and R are equivalent.

# Hypothetical Syllogism – a logical implication

((P=>Q) ^ (Q => R)) => (P => R) named Hypothetical Syllogism

P => Q (if P imples Q and…)
Q => R (if Q implies R…)
Then, P must imply R.

P = I do not wake up
Q = I cannot go to work.
R = I will not get paid.

If I do not wake up, then I cannot go to work. (P => Q)
If I cannot go to work, then I will not get paid. (Q => R)
Therefore, if I do not wake up, then I will not get paid. (P => R)

# Disjunctive Syllogism – a logical implication

(not P ^ (P v Q) => Q named Disjunctive Syllogism formerly known as modus tollendo ponens meaning if not P and P or Q then we can conclude Q.

P v Q (if P or Q is true and….
not P (is true)
then Q must be true.

if P = The car is fast
and Q = The car comes in first

Then the compound statement: “If the car is not fast and (the car is fast or the car comes in first) then the car comes in first” is always true.

Suppose that (P) is false, the car is fast
Suppose that (Q) is true, the car comes in first

Then if the car is not fast and (the car is fast or the car comes in first) then we can conclude that the car comes in first. What happened was that the statement the car is not fast removes the statement the car is fast – all that is left is the “the car comes in first” – if that statement is true then the conclusion can only be that it is true that that the car comes in first.

So you might say as a human, wow the car was not first, but it came in first.

# Simplification – a logical implication

(P ^ Q) => P named Simplification meaning if (P ^ Q) then P
(P ^ Q) => Q named Simplification meaning if (P ^ Q) then P

If the compound statement  (P ^ Q) => P is given and is true (it is always true) and P ^ Q is true then P must be true. (see truth table to follow the reasoning).

The same reasoning applies to (P ^ Q) => Q.

P = I cannot get hold of any money
Q = The bank will not lend me any money

The the compound statement: “if I cannot get hold of any money and the bank will not lend me any money then I cannot get hold of any money” is always true.

Suppose that, it is true that (P) I cannot get hold of any money
Suppose that, it is true that (Q) The bank will not lend me any money

Then we can conclude that it is true that: “I cannot get hold of any money – if I cannot get hold of any money and the bank will not lend me any money”. You can see that just more truth (The bank will not lend me money) to some original truth (I cannot get hold of any money) – So we can take it that the original truth is still true if both the original truth (I cannot get hold of any money) and the added truth (the bank will not lend me any money) are both true.

# Addition – a logical implication

P => (P v Q) meaning, P implies (P or Q)

If the compound statement P=>(P v Q) is given and is true (it is always true) then we can conclude P by just taking P.

If P = the traffic light is green
and Q = the traffic light is red.

Then in English, P=>(P v Q ) will be “If the traffic light is green then the traffic light is green or (otherwise) the traffic light is red”. As mentioned, this compound statement is always true.

So, If the compound expression (P => (P v Q)) is given, we know it is true, so we can always conclude P by just taking P.

Suppose that, it is false that (P) the traffic light is green,
Suppose that, it is true that (Q) the traffic light is red,

Then we can conclude that it is false that the traffic light is green, because P is false.

Suppose that, it is true that (P) the traffic light is green,
Suppose that, it is false that (Q) the traffic light is red,

Then we can conclude that it is true that the traffic light is green, because P is true.

Suppose that it is true that (P) the traffic light is green,
Suppose that it is true that (Q) the traffic light is red,

Then normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that the traffic light is green because P is true.

Suppose that it is true that (P) the traffic false is green,
Suppose that it is true that (Q) the traffic false is red,

Again, normally we would conclude that the traffic light is broken, but in this case, logically we would conclude that it is false that the traffic light is green because P is false.

In the truth table we can see the tautology in P=>(P v Q) – the compound statement is always true – which proves what we have said in the examples.

# Modus tollens – a logical implication

((P => Q) ^ not Q) => not P named Modus tollens meaning the way that denies by denying – denying the consequent Q.

If the compound statement  (P => Q) ^ not Q) => not P is given and is true (it is always true) and P => Q is true then not Q must be true – therefore not P must be true (see truth table to follow the reasoning).

```P => Q (P implies Q true) ^ not Q (true) ======================== then not P is True```

In English

P = an intruder is detected
Q = the alarm goes off

(P=>Q) If an intruder is detected, then the alarm goes off.
(^ not Q) and The alarm does not go off.
(=> not P) then, no intruder is detected

said a little more coherently:

If an intruder is detected, the alarm will go off, and when the alarm does not go off, then, no intruder is detected.

see Wikipedia

# Modus ponens – a logical implication

(P ^ (P => Q)) => Q named Modus ponens meaning affirms by affirming – affirming the antecedent/premise P

If the compound statement (P ^ (P => Q)) => Q is given and is true (it is always true) and P => Q is true then P must be true – therefore Q must be true (see truth table to follow the reasoning).

``` P => Q (If P => Q is true then...) P (must be true) therefore Q must be true.```

In English:
P = today is Tuesday
Q = I will go to work

If today is Tuesday, then I will go to work.
Today is Tuesday.
Therefore, I will go to work.

In instances of modus ponens we assume as premises that p => q is true and p is true. Only one line of the truth table – the last – satisfies these two conditions. On this line, q is also true. Therefore, whenever p => q is true and p is true, q must also be true

see Wikipedia